learnengg

learnengg

さたにあ (黒木優希)

てるろー

@bluecow 🐮

@bluecow 🐮

@BLUECOW009

4 Mar 2025

import * as math from 'mathjs'; // Enhanced mathematical exploration function with SuperPrompt-inspired concepts const advancedMathematicalExplore = (expression, explorationDepth = 3) => { console.log("<prompt_metadata>"); console.log("Type: Advanced Mathematical Analysis"); console.log("Purpose: Multi-dimensional Exploration of Mathematical Expression"); console.log("Paradigm: Holographic Mathematical Reasoning"); console.log("Constraints: Mathematical Rigor with Conceptual Expansion"); console.log("Objective: Discover emergent properties through metamorphic transformations"); console.log("</prompt_metadata>"); console.log("\n<think>"); console.log("Starting expression analysis journey with: " expression); // Phase 1: Core Analysis - Parse and understand the expression structure console.log("\n=== PHASE 1: EXPRESSION TAXONOMY ==="); let expressionTree; try { expressionTree = math.parse(expression); console.log("Expression successfully parsed into structured tree"); // Analyze expression type and complexity const expressionType = determineExpressionType(expressionTree); console.log(`Expression classification: ${expressionType}`); // Extract variables const variables = extractVariables(expressionTree); console.log(`Variables detected: ${variables.length > 0 ? variables.join(', ') : 'none (constant expression)'}`); // Check for special patterns const patterns = detectMathematicalPatterns(expression, expressionTree); if (patterns.length > 0) { console.log("Special patterns detected:"); patterns.forEach(pattern => console.log(`- ${pattern}`)); } } catch (error) { console.log(`Error parsing expression: ${error.message}`); console.log("Proceeding with string-based analysis"); } // Phase 2: Metamorphosis - Transform the expression in different ways console.log("\n=== PHASE 2: EXPRESSION METAMORPHOSIS ==="); const transformations = [ { name: "Simplification", fn: (expr) => math.simplify(expr).toString() }, { name: "Expansion", fn: (expr) => { try { return math.expand(expr).toString(); } catch(e) { return "Not applicable"; } }}, { name: "Factorization", fn: (expr) => { try { return math.rationalize(expr).toString(); } catch(e) { return "Not applicable"; } }} ]; for (const transform of transformations) { try { console.log(`${transform.name}: ${transform.fn(expression)}`); } catch (error) { console.log(`${transform.name} failed: ${error.message}`); } } // Phase 3: Dimensional Transcendence - Explore the expression in different "dimensions" console.log("\n=== PHASE 3: DIMENSIONAL TRANSCENDENCE ==="); const dimensions = [ { name: "Algebraic Domain", explore: (expr, vars) => { if (vars.length === 0) { return "Constant expression: " expr; } // Try to solve for zeros if possible try { return `Looking for zeros where ${expr} = 0`; } catch(e) { return "Cannot solve algebraically"; } } }, { name: "Calculus Domain", explore: (expr, vars) => { if (vars.length === 0) return "No derivatives for constant expressions"; const results = []; for (const v of vars) { try { const derivative = math.derivative(expr, v).toString(); results.push(`d/d${v} = ${derivative}`); } catch(e) { results.push(`d/d${v}: Cannot differentiate`); } } return results.join("\n"); } }, { name: "Numerical Domain", explore: (expr, vars) => { if (vars.length === 0) { try { const result = math.evaluate(expr); return `Evaluates to ${result}`; } catch(e) { return "Cannot evaluate numerically"; } } // If we have one variable, evaluate at different points if (vars.length === 1) { const v = vars[0]; const points = [-10, -1, 0, 1, 10]; const results = []; for (const point of points) { try { const scope = {}; scope[v] = point; const result = math.evaluate(expr, scope); results.push(`f(${point}) = ${result}`); } catch(e) { results.push(`f(${point}) = Error: ${e.message}`); } } return results.join("\n"); } return "Multi-variable expression - selected points evaluation skipped"; } }, { name: "Complex Domain", explore: (expr, vars) => { // Substitute i for complex analysis if applicable if (expr.includes('i')) { try { const withI = expr.replace(/i/g, '1i'); const result = math.evaluate(withI, {i: math.complex(0, 1)}); return `In complex domain: ${result}`; } catch(e) { return `Complex analysis error: ${e.message}`; } } return "No complex variables detected"; } } ]; // Extract variables for dimension exploration let vars = []; try { vars = math.parse(expression).filter(node => node.isSymbolNode).map(node => node.name); // Remove duplicates vars = [...new Set(vars)].filter(v => v !== 'i' && v !== 'e' && v !