cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.8%

Time: 9.0s

Precision: binary64

Cost: 13376

• Unsound rule application detected in e-graph. Results may not be sound.

Math

\[\frac{1 - \cos x}{x \cdot x} \]

↓

\[\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

↓

(FPCore (x) :precision binary64 (* (/ (tan (* x 0.5)) x) (/ (sin x) x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

↓

double code(double x) {
	return (tan((x * 0.5)) / x) * (sin(x) / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (tan((x * 0.5d0)) / x) * (sin(x) / x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

↓

public static double code(double x) {
	return (Math.tan((x * 0.5)) / x) * (Math.sin(x) / x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

↓

def code(x):
	return (math.tan((x * 0.5)) / x) * (math.sin(x) / x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

↓

function code(x)
	return Float64(Float64(tan(Float64(x * 0.5)) / x) * Float64(sin(x) / x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

↓

function tmp = code(x)
	tmp = (tan((x * 0.5)) / x) * (sin(x) / x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{1 - \cos x}{x \cdot x} \]

2. Applied egg-rr 74.4%

   \[\leadsto \frac{\color{blue}{{\sin x}^{2} \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]
   Step-by-step derivation

   [Start] 54.3%	\[ \frac{1 - \cos x}{x \cdot x} \]
   flip-- [=>] 54.1%	\[ \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x \cdot x} \]
   div-inv [=>] 54.0%	\[ \frac{\color{blue}{\left(1 \cdot 1 - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}}{x \cdot x} \]
   metadata-eval [=>] 54.0%	\[ \frac{\left(\color{blue}{1} - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
   1-sub-cos [=>] 74.4%	\[ \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
   pow2 [=>] 74.4%	\[ \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

3. Simplified 74.8%

   \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x \cdot x} \]
   Step-by-step derivation

   [Start] 74.4%	\[ \frac{{\sin x}^{2} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
   unpow2 [=>] 74.4%	\[ \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]
   associate-*l* [=>] 74.4%	\[ \frac{\color{blue}{\sin x \cdot \left(\sin x \cdot \frac{1}{1 + \cos x}\right)}}{x \cdot x} \]
   associate-*r/ [=>] 74.4%	\[ \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]
   *-rgt-identity [=>] 74.4%	\[ \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]
   hang-0p-tan [=>] 74.8%	\[ \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x \cdot x} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]
   Step-by-step derivation

   [Start] 74.8%	\[ \frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x \cdot x} \]
   *-commutative [=>] 74.8%	\[ \frac{\color{blue}{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x \cdot x} \]
   times-frac [=>] 99.8%	\[ \color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x} \cdot \frac{\sin x}{x}} \]
   div-inv [=>] 99.8%	\[ \frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)}}{x} \cdot \frac{\sin x}{x} \]
   metadata-eval [=>] 99.8%	\[ \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{\sin x}{x} \]

5. Final simplification 99.8%

   \[\leadsto \frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	13376

\[\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \]

Alternative 2
Accuracy	99.4%
Cost	7304

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.00012:\\ \;\;\;\;\frac{t_0}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 0.000165:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{-1}{x}}{-x}\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	7241

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00012 \lor \neg \left(x \leq 0.000165\right):\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 4
Accuracy	98.8%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00012 \lor \neg \left(x \leq 0.000165\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 5
Accuracy	78.4%
Cost	969

\[\begin{array}{l} \mathbf{if}\;x \leq -3.45 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\frac{6 - \frac{\frac{72}{x}}{x}}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 6
Accuracy	78.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.25 \lor \neg \left(x \leq 3.2\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \end{array} \]

Alternative 7
Accuracy	78.1%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.45 \lor \neg \left(x \leq 3.4\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Accuracy	8.0%
Cost	64

\[9.92290301275212 \cdot 10^{-8} \]

Alternative 9
Accuracy	9.8%
Cost	64

\[0.027777777777777776 \]

Alternative 10
Accuracy	51.2%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023165 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))