cos2 (problem 3.4.1)

Average Accuracy: 49.9% → 99.8%

Time: 13.0s

Precision: binary64

Cost: 13376

Math

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

↓

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

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)) / Float64(x / 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] / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 49.9%

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

2. Applied egg-rr 51.0%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
   Proof

   [Start] 49.9	\[ \frac{1 - \cos x}{x \cdot x} \]
   associate-/r* [=>] 51.0	\[ \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
   div-inv [=>] 51.0	\[ \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]

3. Applied egg-rr 75.3%

   \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}}}{x} \cdot \frac{1}{x} \]
   Proof

   [Start] 51.0	\[ \frac{1 - \cos x}{x} \cdot \frac{1}{x} \]
   flip-- [=>] 50.8	\[ \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{x} \cdot \frac{1}{x} \]
   div-inv [=>] 50.8	\[ \frac{\color{blue}{\left(1 \cdot 1 - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}}{x} \cdot \frac{1}{x} \]
   metadata-eval [=>] 50.8	\[ \frac{\left(\color{blue}{1} - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}{x} \cdot \frac{1}{x} \]
   1-sub-cos [=>] 75.3	\[ \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{x} \cdot \frac{1}{x} \]

4. Simplified 75.5%

   \[\leadsto \frac{\color{blue}{\sin x \cdot \tan \left(\frac{x}{2}\right)}}{x} \cdot \frac{1}{x} \]
   Proof

   [Start] 75.3	\[ \frac{\left(\sin x \cdot \sin x\right) \cdot \frac{1}{1 + \cos x}}{x} \cdot \frac{1}{x} \]
   *-commutative [<=] 75.3	\[ \frac{\color{blue}{\frac{1}{1 + \cos x} \cdot \left(\sin x \cdot \sin x\right)}}{x} \cdot \frac{1}{x} \]
   associate-*l/ [=>] 75.3	\[ \frac{\color{blue}{\frac{1 \cdot \left(\sin x \cdot \sin x\right)}{1 + \cos x}}}{x} \cdot \frac{1}{x} \]
   *-lft-identity [=>] 75.3	\[ \frac{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}{x} \cdot \frac{1}{x} \]
   associate-*l/ [<=] 75.3	\[ \frac{\color{blue}{\frac{\sin x}{1 + \cos x} \cdot \sin x}}{x} \cdot \frac{1}{x} \]
   *-commutative [=>] 75.3	\[ \frac{\color{blue}{\sin x \cdot \frac{\sin x}{1 + \cos x}}}{x} \cdot \frac{1}{x} \]
   hang-0p-tan [=>] 75.5	\[ \frac{\sin x \cdot \color{blue}{\tan \left(\frac{x}{2}\right)}}{x} \cdot \frac{1}{x} \]

5. Applied egg-rr 99.8%

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

   [Start] 75.5	\[ \frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x} \cdot \frac{1}{x} \]
   un-div-inv [=>] 75.6	\[ \color{blue}{\frac{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}}{x}} \]
   *-commutative [=>] 75.6	\[ \frac{\frac{\color{blue}{\tan \left(\frac{x}{2}\right) \cdot \sin x}}{x}}{x} \]
   *-un-lft-identity [=>] 75.6	\[ \frac{\frac{\tan \left(\frac{x}{2}\right) \cdot \sin x}{\color{blue}{1 \cdot x}}}{x} \]
   times-frac [=>] 99.8	\[ \frac{\color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{1} \cdot \frac{\sin x}{x}}}{x} \]
   tan-quot [=>] 99.8	\[ \frac{\frac{\color{blue}{\frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}}}{1} \cdot \frac{\sin x}{x}}{x} \]
   associate-/l/ [=>] 99.8	\[ \frac{\color{blue}{\frac{\sin \left(\frac{x}{2}\right)}{1 \cdot \cos \left(\frac{x}{2}\right)}} \cdot \frac{\sin x}{x}}{x} \]
   *-un-lft-identity [<=] 99.8	\[ \frac{\frac{\sin \left(\frac{x}{2}\right)}{\color{blue}{\cos \left(\frac{x}{2}\right)}} \cdot \frac{\sin x}{x}}{x} \]
   tan-quot [<=] 99.8	\[ \frac{\color{blue}{\tan \left(\frac{x}{2}\right)} \cdot \frac{\sin x}{x}}{x} \]
   div-inv [=>] 99.8	\[ \frac{\tan \color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \frac{\sin x}{x}}{x} \]
   metadata-eval [=>] 99.8	\[ \frac{\tan \left(x \cdot \color{blue}{0.5}\right) \cdot \frac{\sin x}{x}}{x} \]

6. Applied egg-rr 99.8%

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

   [Start] 99.8	\[ \frac{\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x}}{x} \]
   associate-*r/ [=>] 75.6	\[ \frac{\color{blue}{\frac{\tan \left(x \cdot 0.5\right) \cdot \sin x}{x}}}{x} \]
   associate-/l* [=>] 99.8	\[ \frac{\color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{\frac{x}{\sin x}}}}{x} \]

7. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	13640

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-35}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \]

Alternative 2
Accuracy	99.4%
Cost	13376

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

Alternative 3
Accuracy	99.8%
Cost	13376

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

Alternative 4
Accuracy	99.2%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0052 \lor \neg \left(x \leq 0.0047\right):\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \end{array} \]

Alternative 5
Accuracy	99.3%
Cost	7112

\[\begin{array}{l} t_0 := 1 - \cos x\\ \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;\frac{\frac{t_0}{x}}{x}\\ \mathbf{elif}\;x \leq 0.0047:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{x \cdot x}\\ \end{array} \]

Alternative 6
Accuracy	78.2%
Cost	832

\[\frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2} \]

Alternative 7
Accuracy	78.3%
Cost	713

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

Alternative 8
Accuracy	78.0%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.5 \lor \neg \left(x \leq 3.45\right):\\ \;\;\;\;\frac{6}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 9
Accuracy	52.6%
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))