== 'pi'); } catch (e) { console.log(`Variable extraction error: ${e.message}`); } for (const dimension of dimensions) { console.log(`\n--- ${dimension.name} ---`); try { const result = dimension.explore(expression, vars); console.log(result); } catch (error) { console.log(`Exploration error: ${error.message}`); } } // Phase 4: Recursive Depth - Recursively explore transformations up to specified depth if (explorationDepth > 0) { console.log("\n=== PHASE 4: RECURSIVE EXPLORATION ==="); const exploreRecursively = (expr, depth, path = []) => { if (depth <= 0) return; // Try several transformations and recursively explore promising ones const transformations = [ { name: "Simplify", fn: (e) => math.simplify(e).toString() }, { name: "Derivative", fn: (e) => { if (vars.length > 0) { try { return math.derivative(e, vars[0]).toString(); } catch(err) { return null; } } return null; }} ]; for (const transform of transformations) { try { const newExpr = transform.fn(expr); if (newExpr && newExpr !== expr && !path.includes(newExpr)) { console.log(`${' '.repeat(path.length * 2)}${transform.name} → ${newExpr}`); exploreRecursively(newExpr, depth - 1, [...path, newExpr]); } } catch (e) { // Skip failed transformations silently } } }; exploreRecursively(expression, explorationDepth); } // Phase 5: Synthesis - Integrate findings into cohesive understanding console.log("\n=== PHASE 5: HOLISTIC SYNTHESIS ==="); try { // Check for common mathematical constants if (typeof math.evaluate(expression) === 'number') { const value = math.evaluate(expression); const constants = [ { name: "π (pi)", value: Math.PI }, { name: "e", value: Math.E }, { name: "φ (golden ratio)", value: (1 Math.sqrt(5)) / 2 }, { name: "√2", value: Math.SQRT2 }, { name: "√3", value: Math.sqrt(3) }, { name: "ln(2)", value: Math.LN2 } ]; for (const constant of constants) { const diff = Math.abs(value - constant.value); if (diff < 1e-10) { console.log(`Expression appears to equal ${constant.name}`); } } } // Look for symmetries if (vars.length === 1) { const v = vars[0]; try { const original = math.parse(expression); const negated = math.parse(expression.replace(new RegExp(v, 'g'), `(-${v})`)); const simplified1 = math.simplify(original).toString(); const simplified2 = math.simplify(negated).toString(); if (simplified1 === simplified2) { console.log(`Expression shows even symmetry with respect to ${v}`); } else if (simplified1 === `-${simplified2}` || simplified2 === `-${simplified1}`) { console.log(`Expression shows odd symmetry with respect to ${v}`); } } catch (e) { // Symmetry check failed silently } } } catch (error) { console.log(`Synthesis error: ${error.message}`); } console.log("</think>"); return "Expression analysis complete"; }; // Helper functions function determineExpressionType(node) { if (!node) return "Unknown"; if (node.isConstantNode) return "Constant"; if (node.isSymbolNode) return "Variable"; if (node.isOperatorNode) { const op = node.op; if ([' ', '-'].includes(op)) return "Polynomial-like"; if (['*', '/'].includes(op)) return "Rational expression"; if (op === '^') return "Power expression"; return `Operator (${op})`; } if (node.isFunctionNode) { const fnName = node.fn?.name || "unknown"; if (['sin', 'cos', 'tan'].includes(fnName)) return "Trigonometric"; if (['log', 'ln', 'exp'].includes(fnName)) return "Exponential/Logarithmic"; if (['sqrt', 'cbrt'].includes(fnName)) return "Root function"; return `Function (${fnName})`; } return "Complex expression"; } function extractVariables(node) { if (!node) return []; const variables = new Set(); try { node.traverse(function(node) { if (node.isSymbolNode && !['i', 'e', 'pi'].includes(node.name)) { variables.add(node.name); } }); } catch (e) { // Failed to extract } return Array.from(variables); } function detectMathematicalPatterns(expr, node) { const patterns = []; // Detect some common patterns if (expr.includes('sin') && expr.includes('cos')) { if (expr.includes('sin^2') && expr.includes('cos^2')) { patterns.push("Possible trigonometric identity (sin² cos²)"); } } if (expr.includes('^2') && expr.includes('sqrt')) { patterns.push("Possible square-root/square relationship"); } // Quadratic pattern if (expr.match(/x\s*\^\s*2/) && expr.match(/\d \s*\*\s*x/) && expr.match(/[ \-]\s*\d $/)) { patterns.push("Possible quadratic expression"); } return patterns; } // Test the enhanced explorer with interesting expressions console.log("EXPLORING PYTHAGOREAN IDENTITY:"); advancedMathematicalExplore("sin(x)^2 cos(x)^2"); console.log("\n\nEXPLORING COMPLEX EXPRESSION:"); advancedMathematicalExplore("(e^(i*x) e^(-i*x))/2"); console.log("\n\nEXPLORING EULER'S IDENTITY:"); advancedMathematicalExplore("e^(i*pi) 1");

