cos2 (problem 3.4.1)

Percentage Accurate: 51.2% → 99.8%

Time: 10.8s

Alternatives: 7

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: 51.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.5%

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

   1. flip-- N/A

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

   2. associate-/l/ N/A

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

   3. metadata-eval N/A

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

   4. 1-sub-cos N/A

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

   5. times-frac N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. hang-0p-tan N/A

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

   11. tan-lowering-tan.f64 N/A

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

   12. /-lowering-/.f64 78.6

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

4. Applied egg-rr 78.6%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. tan-lowering-tan.f64 N/A

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

   7. div-inv N/A

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

   8. *-lowering-*.f64 N/A

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

   9. metadata-eval N/A

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

   10. /-lowering-/.f64 N/A

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

   11. sin-lowering-sin.f64 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 2: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.031:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.031)
   (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 <= 0.031) {
		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 <= 0.031)
		tmp = fma(x, Float64(x * fma(x, Float64(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, 0.031], N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $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 < 0.031

   1. Initial program 36.6%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. metadata-eval N/A

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

      4. 1-sub-cos N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. hang-0p-tan N/A

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

      11. tan-lowering-tan.f64 N/A

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

      12. /-lowering-/.f64 72.8

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

   4. Applied egg-rr 72.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 72.8

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

   6. Applied egg-rr 72.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   8. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 65.6

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

   9. Simplified 65.6%

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

   if 0.031 < x

   1. Initial program 98.0%

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

   4. Applied egg-rr 97.9%

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

      1. associate-*l/ N/A

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

      2. frac-times N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. neg-mul-1 N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. /-lowering-/.f64 N/A

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

      14. --lowering--.f64 N/A

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

      15. cos-lowering-cos.f64 99.5

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

   6. Applied egg-rr 99.5%

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

Alternative 3: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.031:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.031)
   (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 <= 0.031) {
		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 <= 0.031)
		tmp = fma(x, Float64(x * fma(x, Float64(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, 0.031], N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $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 < 0.031

   1. Initial program 36.6%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. metadata-eval N/A

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

      4. 1-sub-cos N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. hang-0p-tan N/A

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

      11. tan-lowering-tan.f64 N/A

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

      12. /-lowering-/.f64 72.8

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

   4. Applied egg-rr 72.8%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 72.8

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

   6. Applied egg-rr 72.8%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   8. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 65.6

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

   9. Simplified 65.6%

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

   if 0.031 < x

   1. Initial program 98.0%

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

Alternative 4: 63.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, x, \frac{-1}{x}\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.5e+29)
   (fma x (* x (fma x (* x 0.001388888888888889) -0.041666666666666664)) 0.5)
   (/ (fma (/ 1.0 (* x x)) x (/ -1.0 x)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.5e+29) {
		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;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.5e+29)
		tmp = fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), -0.041666666666666664)), 0.5);
	else
		tmp = Float64(fma(Float64(1.0 / Float64(x * x)), x, Float64(-1.0 / x)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.5e+29], N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * x + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5e29

   1. Initial program 38.8%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. metadata-eval N/A

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

      4. 1-sub-cos N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. hang-0p-tan N/A

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

      11. tan-lowering-tan.f64 N/A

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

      12. /-lowering-/.f64 73.7

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

   4. Applied egg-rr 73.7%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 73.7

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   8. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 63.5

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

   9. Simplified 63.5%

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

   if 1.5e29 < x

   1. Initial program 97.8%

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

   4. Applied egg-rr 97.7%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*l/ N/A

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

      4. metadata-eval N/A

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

      5. /-rgt-identity N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. frac-add N/A

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

      9. *-lft-identity N/A

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

      10. /-lowering-/.f64 N/A

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 55.9

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

   9. Simplified 55.9%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x \cdot x}, x, \frac{-1}{x}\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.5 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.001388888888888889, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 6.5e+38)
   (fma x (* x (fma x (* x 0.001388888888888889) -0.041666666666666664)) 0.5)
   0.0))

C

double code(double x) {
	double tmp;
	if (x <= 6.5e+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 <= 6.5e+38)
		tmp = fma(x, Float64(x * fma(x, Float64(x * 0.001388888888888889), -0.041666666666666664)), 0.5);
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 6.5e+38], N[(x * N[(x * N[(x * N[(x * 0.001388888888888889), $MachinePrecision] + -0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 6.5e38

   1. Initial program 39.4%

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

      1. flip-- N/A

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

      2. associate-/l/ N/A

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

      3. metadata-eval N/A

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

      4. 1-sub-cos N/A

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

      5. times-frac N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. hang-0p-tan N/A

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

      11. tan-lowering-tan.f64 N/A

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

      12. /-lowering-/.f64 74.0

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

   4. Applied egg-rr 74.0%

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

      1. associate-*l/ N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. tan-lowering-tan.f64 N/A

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

      7. div-inv N/A

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

      8. *-lowering-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sin-lowering-sin.f64 74.0

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

   6. Applied egg-rr 74.0%

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

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   8. 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. unpow2 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 63.0

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

   9. Simplified 63.0%

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

   if 6.5e38 < x

   1. Initial program 97.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. Simplified 57.5%

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

      1. metadata-eval N/A

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

      2. div0 57.5

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

   7. Applied egg-rr 57.5%

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

Alternative 6: 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 42.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 60.6%

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

   if 8.49999999999999992e76 < x

   1. Initial program 97.4%

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

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

      1. metadata-eval N/A

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

      2. div0 71.2

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

   7. Applied egg-rr 71.2%

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

Alternative 7: 28.0% 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 50.5%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. Simplified 26.5%

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

   1. metadata-eval N/A

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

   2. div0 27.1

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

7. Applied egg-rr 27.1%

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

Reproduce

herbie shell --seed 2024194 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))