cos2 (problem 3.4.1)

Percentage Accurate: 51.9% → 99.3%

Time: 10.7s

Alternatives: 10

Speedup: 15.2×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.5e-11) 0.5 (* (tan (* x 0.5)) (/ (sin x) (* x x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.5e-11) {
		tmp = 0.5;
	} else {
		tmp = tan((x * 0.5)) * (sin(x) / (x * x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d-11) then
        tmp = 0.5d0
    else
        tmp = tan((x * 0.5d0)) * (sin(x) / (x * x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.5e-11) {
		tmp = 0.5;
	} else {
		tmp = Math.tan((x * 0.5)) * (Math.sin(x) / (x * x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.5e-11:
		tmp = 0.5
	else:
		tmp = math.tan((x * 0.5)) * (math.sin(x) / (x * x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.5e-11)
		tmp = 0.5;
	else
		tmp = Float64(tan(Float64(x * 0.5)) * Float64(sin(x) / Float64(x * x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.5e-11)
		tmp = 0.5;
	else
		tmp = tan((x * 0.5)) * (sin(x) / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.5e-11], 0.5, N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000009e-11

   1. Initial program 33.8%

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

   2. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{0.5} \]

   if 2.50000000000000009e-11 < x

   1. Initial program 97.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. flip-- 96.9%

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

      2. div-inv 96.9%

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

      3. metadata-eval 96.9%

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

      4. 1-sub-cos 98.7%

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

      5. pow2 98.7%

         \[\leadsto \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

   3. Applied egg-rr 98.7%

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

      1. unpow2 98.7%

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

      2. associate-*l* 98.8%

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

      3. associate-*r/ 98.6%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]

      4. *-rgt-identity 98.6%

         \[\leadsto \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]

      5. hang-0p-tan 99.6%

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

   5. Simplified 99.6%

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

      1. associate-/l* 99.5%

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

      2. associate-/r/ 99.5%

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

      3. div-inv 99.5%

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

      4. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(x \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\tan \left(x \cdot 0.5\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{\tan \left(x \cdot 0.5\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.5e-11) 0.5 (* (sin x) (/ (tan (* x 0.5)) (* x x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.5e-11) {
		tmp = 0.5;
	} else {
		tmp = sin(x) * (tan((x * 0.5)) / (x * x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d-11) then
        tmp = 0.5d0
    else
        tmp = sin(x) * (tan((x * 0.5d0)) / (x * x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.5e-11) {
		tmp = 0.5;
	} else {
		tmp = Math.sin(x) * (Math.tan((x * 0.5)) / (x * x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.5e-11:
		tmp = 0.5
	else:
		tmp = math.sin(x) * (math.tan((x * 0.5)) / (x * x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.5e-11)
		tmp = 0.5;
	else
		tmp = Float64(sin(x) * Float64(tan(Float64(x * 0.5)) / Float64(x * x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.5e-11)
		tmp = 0.5;
	else
		tmp = sin(x) * (tan((x * 0.5)) / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.5e-11], 0.5, N[(N[Sin[x], $MachinePrecision] * N[(N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.50000000000000009e-11

   1. Initial program 33.8%

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

   2. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{0.5} \]

   if 2.50000000000000009e-11 < x

   1. Initial program 97.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. flip-- 96.9%

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

      2. div-inv 96.9%

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

      3. metadata-eval 96.9%

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

      4. 1-sub-cos 98.7%

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

      5. pow2 98.7%

         \[\leadsto \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

   3. Applied egg-rr 98.7%

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

      1. unpow2 98.7%

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

      2. associate-*l* 98.8%

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

      3. associate-*r/ 98.6%

         \[\leadsto \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]

      4. *-rgt-identity 98.6%

         \[\leadsto \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]

      5. hang-0p-tan 99.6%

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

   5. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. times-frac 99.5%

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

      3. div-inv 99.5%

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

      4. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. expm1-log1p-u 99.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}\right)\right)} \]

      2. expm1-udef 47.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}\right)} - 1} \]

      3. frac-times 47.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\tan \left(x \cdot 0.5\right) \cdot \sin x}{x \cdot x}}\right)} - 1 \]

      4. div-inv 47.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \cdot \frac{1}{x \cdot x}}\right)} - 1 \]

      5. pow2 47.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \cdot \frac{1}{\color{blue}{{x}^{2}}}\right)} - 1 \]

