cos2 (problem 3.4.1)

Percentage Accurate: 50.4% → 99.8%

Time: 9.3s

Alternatives: 8

Speedup: 120.0×

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: 50.4% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. 1-sub-cos N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. hang-0p-tan N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-/.f64 78.3

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

4. Applied rewrites 78.3%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. associate-/r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 75.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.112)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (- (/ 1.0 x) (/ (cos x) x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.112) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((1.0 / x) - (cos(x) / x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.112)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(cos(x) / x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.112], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 35.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right), {x}^{2}, \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{40320} \cdot {x}^{2} + \frac{1}{720}}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{40320}, {x}^{2}, \frac{1}{720}\right)}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      15. lower-*.f64 66.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.112000000000000002 < x

   1. Initial program 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. distribute-rgt-neg-in N/A

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. frac-2neg N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{\cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.105:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.105)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.105) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.105)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.105], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.104999999999999996

   1. Initial program 35.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right), {x}^{2}, \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{40320} \cdot {x}^{2} + \frac{1}{720}}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{40320}, {x}^{2}, \frac{1}{720}\right)}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      15. lower-*.f64 66.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.104999999999999996 < x

   1. Initial program 98.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   4. Applied rewrites 98.9%

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

Alternative 4: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.105:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.105)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.105) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.105)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.105], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.104999999999999996

   1. Initial program 35.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2}} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right), {x}^{2}, \frac{1}{2}\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}\right) \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720} + \frac{-1}{40320} \cdot {x}^{2}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{40320} \cdot {x}^{2} + \frac{1}{720}}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{40320}, {x}^{2}, \frac{1}{720}\right)}, {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, \color{blue}{x \cdot x}, \frac{1}{720}\right), {x}^{2}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{40320}, x \cdot x, \frac{1}{720}\right), x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      15. lower-*.f64 66.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.104999999999999996 < x

   1. Initial program 98.2%

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

Alternative 5: 63.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e+38)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 5e+38) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e+38)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 5e+38], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999997e38

   1. Initial program 39.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      10. lower-*.f64 64.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 4.9999999999999997e38 < x

   1. Initial program 98.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 53.9%

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

Alternative 6: 63.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = fma(-0.041666666666666664, (x * x), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.5)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 37.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      3. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      4. lower-*.f64 65.4

         \[\leadsto \mathsf{fma}\left(-0.041666666666666664, \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)} \]

   if 3.5 < x

   1. Initial program 98.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 48.3%

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

Alternative 7: 78.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\mathsf{fma}\left(-0.16666666666666666, x \cdot x, -2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ -1.0 (fma -0.16666666666666666 (* x x) -2.0)))

C

double code(double x) {
	return -1.0 / fma(-0.16666666666666666, (x * x), -2.0);
}

Julia

function code(x)
	return Float64(-1.0 / fma(-0.16666666666666666, Float64(x * x), -2.0))
end

Wolfram

code[x_] := N[(-1.0 / N[(-0.16666666666666666 * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.2%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

4. Applied rewrites 52.2%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{6} \cdot {x}^{2} - 2}} \]
6. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{6} \cdot {x}^{2} + \left(\mathsf{neg}\left(2\right)\right)}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {x}^{2}, \mathsf{neg}\left(2\right)\right)}} \]

   3. unpow2 N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, \mathsf{neg}\left(2\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{-1}{\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{x \cdot x}, \mathsf{neg}\left(2\right)\right)} \]

   5. metadata-eval 77.3

      \[\leadsto \frac{-1}{\mathsf{fma}\left(-0.16666666666666666, x \cdot x, \color{blue}{-2}\right)} \]

7. Applied rewrites 77.3%

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666, x \cdot x, -2\right)}} \]
8. Add Preprocessing

Alternative 8: 51.9% accurate, 120.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 52.2%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Applied rewrites 50.6%

   \[\leadsto \color{blue}{0.5} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))