cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.8%

Time: 9.4s

Alternatives: 10

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: 50.5% 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, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(\sin x\_m \cdot \tan \left(0.5 \cdot x\_m\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0005)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (* (pow x_m -2.0) (* (sin x_m) (tan (* 0.5 x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0005) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = pow(x_m, -2.0) * (sin(x_m) * tan((0.5 * x_m)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0005)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64((x_m ^ -2.0) * Float64(sin(x_m) * tan(Float64(0.5 * x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0005], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000001e-4

   1. Initial program 32.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

   if 5.0000000000000001e-4 < x

   1. Initial program 98.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
   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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. frac-times N/A

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

      21. metadata-eval N/A

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

      22. pow2 N/A

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

      23. pow-flip N/A

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

      24. lower-pow.f64 N/A

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

      25. metadata-eval 99.7

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

   6. Applied rewrites 99.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0005:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(\sin x \cdot \tan \left(0.5 \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x\_m \cdot \tan \left(0.5 \cdot x\_m\right)}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.002)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (/ (/ (* (sin x_m) (tan (* 0.5 x_m))) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.002) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = ((sin(x_m) * tan((0.5 * x_m))) / x_m) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.002)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(sin(x_m) * tan(Float64(0.5 * 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, 0.002], 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[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-3

   1. Initial program 32.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

   if 2e-3 < 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. 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 99.6

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

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
   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. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 99.7

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

      10. lift-/.f64 N/A

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

      11. clear-num N/A

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

      12. associate-/r/ N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 99.7

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

   6. Applied rewrites 99.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \tan \left(0.5 \cdot x\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.035:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x\_m} - \frac{1}{x\_m} \cdot \cos x\_m}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.035)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (/ (- (/ 1.0 x_m) (* (/ 1.0 x_m) (cos x_m))) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.035) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = ((1.0 / x_m) - ((1.0 / x_m) * cos(x_m))) / x_m;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.035)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / x_m) - Float64(Float64(1.0 / x_m) * cos(x_m))) / x_m);
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.035], 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[(N[(1.0 / x$95$m), $MachinePrecision] - N[(N[(1.0 / x$95$m), $MachinePrecision] * N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.035000000000000003

   1. Initial program 32.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

   if 0.035000000000000003 < x

   1. Initial program 99.2%

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

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
   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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. lift-/.f64 N/A

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

      6. div-sub N/A

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

      7. associate-/r* N/A

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

      8. metadata-eval N/A

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

      9. frac-times N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. lift-neg.f64 N/A

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

      16. div-inv N/A

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

      17. lift-/.f64 N/A

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

      18. frac-2neg N/A

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

      19. lift-neg.f64 N/A

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

      20. associate-/l/ N/A

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

      21. neg-mul-1 N/A

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

   6. Applied rewrites 99.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.035:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - \frac{1}{x} \cdot \cos x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.026:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.026)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -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 <= 0.026) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -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 <= 0.026)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -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, 0.026], 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[(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 < 0.0259999999999999988

   1. Initial program 32.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

   if 0.0259999999999999988 < x

   1. Initial program 99.2%

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

   4. Applied rewrites 99.2%

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

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.026:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.026)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -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 <= 0.026) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -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 <= 0.026)
		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 - 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, 0.026], 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[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 < 0.0259999999999999988

   1. Initial program 32.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

   if 0.0259999999999999988 < 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: 77.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-0.16666666666666666 \cdot \left(x\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 4.8)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (/ -1.0 (* -0.16666666666666666 (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.8) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 0.5);
	} else {
		tmp = -1.0 / (-0.16666666666666666 * (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.8)
		tmp = fma(fma(0.001388888888888889, Float64(x_m * x_m), -0.041666666666666664), Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(-1.0 / Float64(-0.16666666666666666 * 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, 4.8], 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[(-1.0 / N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999982

   1. Initial program 32.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   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 69.9

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

   5. Applied rewrites 69.9%

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

   if 4.79999999999999982 < x

   1. Initial program 99.2%

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

   4. Applied rewrites 99.2%

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

   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 60.2

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

   7. Applied rewrites 60.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 60.2%

       \[\leadsto \frac{-1}{-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{-0.16666666666666666 \cdot \left(x\_m \cdot x\_m\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.3)
   (fma -0.041666666666666664 (* x_m x_m) 0.5)
   (/ -1.0 (* -0.16666666666666666 (* x_m x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.3) {
		tmp = fma(-0.041666666666666664, (x_m * x_m), 0.5);
	} else {
		tmp = -1.0 / (-0.16666666666666666 * (x_m * x_m));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.3)
		tmp = fma(-0.041666666666666664, Float64(x_m * x_m), 0.5);
	else
		tmp = Float64(-1.0 / Float64(-0.16666666666666666 * 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, 3.3], N[(-0.041666666666666664 * N[(x$95$m * x$95$m), $MachinePrecision] + 0.5), $MachinePrecision], N[(-1.0 / N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 32.3%

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

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

   5. Applied rewrites 69.5%

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

   if 3.2999999999999998 < x

   1. Initial program 99.2%

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

   4. Applied rewrites 99.2%

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

   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 60.2

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

   7. Applied rewrites 60.2%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 60.2%

       \[\leadsto \frac{-1}{-0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.4e+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 <= 1.4e+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 <= 1.4d+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 <= 1.4e+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 <= 1.4e+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 <= 1.4e+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 <= 1.4e+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, 1.4e+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 < 1.4e77

   1. Initial program 36.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 1.4e77 < x

   1. Initial program 99.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 70.6%

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

Alternative 9: 77.6% accurate, 5.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ -1.0 (fma -0.16666666666666666 (* x_m x_m) -2.0)))

C

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

Julia

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(-1.0 / N[(-0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.7%

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

4. Applied rewrites 47.7%

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

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 79.6

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

7. Applied rewrites 79.6%

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

Alternative 10: 51.8% 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 47.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.8%

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

Reproduce

herbie shell --seed 2024242 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))