cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.6%

Time: 8.3s

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: 50.3% 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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x\_m} \cdot \left(\cos x\_m + -1\right)}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (/ (* (/ -1.0 x_m) (+ (cos x_m) -1.0)) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0056) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = ((-1.0 / x_m) * (cos(x_m) + -1.0)) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0056)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(-1.0 / x_m) * Float64(cos(x_m) + -1.0)) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0056], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-1.0 / x$95$m), $MachinePrecision] * N[(N[Cos[x$95$m], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00559999999999999994

   1. Initial program 30.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

   if 0.00559999999999999994 < x

   1. Initial program 99.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. frac-sub N/A

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

      7. lift-*.f64 N/A

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

      8. cube-unmult N/A

         \[\leadsto \frac{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}{\color{blue}{{x}^{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}{{x}^{3}}} \]

      10. inv-pow N/A

          \[\leadsto \frac{\color{blue}{{x}^{-1}} \cdot \left(x \cdot x\right) - x \cdot \cos x}{{x}^{3}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{{x}^{-1} \cdot \color{blue}{\left(x \cdot x\right)} - x \cdot \cos x}{{x}^{3}} \]

      12. pow2 N/A

          \[\leadsto \frac{{x}^{-1} \cdot \color{blue}{{x}^{2}} - x \cdot \cos x}{{x}^{3}} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{\color{blue}{{x}^{\left(-1 + 2\right)}} - x \cdot \cos x}{{x}^{3}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{{x}^{\color{blue}{1}} - x \cdot \cos x}{{x}^{3}} \]

      15. unpow1 N/A

          \[\leadsto \frac{\color{blue}{x} - x \cdot \cos x}{{x}^{3}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - x \cdot \cos x}}{{x}^{3}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x - \color{blue}{x \cdot \cos x}}{{x}^{3}} \]

      18. lower-pow.f64 82.5

          \[\leadsto \frac{x - x \cdot \cos x}{\color{blue}{{x}^{3}}} \]

   4. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\frac{x - x \cdot \cos x}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \frac{x - x \cdot \cos x}{{x}^{3}}} \]

      2. *-inverses N/A

         \[\leadsto \color{blue}{\frac{x}{x}} \cdot \frac{x - x \cdot \cos x}{{x}^{3}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x}{x} \cdot \color{blue}{\frac{x - x \cdot \cos x}{{x}^{3}}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(x - x \cdot \cos x\right)}{x \cdot {x}^{3}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(x - x \cdot \cos x\right)}}{x \cdot {x}^{3}} \]

      6. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{x \cdot x - x \cdot \left(x \cdot \cos x\right)}}{x \cdot {x}^{3}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x} - x \cdot \left(x \cdot \cos x\right)}{x \cdot {x}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x - x \cdot \color{blue}{\left(x \cdot \cos x\right)}}{x \cdot {x}^{3}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{x \cdot x - \color{blue}{\left(x \cdot x\right) \cdot \cos x}}{x \cdot {x}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x - \color{blue}{\left(x \cdot x\right)} \cdot \cos x}{x \cdot {x}^{3}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x - \color{blue}{\left(x \cdot x\right) \cdot \cos x}}{x \cdot {x}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{x \cdot x - \left(x \cdot x\right) \cdot \cos x}{x \cdot \color{blue}{{x}^{3}}} \]

      13. cube-mult N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. lift--.f64 N/A

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

      18. associate-/l/ N/A

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

   6. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/r* N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.1

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 99.1

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

   8. Applied rewrites 99.1%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ (tan (* 0.5 x_m)) (* x_m (/ x_m (sin x_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Tan[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] / N[(x$95$m * N[(x$95$m / N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.4%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 80.1

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

4. Applied rewrites 80.1%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
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. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. associate-/r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 99.7

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

6. Applied rewrites 99.7%

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

Alternative 3: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x\_m \cdot x\_m} \cdot \mathsf{fma}\left({x\_m}^{-1}, x\_m, -1\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.5e+22)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (* (/ -1.0 (* x_m x_m)) (fma (pow x_m -1.0) x_m -1.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.5e+22) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = (-1.0 / (x_m * x_m)) * fma(pow(x_m, -1.0), x_m, -1.0);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.5e+22)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(-1.0 / Float64(x_m * x_m)) * fma((x_m ^ -1.0), x_m, -1.0));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 3.5e+22], N[(N[(0.001388888888888889 * N[(x$95$m * x$95$m), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(-1.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(N[Power[x$95$m, -1.0], $MachinePrecision] * x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5e22

   1. Initial program 32.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 69.5

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

   5. Applied rewrites 69.5%

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

   if 3.5e22 < x

   1. Initial program 99.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. frac-sub N/A

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

      7. lift-*.f64 N/A

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

      8. cube-unmult N/A

         \[\leadsto \frac{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}{\color{blue}{{x}^{3}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{x} \cdot \left(x \cdot x\right) - x \cdot \cos x}{{x}^{3}}} \]

      10. inv-pow N/A

          \[\leadsto \frac{\color{blue}{{x}^{-1}} \cdot \left(x \cdot x\right) - x \cdot \cos x}{{x}^{3}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{{x}^{-1} \cdot \color{blue}{\left(x \cdot x\right)} - x \cdot \cos x}{{x}^{3}} \]

      12. pow2 N/A

          \[\leadsto \frac{{x}^{-1} \cdot \color{blue}{{x}^{2}} - x \cdot \cos x}{{x}^{3}} \]

