cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.8%

Time: 8.6s

Alternatives: 10

Speedup: 4.6×

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: 51.3% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. 1-sub-cos N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. hang-0p-tan N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-/.f64 77.2

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

4. Applied rewrites 77.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\cos x - 1}\right)}^{-1} \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (* (pow (/ x (- (cos x) 1.0)) -1.0) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.095) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = pow((x / (cos(x) - 1.0)), -1.0) * (-1.0 / x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.095)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64((Float64(x / Float64(cos(x) - 1.0)) ^ -1.0) * Float64(-1.0 / x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.095], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Power[N[(x / N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 33.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 0.095000000000000001 < x

   1. Initial program 97.4%

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

   4. Applied rewrites 99.3%

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

Alternative 3: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-1}}{\frac{x}{1 - \cos x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (pow x -1.0) (/ x (- 1.0 (cos x))))))

C

double code(double x) {
	double tmp;
	if (x <= 0.095) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = pow(x, -1.0) / (x / (1.0 - cos(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.095)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64((x ^ -1.0) / Float64(x / Float64(1.0 - cos(x))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.095], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Power[x, -1.0], $MachinePrecision] / N[(x / N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 33.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 0.095000000000000001 < x

   1. Initial program 97.4%

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

   4. Applied rewrites 99.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. clear-num N/A

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

      10. frac-2neg N/A

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

      11. mul-1-neg N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-1}}{\frac{x}{1 - \cos x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (* (pow x -2.0) (- 1.0 (cos x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.095) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = pow(x, -2.0) * (1.0 - cos(x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.095)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64((x ^ -2.0) * Float64(1.0 - cos(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.095], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[Power[x, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 33.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 0.095000000000000001 < x

   1. Initial program 97.4%

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

   4. Applied rewrites 99.4%

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

Alternative 5: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.095) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.095)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.095], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 33.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 0.095000000000000001 < x

   1. Initial program 97.4%

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

   4. Applied rewrites 99.3%

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

Alternative 6: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.48015873015873 \cdot 10^{-5}, x \cdot x, 0.001388888888888889\right), x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.095)
   (fma
    (fma
     (fma -2.48015873015873e-5 (* x x) 0.001388888888888889)
     (* x x)
     -0.041666666666666664)
    (* x x)
    0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.095) {
		tmp = fma(fma(fma(-2.48015873015873e-5, (x * x), 0.001388888888888889), (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.095)
		tmp = fma(fma(fma(-2.48015873015873e-5, Float64(x * x), 0.001388888888888889), Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.095], N[(N[(N[(-2.48015873015873e-5 * N[(x * x), $MachinePrecision] + 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 33.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. *-commutative N/A

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 68.8

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

   5. Applied rewrites 68.8%

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

   if 0.095000000000000001 < x

   1. Initial program 97.4%

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

Alternative 7: 77.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 78.9

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

7. Applied rewrites 78.9%

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

8. Final simplification 78.9%

   \[\leadsto {\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 2\right)\right)}^{-1} \]
9. Add Preprocessing

Alternative 8: 63.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.2e+38)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.2e+38) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2e+38)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.2e+38], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999985e38

   1. Initial program 36.6%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 66.3

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

   5. Applied rewrites 66.3%

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

   if 3.19999999999999985e38 < x

   1. Initial program 97.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 58.7%

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

Alternative 9: 62.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 34.1%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.2

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

   5. Applied rewrites 68.2%

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

   if 3.5 < x

   1. Initial program 97.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 51.1%

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

Alternative 10: 51.2% accurate, 120.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 48.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 53.9%

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

Reproduce

herbie shell --seed 2024306 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))