      6. pow-flip 47.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \cdot \color{blue}{{x}^{\left(-2\right)}}\right)} - 1 \]

      7. metadata-eval 47.2%

         \[\leadsto e^{\mathsf{log1p}\left(\left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \cdot {x}^{\color{blue}{-2}}\right)} - 1 \]

   9. Applied egg-rr 47.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \cdot {x}^{-2}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 99.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \cdot {x}^{-2}\right)\right)} \]

       2. expm1-log1p 99.6%

          \[\leadsto \color{blue}{\left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \cdot {x}^{-2}} \]

       3. *-commutative 99.6%

          \[\leadsto \color{blue}{{x}^{-2} \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right)} \]

       4. metadata-eval 99.6%

          \[\leadsto {x}^{\color{blue}{\left(2 \cdot -1\right)}} \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \]

       5. pow-sqr 99.4%

          \[\leadsto \color{blue}{\left({x}^{-1} \cdot {x}^{-1}\right)} \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \]

       6. unpow-1 99.4%

          \[\leadsto \left(\color{blue}{\frac{1}{x}} \cdot {x}^{-1}\right) \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \]

       7. unpow-1 99.4%

          \[\leadsto \left(\frac{1}{x} \cdot \color{blue}{\frac{1}{x}}\right) \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right) \]

       8. associate-*l* 99.3%

          \[\leadsto \color{blue}{\frac{1}{x} \cdot \left(\frac{1}{x} \cdot \left(\tan \left(x \cdot 0.5\right) \cdot \sin x\right)\right)} \]

       9. associate-*l* 99.3%

          \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\left(\frac{1}{x} \cdot \tan \left(x \cdot 0.5\right)\right) \cdot \sin x\right)} \]

       10. *-commutative 99.3%

           \[\leadsto \frac{1}{x} \cdot \left(\color{blue}{\left(\tan \left(x \cdot 0.5\right) \cdot \frac{1}{x}\right)} \cdot \sin x\right) \]

       11. associate-*r/ 99.4%

           \[\leadsto \frac{1}{x} \cdot \left(\color{blue}{\frac{\tan \left(x \cdot 0.5\right) \cdot 1}{x}} \cdot \sin x\right) \]

       12. *-rgt-identity 99.4%

           \[\leadsto \frac{1}{x} \cdot \left(\frac{\color{blue}{\tan \left(x \cdot 0.5\right)}}{x} \cdot \sin x\right) \]

       13. *-commutative 99.4%

           \[\leadsto \frac{1}{x} \cdot \color{blue}{\left(\sin x \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}\right)} \]

       14. associate-*r* 99.5%

           \[\leadsto \color{blue}{\left(\frac{1}{x} \cdot \sin x\right) \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}} \]

       15. *-commutative 99.5%

           \[\leadsto \color{blue}{\left(\sin x \cdot \frac{1}{x}\right)} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x} \]

       16. associate-*l* 99.5%

           \[\leadsto \color{blue}{\sin x \cdot \left(\frac{1}{x} \cdot \frac{\tan \left(x \cdot 0.5\right)}{x}\right)} \]

       17. times-frac 99.4%

           \[\leadsto \sin x \cdot \color{blue}{\frac{1 \cdot \tan \left(x \cdot 0.5\right)}{x \cdot x}} \]

       18. *-commutative 99.4%

           \[\leadsto \sin x \cdot \frac{\color{blue}{\tan \left(x \cdot 0.5\right) \cdot 1}}{x \cdot x} \]

   11. Simplified 99.4%

       \[\leadsto \color{blue}{\sin x \cdot \frac{\tan \left(0.5 \cdot x\right)}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \frac{\tan \left(x \cdot 0.5\right)}{x \cdot x}\\ \end{array} \]

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* (/ (tan (* x 0.5)) x) (/ (sin x) x)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = (tan((x * 0.5d0)) / x) * (sin(x) / x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
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 47.8%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Step-by-step derivation

   1. flip-- 47.5%

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

   2. div-inv 47.5%

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

   3. metadata-eval 47.5%

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

   4. 1-sub-cos 73.0%

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

   5. pow2 73.0%

      \[\leadsto \frac{\color{blue}{{\sin x}^{2}} \cdot \frac{1}{1 + \cos x}}{x \cdot x} \]

