cos2 (problem 3.4.1)

Percentage Accurate: 51.2% → 99.7%

Time: 8.3s

Alternatives: 7

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: 51.2% 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: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{-x} \cdot \left(\frac{\sin x}{x} \cdot \tan \left(0.5 \cdot x\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ -1.0 (- x)) (* (/ (sin x) x) (tan (* 0.5 x)))))

C

double code(double x) {
	return (-1.0 / -x) * ((sin(x) / x) * tan((0.5 * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((-1.0d0) / -x) * ((sin(x) / x) * tan((0.5d0 * x)))
end function

Java

public static double code(double x) {
	return (-1.0 / -x) * ((Math.sin(x) / x) * Math.tan((0.5 * x)));
}

Python

def code(x):
	return (-1.0 / -x) * ((math.sin(x) / x) * math.tan((0.5 * x)))

Julia

function code(x)
	return Float64(Float64(-1.0 / Float64(-x)) * Float64(Float64(sin(x) / x) * tan(Float64(0.5 * x))))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / -x) * ((sin(x) / x) * tan((0.5 * x)));
end

Wolfram

code[x_] := N[(N[(-1.0 / (-x)), $MachinePrecision] * N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[Tan[N[(0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.9%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. hang-0p-tan N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-/.f64 75.9

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

4. Applied rewrites 75.9%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. neg-mul-1 N/A

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

   4. lift-*.f64 N/A

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

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

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

   6. lift-neg.f64 N/A

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

   7. times-frac N/A

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

   8. lift-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\frac{-1}{x} \cdot \frac{\sin x}{-x}\right)} \cdot \tan \left(\frac{x}{2}\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.7

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

   \[\leadsto \frac{-1}{-x} \cdot \left(\frac{\sin x}{x} \cdot \tan \left(0.5 \cdot x\right)\right) \]
10. Add Preprocessing

Alternative 2: 75.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x + -1\right) \cdot \frac{-1}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.102)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (* (+ (cos x) -1.0) (/ -1.0 x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.102) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((cos(x) + -1.0) * (-1.0 / x)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.102)
		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(cos(x) + -1.0) * Float64(-1.0 / x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.102], 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[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.101999999999999993

   1. Initial program 33.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   if 0.101999999999999993 < x

   1. Initial program 96.5%

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

   4. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. lift-neg.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. lift-neg.f64 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. frac-2neg N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.1

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

      10. lift-neg.f64 N/A

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

      11. neg-sub0 N/A

          \[\leadsto \frac{\frac{-1}{x} \cdot \color{blue}{\left(0 - \left(1 - \cos x\right)\right)}}{x} \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\frac{-1}{x} \cdot \left(0 - \color{blue}{\left(1 - \cos x\right)}\right)}{x} \]

      13. associate--r- N/A

          \[\leadsto \frac{\frac{-1}{x} \cdot \color{blue}{\left(\left(0 - 1\right) + \cos x\right)}}{x} \]

      14. metadata-eval N/A

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

      15. lower-+.f64 99.1

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

   8. Applied rewrites 99.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\cos x + -1\right) \cdot \frac{-1}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.102)
   (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 <= 0.102) {
		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 <= 0.102)
		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, 0.102], 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 < 0.101999999999999993

   1. Initial program 33.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   if 0.101999999999999993 < x

   1. Initial program 96.5%

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

   4. Applied rewrites 99.1%

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

Alternative 4: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.102:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.102)
   (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 <= 0.102) {
		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 <= 0.102)
		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, 0.102], 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 < 0.101999999999999993

   1. Initial program 33.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.7

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

   5. Applied rewrites 68.7%

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

   if 0.101999999999999993 < x

   1. Initial program 96.5%

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

Alternative 5: 64.0% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.2e+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 <= 7.2e+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 <= 7.2e+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, 7.2e+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 < 7.19999999999999938e38

   1. Initial program 34.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 67.6

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

   5. Applied rewrites 67.6%

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

   if 7.19999999999999938e38 < x

   1. Initial program 96.4%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 64.1%

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

Alternative 6: 64.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.4 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.4e+76) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 5.4e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.4d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.4e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.4e+76:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.4e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.4e+76)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.4e+76], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.3999999999999998e76

   1. Initial program 37.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.0%

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

   if 5.3999999999999998e76 < x

   1. Initial program 96.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 74.1%

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

Alternative 7: 51.2% 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 49.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.0%

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

Reproduce

herbie shell --seed 2024294 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))