cos2 (problem 3.4.1)

Percentage Accurate: 51.1% → 99.6%

Time: 9.4s

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: 51.1% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.032:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m} \cdot \frac{1}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.032)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    0.5)
   (* (/ (- 1.0 (cos x_m)) x_m) (/ 1.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.032) {
		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) * (1.0 / x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.032)
		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) * Float64(1.0 / x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.032], 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] * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 37.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 38.5

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

   4. Applied rewrites 38.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.8

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

   7. Applied rewrites 64.8%

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

   if 0.032000000000000001 < x

   1. Initial program 96.7%

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

   4. Applied rewrites 96.7%

      \[\leadsto \color{blue}{\frac{1}{x \cdot x} \cdot \left(1 - \cos x\right)} \]
   5. 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. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lift--.f64 N/A

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

      6. div-sub N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. remove-double-neg N/A

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

      12. neg-mul-1 N/A

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. rgt-mult-inverse N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l* N/A

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

      19. *-commutative N/A

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

      20. lift-/.f64 N/A

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

      21. un-div-inv N/A

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

   6. Applied rewrites 99.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x} \cdot \frac{1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.032:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.032)
   (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 <= 0.032) {
		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 <= 0.032)
		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, 0.032], 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 < 0.032000000000000001

   1. Initial program 37.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 38.5

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

   4. Applied rewrites 38.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.8

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

   7. Applied rewrites 64.8%

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

   if 0.032000000000000001 < x

   1. Initial program 96.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.032:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.032)
   (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 <= 0.032) {
		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 <= 0.032)
		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, 0.032], 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 < 0.032000000000000001

   1. Initial program 37.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 38.5

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

   4. Applied rewrites 38.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.8

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

   7. Applied rewrites 64.8%

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

   if 0.032000000000000001 < x

   1. Initial program 96.7%

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

Alternative 4: 77.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.7%

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

4. Applied rewrites 56.2%

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

5. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{x}}{\color{blue}{\frac{2 + \frac{1}{6} \cdot {x}^{2}}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{1}{x}}{\frac{\color{blue}{\frac{1}{6} \cdot {x}^{2} + 2}}{x}} \]

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 76.6

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

7. Applied rewrites 76.6%

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

Alternative 5: 75.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 7 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x\_m \cdot x\_m, -0.041666666666666664\right), x\_m \cdot x\_m, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 7e+38)
   (fma
    (fma 0.001388888888888889 (* x_m x_m) -0.041666666666666664)
    (* x_m x_m)
    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 <= 7e+38) {
		tmp = fma(fma(0.001388888888888889, (x_m * x_m), -0.041666666666666664), (x_m * x_m), 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 <= 7e+38)
		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 - 1.0) / 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, 7e+38], 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 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.00000000000000003e38

   1. Initial program 41.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/r/ N/A

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

      8. clear-num N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-neg.f64 42.5

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

   4. Applied rewrites 42.5%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   6. 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 60.9

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

   7. Applied rewrites 60.9%

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

   if 7.00000000000000003e38 < x

   1. Initial program 96.3%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 57.7%

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

Alternative 6: 75.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 9.6 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 9.6e+76) 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 <= 9.6e+76) {
		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 <= 9.6d+76) 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 <= 9.6e+76) {
		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 <= 9.6e+76:
		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 <= 9.6e+76)
		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 <= 9.6e+76)
		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, 9.6e+76], 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 < 9.5999999999999999e76

   1. Initial program 44.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.0%

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

   if 9.5999999999999999e76 < x

   1. Initial program 95.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 69.7%

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

Alternative 7: 51.3% 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 54.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 47.5%

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

Reproduce

herbie shell --seed 2024240 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))