3. Applied egg-rr 73.0%

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

   1. unpow2 73.0%

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

   2. associate-*l* 73.0%

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

   3. associate-*r/ 72.9%

      \[\leadsto \frac{\sin x \cdot \color{blue}{\frac{\sin x \cdot 1}{1 + \cos x}}}{x \cdot x} \]

   4. *-rgt-identity 72.9%

      \[\leadsto \frac{\sin x \cdot \frac{\color{blue}{\sin x}}{1 + \cos x}}{x \cdot x} \]

   5. hang-0p-tan 73.2%

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

5. Simplified 73.2%

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

   1. *-commutative 73.2%

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

   2. times-frac 99.8%

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

   3. div-inv 99.8%

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

   4. metadata-eval 99.8%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{-x} \cdot \left(\cos x + -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.00015) 0.5 (* (/ (/ 1.0 x) (- x)) (+ (cos x) -1.0))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.00015) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 / x) / -x) * (cos(x) + -1.0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00015d0) then
        tmp = 0.5d0
    else
        tmp = ((1.0d0 / x) / -x) * (cos(x) + (-1.0d0))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.00015) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 / x) / -x) * (Math.cos(x) + -1.0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.00015:
		tmp = 0.5
	else:
		tmp = ((1.0 / x) / -x) * (math.cos(x) + -1.0)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.00015)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(1.0 / x) / Float64(-x)) * Float64(cos(x) + -1.0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00015)
		tmp = 0.5;
	else
		tmp = ((1.0 / x) / -x) * (cos(x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.00015], 0.5, N[(N[(N[(1.0 / x), $MachinePrecision] / (-x)), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.49999999999999987e-4

   1. Initial program 33.6%

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

   2. Taylor expanded in x around 0 68.5%

      \[\leadsto \color{blue}{0.5} \]

   if 1.49999999999999987e-4 < x

   1. Initial program 99.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. frac-2neg 99.4%

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

      2. div-inv 99.3%

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

      3. distribute-rgt-neg-in 99.3%

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

   3. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\left(-\left(1 - \cos x\right)\right) \cdot \frac{1}{x \cdot \left(-x\right)}} \]
   4. Step-by-step derivation

      1. distribute-lft-neg-out 99.3%

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

      2. associate-/r* 99.3%

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

   5. Simplified 99.3%

      \[\leadsto \color{blue}{-\left(1 - \cos x\right) \cdot \frac{\frac{1}{x}}{-x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{-x} \cdot \left(\cos x + -1\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.00015) 0.5 (/ (- 1.0 (cos x)) (* x x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.00015) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00015d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.00015) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.00015:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.00015)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00015)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.00015], 0.5, N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.49999999999999987e-4

   1. Initial program 33.6%

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

   2. Taylor expanded in x around 0 68.5%

      \[\leadsto \color{blue}{0.5} \]

   if 1.49999999999999987e-4 < x

   1. Initial program 99.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.00015) 0.5 (/ (/ (- 1.0 (cos x)) x) x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.00015) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00015d0) then
        tmp = 0.5d0
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.00015) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.00015:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.00015)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00015)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.00015], 0.5, N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.49999999999999987e-4

   1. Initial program 33.6%

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

   2. Taylor expanded in x around 0 68.5%

      \[\leadsto \color{blue}{0.5} \]

   if 1.49999999999999987e-4 < x

   1. Initial program 99.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. associate-/r* 99.3%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
   4. Step-by-step derivation

      1. un-div-inv 99.3%

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

   5. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00015:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]

Alternative 7: 78.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{1}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2\right)} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (/ 1.0 (* x (+ (* x 0.16666666666666666) (* (/ 1.0 x) 2.0)))))

C

x = abs(x);
double code(double x) {
	return 1.0 / (x * ((x * 0.16666666666666666) + ((1.0 / x) * 2.0)));
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (x * ((x * 0.16666666666666666d0) + ((1.0d0 / x) * 2.0d0)))
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 1.0 / (x * ((x * 0.16666666666666666) + ((1.0 / x) * 2.0)));
}

Python

x = abs(x)
def code(x):
	return 1.0 / (x * ((x * 0.16666666666666666) + ((1.0 / x) * 2.0)))