@bluecow 🐮

@BLUECOW009

4 Mar 2025

EXPLORING PYTHAGOREAN IDENTITY: <prompt_metadata> Type: Advanced Mathematical Analysis Purpose: Multi-dimensional Exploration of Mathematical Expression Paradigm: Holographic Mathematical Reasoning Constraints: Mathematical Rigor with Conceptual Expansion Objective: Discover emergent properties through metamorphic transformations </prompt_metadata> <think> Starting expression analysis journey with: sin(x)^2 cos(x)^2 === PHASE 1: EXPRESSION TAXONOMY === Expression successfully parsed into structured tree Expression classification: Polynomial-like Variables detected: sin, x, cos === PHASE 2: EXPRESSION METAMORPHOSIS === Simplification: sin(x) ^ 2 cos(x) ^ 2 Expansion: Not applicable Factorization: Not applicable === PHASE 3: DIMENSIONAL TRANSCENDENCE === --- Algebraic Domain --- Looking for zeros where sin(x)^2 cos(x)^2 = 0 --- Calculus Domain --- d/dsin = 0 d/dx = 0 d/dcos = 0 --- Numerical Domain --- Multi-variable expression - selected points evaluation skipped --- Complex Domain --- Complex analysis error: Undefined function s1in === PHASE 4: RECURSIVE EXPLORATION === Simplify → sin(x) ^ 2 cos(x) ^ 2 Derivative → 0 Derivative → 0 === PHASE 5: HOLISTIC SYNTHESIS === Synthesis error: Undefined symbol x </think> EXPLORING COMPLEX EXPRESSION: <prompt_metadata> Type: Advanced Mathematical Analysis Purpose: Multi-dimensional Exploration of Mathematical Expression Paradigm: Holographic Mathematical Reasoning Constraints: Mathematical Rigor with Conceptual Expansion Objective: Discover emergent properties through metamorphic transformations </prompt_metadata> <think> Starting expression analysis journey with: (e^(i*x) e^(-i*x))/2 === PHASE 1: EXPRESSION TAXONOMY === Expression successfully parsed into structured tree Expression classification: Rational expression Variables detected: x === PHASE 2: EXPRESSION METAMORPHOSIS === Simplification: (e ^ (i * x) e ^ (-(i * x))) / 2 Expansion: Not applicable Factorization: Not applicable === PHASE 3: DIMENSIONAL TRANSCENDENCE === --- Algebraic Domain --- Looking for zeros where (e^(i*x) e^(-i*x))/2 = 0 --- Calculus Domain --- d/dx = i * (e ^ (i * x) - 1 / e ^ (i * x)) / 2 --- Numerical Domain --- f(-10) = -0.8390715290764524 f(-1) = 0.5403023058681398 f(0) = 1 f(1) = 0.5403023058681398 f(10) = -0.8390715290764524 --- Complex Domain --- Complex analysis error: Undefined symbol x === PHASE 4: RECURSIVE EXPLORATION === Simplify → (e ^ (i * x) e ^ (-(i * x))) / 2 Derivative → i * (e ^ (i * x) - 1 / e ^ (i * x)) / 2 Derivative → i ^ 2 * (e ^ (i * x) 1 / e ^ (i * x)) / 2 Derivative → i * (e ^ (i * x) - 1 / e ^ (i * x)) / 2 Derivative → i ^ 2 * (e ^ (i * x) 1 / e ^ (i * x)) / 2 Derivative → i ^ 3 * (e ^ (i * x) - 1 / e ^ (i * x)) / 2 === PHASE 5: HOLISTIC SYNTHESIS === Synthesis error: Undefined symbol x </think> EXPLORING EULER'S IDENTITY: <prompt_metadata> Type: Advanced Mathematical Analysis Purpose: Multi-dimensional Exploration of Mathematical Expression Paradigm: Holographic Mathematical Reasoning Constraints: Mathematical Rigor with Conceptual Expansion Objective: Discover emergent properties through metamorphic transformations </prompt_metadata> <think> Starting expression analysis journey with: e^(i*pi) 1 === PHASE 1: EXPRESSION TAXONOMY === Expression successfully parsed into structured tree Expression classification: Polynomial-like Variables detected: none (constant expression) === PHASE 2: EXPRESSION METAMORPHOSIS === Simplification: e ^ (i * pi) 1 Expansion: Not applicable Factorization: Not applicable === PHASE 3: DIMENSIONAL TRANSCENDENCE === --- Algebraic Domain --- Constant expression: e^(i*pi) 1 --- Calculus Domain --- No derivatives for constant expressions --- Numerical Domain --- Evaluates to 0 --- Complex Domain --- Complex analysis error: Undefined symbol p1i === PHASE 4: RECURSIVE EXPLORATION === Simplify → e ^ (i * pi) 1 === PHASE 5: HOLISTIC SYNTHESIS === </think>

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