cos2 (problem 3.4.1)

Percentage Accurate: 50.7% → 99.8%

Time: 10.2s

Alternatives: 11

Speedup: 17.1×

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: 50.7% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

   1. lift-cos.f64 N/A

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

   2. flip-- N/A

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

   3. lift-*.f64 N/A

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

   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 81.3

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

4. Applied rewrites 81.3%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. frac-2neg N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. frac-2neg N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 99.8

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

   13. lift-/.f64 N/A

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

   14. div-inv N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} \cdot \left(1 - \cos x\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -0.041666666666666664) 0.5)
   (/ (* (/ 1.0 x) (- 1.0 (cos x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = ((1.0 / x) * (1.0 - cos(x))) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 / x) * Float64(1.0 - cos(x))) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.033000000000000002

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 0.033000000000000002 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. neg-mul-1 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-neg-in N/A

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

      19. sub-neg N/A

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

      20. lower--.f64 N/A

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

      21. metadata-eval 99.4

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

   6. Applied rewrites 99.4%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

   8. Applied rewrites 99.4%

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

Alternative 3: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x}}{x} \cdot \left(\cos x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -0.041666666666666664) 0.5)
   (* (/ (/ -1.0 x) x) (+ (cos x) -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = ((-1.0 / x) / x) * (cos(x) + -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(-1.0 / x) / x) * Float64(cos(x) + -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-1.0 / x), $MachinePrecision] / x), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.033000000000000002

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 0.033000000000000002 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.8%

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

      1. associate-/r* N/A

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

      2. lift-/.f64 N/A

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

      3. lower-/.f64 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 4: 74.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -0.041666666666666664) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $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 < 0.033000000000000002

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 0.033000000000000002 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. frac-times N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-commutative N/A

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

      10. lower-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*l/ N/A

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

      13. *-commutative N/A

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

      14. neg-mul-1 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. distribute-neg-in N/A

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

      19. sub-neg N/A

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

      20. lower--.f64 N/A

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

      21. metadata-eval 99.4

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

   6. Applied rewrites 99.4%

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

Alternative 5: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -0.041666666666666664) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $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 < 0.033000000000000002

   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.9

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

   5. Applied rewrites 65.9%

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

   if 0.033000000000000002 < x

   1. Initial program 98.8%

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

Alternative 6: 63.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 700000:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x}, x, x \cdot \frac{-1}{x}\right)}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 700000.0)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -0.041666666666666664) 0.5)
   (/ (fma (/ 1.0 x) x (* x (/ -1.0 x))) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 700000.0) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = fma((1.0 / x), x, (x * (-1.0 / x))) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 700000.0)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = Float64(fma(Float64(1.0 / x), x, Float64(x * Float64(-1.0 / x))) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 700000.0], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] * x + N[(x * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7e5

   1. Initial program 36.9%

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

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if 7e5 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 50.4%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-/.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. associate-*l/ N/A

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

       9. lift-*.f64 N/A

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

       10. associate-/r* N/A

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

       11. lift-/.f64 N/A

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

       12. frac-add N/A

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

       13. lift-*.f64 N/A

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

       14. lower-/.f64 N/A

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

   11. Applied rewrites 51.2%

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

Alternative 7: 63.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, \frac{1}{x}, \frac{-1}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+22)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -0.041666666666666664) 0.5)
   (fma (/ 1.0 x) (/ 1.0 x) (/ -1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+22) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = fma((1.0 / x), (1.0 / x), (-1.0 / (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+22)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = fma(Float64(1.0 / x), Float64(1.0 / x), Float64(-1.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+22], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e22

   1. Initial program 37.9%

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

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 64.6

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

   5. Applied rewrites 64.6%

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

   if 1.4e22 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 52.3%

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

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. +-commutative N/A

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

       6. lift-/.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-/r* N/A

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

       9. lift-/.f64 N/A

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

       10. div-inv N/A

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

       11. lift-/.f64 N/A

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

       12. lower-fma.f64 N/A

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

       13. lift-/.f64 N/A

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

       14. lift-/.f64 N/A

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

       15. frac-times N/A

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

       16. metadata-eval N/A

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

       17. lift-*.f64 N/A

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

       18. lower-/.f64 53.0

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

   11. Applied rewrites 53.0%

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

Alternative 8: 63.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{x}, \frac{-1}{x}, \frac{1}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.8e+14)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -0.041666666666666664) 0.5)
   (fma (/ 1.0 x) (/ -1.0 x) (/ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 4.8e+14) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = fma((1.0 / x), (-1.0 / x), (1.0 / (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.8e+14)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = fma(Float64(1.0 / x), Float64(-1.0 / x), Float64(1.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.8e+14], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision] + N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.8e14

   1. Initial program 36.9%

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

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if 4.8e14 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*r/ N/A

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

      14. lift-*.f64 N/A

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

      15. times-frac N/A

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

      16. lift-/.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. lower-fma.f64 N/A

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

      19. lift-/.f64 N/A

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

      20. lift-/.f64 N/A

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

   6. Applied rewrites 98.6%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 50.4%

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

Alternative 9: 63.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.6 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.6e+38)
   (fma (* x x) (fma (* x x) 0.001388888888888889 -0.041666666666666664) 0.5)
   0.0))

C

double code(double x) {
	double tmp;
	if (x <= 4.6e+38) {
		tmp = fma((x * x), fma((x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.6e+38)
		tmp = fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, -0.041666666666666664), 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.6e+38], N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.6000000000000002e38

   1. Initial program 39.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if 4.6000000000000002e38 < x

   1. Initial program 98.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 57.2%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. div0 57.2

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 57.2%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 63.5% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 8.5e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 8.5e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 8.5d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 8.5e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 8.5e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 8.5e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 8.5e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 8.5e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 8.49999999999999992e76

   1. Initial program 41.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 61.4%

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

   if 8.49999999999999992e76 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.3%

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

      1. metadata-eval N/A

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

      2. lift-*.f64 N/A

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

      3. div0 63.3

         \[\leadsto \color{blue}{0} \]

   7. Applied rewrites 63.3%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 27.5% accurate, 120.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 53.1%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 24.8%

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

   1. metadata-eval N/A

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

   2. lift-*.f64 N/A

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

   3. div0 25.4

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 25.4%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))