Julia

x = abs(x)
function code(x)
	return Float64(1.0 / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(Float64(1.0 / x) * 2.0))))
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 1.0 / (x * ((x * 0.16666666666666666) + ((1.0 / x) * 2.0)));
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(1.0 / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.8%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Step-by-step derivation

   1. associate-/r* 48.6%

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

   2. div-inv 48.6%

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

3. Applied egg-rr 48.6%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
4. Step-by-step derivation

   1. clear-num 48.6%

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

   2. frac-times 48.6%

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

   3. metadata-eval 48.6%

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

5. Applied egg-rr 48.6%

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

6. Taylor expanded in x around 0 77.9%

   \[\leadsto \frac{1}{\color{blue}{\left(0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)} \cdot x} \]

7. Final simplification 77.9%

   \[\leadsto \frac{1}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2\right)} \]

Alternative 8: 78.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \frac{\frac{1}{x}}{x \cdot 0.16666666666666666 + \frac{1}{x} \cdot 2} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (/ (/ 1.0 x) (+ (* x 0.16666666666666666) (* (/ 1.0 x) 2.0))))

C

x = abs(x);
double code(double x) {
	return (1.0 / x) / ((x * 0.16666666666666666) + ((1.0 / x) * 2.0));
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / x) / ((x * 0.16666666666666666d0) + ((1.0d0 / x) * 2.0d0))
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return (1.0 / x) / ((x * 0.16666666666666666) + ((1.0 / x) * 2.0));
}

Python

x = abs(x)
def code(x):
	return (1.0 / x) / ((x * 0.16666666666666666) + ((1.0 / x) * 2.0))

Julia

x = abs(x)
function code(x)
	return Float64(Float64(1.0 / x) / Float64(Float64(x * 0.16666666666666666) + Float64(Float64(1.0 / x) * 2.0)))
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = (1.0 / x) / ((x * 0.16666666666666666) + ((1.0 / x) * 2.0));
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[(1.0 / x), $MachinePrecision] / N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(N[(1.0 / x), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.8%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Step-by-step derivation

   1. associate-/r* 48.6%

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

   2. div-inv 48.6%

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

3. Applied egg-rr 48.6%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
4. Step-by-step derivation

   1. *-commutative 48.6%

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

   2. clear-num 48.6%

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

   3. un-div-inv 48.6%

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

5. Applied egg-rr 48.6%

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

6. Taylor expanded in x around 0 78.1%

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

7. Final simplification 78.1%

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

Alternative 9: 77.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 3.45) 0.5 (/ 6.0 (* x x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 3.45) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.45d0) then
        tmp = 0.5d0
    else
        tmp = 6.0d0 / (x * x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 3.45) {
		tmp = 0.5;
	} else {
		tmp = 6.0 / (x * x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 3.45:
		tmp = 0.5
	else:
		tmp = 6.0 / (x * x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 3.45)
		tmp = 0.5;
	else
		tmp = Float64(6.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.45)
		tmp = 0.5;
	else
		tmp = 6.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 3.45], 0.5, N[(6.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.4500000000000002

   1. Initial program 33.6%

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

   2. Taylor expanded in x around 0 68.5%

      \[\leadsto \color{blue}{0.5} \]

   if 3.4500000000000002 < x

   1. Initial program 99.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Step-by-step derivation

      1. associate-/r* 99.3%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
   4. Step-by-step derivation

      1. *-commutative 99.2%

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

      2. clear-num 99.2%

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

      3. un-div-inv 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around 0 51.3%

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

   7. Taylor expanded in x around inf 51.3%

      \[\leadsto \color{blue}{\frac{6}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 51.3%

         \[\leadsto \frac{6}{\color{blue}{x \cdot x}} \]

   9. Simplified 51.3%

      \[\leadsto \color{blue}{\frac{6}{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.45:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{6}{x \cdot x}\\ \end{array} \]

Alternative 10: 50.6% accurate, 107.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0.5 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 0.5)

C

x = abs(x);
double code(double x) {
	return 0.5;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 0.5;
}

Python

x = abs(x)
def code(x):
	return 0.5

Julia

x = abs(x)
function code(x)
	return 0.5
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 0.5;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 0.5

Derivation

1. Initial program 47.8%

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

2. Taylor expanded in x around 0 54.8%

   \[\leadsto \color{blue}{0.5} \]

3. Final simplification 54.8%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023200 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))