cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.8%

Time: 8.7s

Alternatives: 9

Speedup: 5.2×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. hang-0p-tan N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-/.f64 80.4

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

4. Applied rewrites 80.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00018:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-1} - {x}^{-1} \cdot \cos x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00018) 0.5 (/ (- (pow x -1.0) (* (pow x -1.0) (cos x))) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00018) {
		tmp = 0.5;
	} else {
		tmp = (pow(x, -1.0) - (pow(x, -1.0) * cos(x))) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00018d0) then
        tmp = 0.5d0
    else
        tmp = ((x ** (-1.0d0)) - ((x ** (-1.0d0)) * cos(x))) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00018) {
		tmp = 0.5;
	} else {
		tmp = (Math.pow(x, -1.0) - (Math.pow(x, -1.0) * Math.cos(x))) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00018:
		tmp = 0.5
	else:
		tmp = (math.pow(x, -1.0) - (math.pow(x, -1.0) * math.cos(x))) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00018)
		tmp = 0.5;
	else
		tmp = Float64(Float64((x ^ -1.0) - Float64((x ^ -1.0) * cos(x))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00018)
		tmp = 0.5;
	else
		tmp = ((x ^ -1.0) - ((x ^ -1.0) * cos(x))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00018], 0.5, N[(N[(N[Power[x, -1.0], $MachinePrecision] - N[(N[Power[x, -1.0], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.80000000000000011e-4

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.0%

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

   if 1.80000000000000011e-4 < x

   1. Initial program 98.7%

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

   4. Applied rewrites 98.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-*.f64 N/A

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

      6. clear-num N/A

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

      7. lift--.f64 N/A

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

      8. sub-div N/A

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

      9. frac-2neg N/A

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

      10. metadata-eval N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-lft-neg-out N/A

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

      13. lift-neg.f64 N/A

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

      14. associate-/l/ N/A

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

      15. lift-/.f64 N/A

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

      16. frac-2neg N/A

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

      17. lift-*.f64 N/A

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

      18. distribute-lft-neg-out N/A

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

      19. lift-neg.f64 N/A

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

      20. neg-mul-1 N/A

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

      21. *-commutative N/A

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

      22. frac-times N/A

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

   6. Applied rewrites 99.3%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00018:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{-1} - {x}^{-1} \cdot \cos x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00013:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-\cos x, \frac{-1}{x}, \frac{-1}{x}\right)}{-x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00013) 0.5 (/ (fma (- (cos x)) (/ -1.0 x) (/ -1.0 x)) (- x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00013) {
		tmp = 0.5;
	} else {
		tmp = fma(-cos(x), (-1.0 / x), (-1.0 / x)) / -x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00013)
		tmp = 0.5;
	else
		tmp = Float64(fma(Float64(-cos(x)), Float64(-1.0 / x), Float64(-1.0 / x)) / Float64(-x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.00013], 0.5, N[(N[((-N[Cos[x], $MachinePrecision]) * N[(-1.0 / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999989e-4

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.0%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. associate-/r* N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/r* N/A

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

      11. frac-2neg N/A

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

      12. sub-div N/A

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

      13. frac-2neg N/A

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

      14. metadata-eval N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

   4. Applied rewrites 99.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift-neg.f64 N/A

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

      9. div-inv N/A

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

      10. metadata-eval N/A

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

      11. lift-neg.f64 N/A

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

      12. frac-2neg N/A

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

      13. lift-/.f64 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 99.3

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

   6. Applied rewrites 99.3%

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

Alternative 4: 75.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00013:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00013) 0.5 (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00013) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00013d0) then
        tmp = 0.5d0
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00013) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00013:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00013)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00013)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00013], 0.5, N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999989e-4

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.0%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.7%

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

   4. Applied rewrites 99.2%

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

Alternative 5: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00013:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00013) 0.5 (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00013) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00013d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00013) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00013:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00013)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00013)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00013], 0.5, N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.29999999999999989e-4

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.0%

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

   if 1.29999999999999989e-4 < x

   1. Initial program 98.7%

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

Alternative 6: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5) 0.5 (pow (* 0.16666666666666666 (* x x)) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = pow((0.16666666666666666 * (x * x)), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.5d0) then
        tmp = 0.5d0
    else
        tmp = (0.16666666666666666d0 * (x * x)) ** (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = 0.5;
	} else {
		tmp = Math.pow((0.16666666666666666 * (x * x)), -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.5:
		tmp = 0.5
	else:
		tmp = math.pow((0.16666666666666666 * (x * x)), -1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.5)
		tmp = 0.5;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * x)) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.5)
		tmp = 0.5;
	else
		tmp = (0.16666666666666666 * (x * x)) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], 0.5, N[Power[N[(0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 40.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.0%

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

   if 3.5 < x

   1. Initial program 98.7%

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

   4. Applied rewrites 98.5%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 56.7

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

   7. Applied rewrites 56.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 56.7%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;{\left(0.16666666666666666 \cdot \left(x \cdot x\right)\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3e+76) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.3e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.3d+76) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.3e+76) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.3e+76:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.3e+76)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.3e+76)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.3e+76], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.3000000000000001e76

   1. Initial program 43.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 58.6%

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

   if 3.3000000000000001e76 < x

   1. Initial program 99.0%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 64.8%

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

Alternative 8: 78.6% accurate, 5.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

4. Applied rewrites 53.5%

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

5. Taylor expanded in x around 0

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

   1. sub-neg N/A

      \[\leadsto \frac{-1}{\color{blue}{\frac{-1}{6} \cdot {x}^{2} + \left(\mathsf{neg}\left(2\right)\right)}} \]

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval 76.3

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

7. Applied rewrites 76.3%

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

Alternative 9: 51.2% accurate, 120.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

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

Java

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

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

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

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.7%

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

Reproduce

herbie shell --seed 2024299 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))