cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.8%

Time: 6.4s

Alternatives: 5

Speedup: 120.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift--.f64 N/A

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

   4. flip-- N/A

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

   5. distribute-neg-frac N/A

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

   6. associate-/l/ N/A

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

   7. lower-/.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. 1-sub-cos N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. lift-*.f64 N/A

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

   20. distribute-lft-neg-in N/A

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

   21. lower-*.f64 N/A

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

4. Applied rewrites 75.4%

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

5. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*l/ N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f64 N/A

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

   7. hang-0p-tan N/A

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

   8. *-rgt-identity N/A

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

   9. associate-/l* N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 N/A

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

   16. lower-sin.f64 76.0

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

7. Applied rewrites 76.0%

   \[\leadsto \color{blue}{\frac{\tan \left(0.5 \cdot x\right)}{x \cdot x} \cdot \sin x} \]
8. Step-by-step derivation

9. Applied rewrites 99.8%

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

Alternative 2: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0055:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0055)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0055) {
		tmp = fma((x * x), -0.041666666666666664, 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0055)
		tmp = fma(Float64(x * x), -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, 0.0055], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 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 < 0.0054999999999999997

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{-1}{24}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if 0.0054999999999999997 < x

   1. Initial program 99.5%

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

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

Alternative 3: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0055:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0055)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0055) {
		tmp = fma((x * x), -0.041666666666666664, 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0055)
		tmp = fma(Float64(x * x), -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, 0.0055], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 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 < 0.0054999999999999997

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \frac{-1}{24}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.5

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

   5. Applied rewrites 66.5%

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

   if 0.0054999999999999997 < x

   1. Initial program 99.5%

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

Alternative 4: 63.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.46 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.46e+77) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.46e+77) {
		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 <= 1.46d+77) 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 <= 1.46e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.46e+77:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.46e+77)
		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 <= 1.46e+77)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.46e+77], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.46e77

   1. Initial program 40.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.0%

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

   if 1.46e77 < x

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.3%

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

Alternative 5: 51.4% 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 51.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.3%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))