      13. pow-prod-up N/A

          \[\leadsto \frac{\color{blue}{{x}^{\left(-1 + 2\right)}} - x \cdot \cos x}{{x}^{3}} \]

      14. metadata-eval N/A

          \[\leadsto \frac{{x}^{\color{blue}{1}} - x \cdot \cos x}{{x}^{3}} \]

      15. unpow1 N/A

          \[\leadsto \frac{\color{blue}{x} - x \cdot \cos x}{{x}^{3}} \]

      16. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{x - x \cdot \cos x}}{{x}^{3}} \]

      17. lower-*.f64 N/A

          \[\leadsto \frac{x - \color{blue}{x \cdot \cos x}}{{x}^{3}} \]

      18. lower-pow.f64 80.8

          \[\leadsto \frac{x - x \cdot \cos x}{\color{blue}{{x}^{3}}} \]

   4. Applied rewrites 80.8%

      \[\leadsto \color{blue}{\frac{x - x \cdot \cos x}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \color{blue}{1 \cdot \frac{x - x \cdot \cos x}{{x}^{3}}} \]

      2. *-inverses N/A

         \[\leadsto \color{blue}{\frac{x}{x}} \cdot \frac{x - x \cdot \cos x}{{x}^{3}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{x}{x} \cdot \color{blue}{\frac{x - x \cdot \cos x}{{x}^{3}}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x \cdot \left(x - x \cdot \cos x\right)}{x \cdot {x}^{3}}} \]

      5. lift--.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(x - x \cdot \cos x\right)}}{x \cdot {x}^{3}} \]

      6. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{x \cdot x - x \cdot \left(x \cdot \cos x\right)}}{x \cdot {x}^{3}} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot x} - x \cdot \left(x \cdot \cos x\right)}{x \cdot {x}^{3}} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot x - x \cdot \color{blue}{\left(x \cdot \cos x\right)}}{x \cdot {x}^{3}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{x \cdot x - \color{blue}{\left(x \cdot x\right) \cdot \cos x}}{x \cdot {x}^{3}} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x - \color{blue}{\left(x \cdot x\right)} \cdot \cos x}{x \cdot {x}^{3}} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{x \cdot x - \color{blue}{\left(x \cdot x\right) \cdot \cos x}}{x \cdot {x}^{3}} \]

      12. lift-pow.f64 N/A

          \[\leadsto \frac{x \cdot x - \left(x \cdot x\right) \cdot \cos x}{x \cdot \color{blue}{{x}^{3}}} \]

      13. cube-mult N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l* N/A

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

      16. lift-*.f64 N/A

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

      17. lift--.f64 N/A

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

      18. associate-/l/ N/A

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

   6. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. *-lft-identity N/A

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

      4. *-inverses N/A

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

      5. associate-/r/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/r/ N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{x \cdot x} \cdot \mathsf{fma}\left(\color{blue}{\frac{1}{x}}, x, -1\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 50.7

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

   11. Applied rewrites 50.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x \cdot x} \cdot \mathsf{fma}\left({x}^{-1}, x, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0056) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0056)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / x_m) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0056], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00559999999999999994

   1. Initial program 30.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

   if 0.00559999999999999994 < x

   1. Initial program 99.2%

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

   4. Applied rewrites 99.0%

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

Alternative 5: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0056) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0056)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0056], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00559999999999999994

   1. Initial program 30.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 71.2

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

   5. Applied rewrites 71.2%

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

   if 0.00559999999999999994 < x

   1. Initial program 99.2%

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

Alternative 6: 76.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5.8 \cdot 10^{+51}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x\_m \cdot \frac{1}{x\_m}}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5.8e+51) 0.5 (/ (- 1.0 (* x_m (/ 1.0 x_m))) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5.8e+51) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - (x_m * (1.0 / x_m))) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5.8d+51) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - (x_m * (1.0d0 / x_m))) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 5.8e+51) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - (x_m * (1.0 / x_m))) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 5.8e+51:
		tmp = 0.5
	else:
		tmp = (1.0 - (x_m * (1.0 / x_m))) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5.8e+51)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - Float64(x_m * Float64(1.0 / x_m))) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 5.8e+51)
		tmp = 0.5;
	else
		tmp = (1.0 - (x_m * (1.0 / x_m))) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 5.8e+51], 0.5, N[(N[(1.0 - N[(x$95$m * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.7999999999999997e51

   1. Initial program 35.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.8%

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

   if 5.7999999999999997e51 < x

   1. Initial program 99.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift--.f64 N/A

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

      5. div-sub N/A

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

      6. sub-div N/A

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

      7. frac-sub N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

          \[\leadsto \frac{{x}^{\color{blue}{0}} - x \cdot \frac{1}{x}}{x \cdot x} \]

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 56.2

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

   7. Applied rewrites 56.2%

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

Alternative 7: 76.4% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.12 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.12e+77) 0.5 (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.12e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.12d+77) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.12e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.12e+77:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.12e+77)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.12e+77)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.12e+77], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1199999999999999e77

   1. Initial program 37.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.7%

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

   if 1.1199999999999999e77 < x

   1. Initial program 99.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 63.5%

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

Alternative 8: 52.2% accurate, 120.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

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

Java

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

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 48.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.5%

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

Reproduce

herbie shell --seed 2024317 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))