cos2 (problem 3.4.1)

Percentage Accurate: 51.9% → 99.8%

Time: 8.7s

Alternatives: 7

Speedup: 4.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.5%

   \[\frac{1 - \cos x}{x \cdot x} \]
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 68.8

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

4. Applied rewrites 68.8%

   \[\leadsto \color{blue}{\frac{\sin x}{x \cdot x} \cdot \tan \left(\frac{x}{2}\right)} \]
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. lift-*.f64 N/A

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

   4. associate-/r* N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.9

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000125:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000125) 0.5 (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000125) {
		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 <= 0.000125d0) 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 <= 0.000125) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000125:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000125)
		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 <= 0.000125)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000125], 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 < 1.25e-4

   1. Initial program 28.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.4%

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

   if 1.25e-4 < x

   1. Initial program 98.1%

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

   4. Applied rewrites 99.5%

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

Alternative 3: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000125:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000125) 0.5 (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000125) {
		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 <= 0.000125d0) 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 <= 0.000125) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000125:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000125)
		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 <= 0.000125)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000125], 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 < 1.25e-4

   1. Initial program 28.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 73.4%

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

   if 1.25e-4 < x

   1. Initial program 98.1%

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

Alternative 4: 62.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+78}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x \cdot x}}{x} - \frac{1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5.8e+78) 0.5 (- (/ (/ x (* x x)) x) (/ 1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 5.8e+78) {
		tmp = 0.5;
	} else {
		tmp = ((x / (x * x)) / x) - (1.0 / (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5.8d+78) then
        tmp = 0.5d0
    else
        tmp = ((x / (x * x)) / x) - (1.0d0 / (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5.8e+78) {
		tmp = 0.5;
	} else {
		tmp = ((x / (x * x)) / x) - (1.0 / (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5.8e+78:
		tmp = 0.5
	else:
		tmp = ((x / (x * x)) / x) - (1.0 / (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5.8e+78)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(x / Float64(x * x)) / x) - Float64(1.0 / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5.8e+78)
		tmp = 0.5;
	else
		tmp = ((x / (x * x)) / x) - (1.0 / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5.8e+78], 0.5, N[(N[(N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] - N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.80000000000000034e78

   1. Initial program 31.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.2%

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

   if 5.80000000000000034e78 < x

   1. Initial program 98.1%

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

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

      7. associate-/r* N/A

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

      8. frac-sub N/A

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

      9. neg-mul-1 N/A

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

      10. div-sub N/A

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

      11. lower--.f64 N/A

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

   4. Applied rewrites 6.0%

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

   5. Taylor expanded in x around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-neg.f64 N/A

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

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

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

      9. lift-*.f64 N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 75.8

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

   9. Applied rewrites 75.8%

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

   10. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f64 76.0

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

   12. Applied rewrites 76.0%

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

Alternative 5: 62.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{x} \cdot x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 7.6e+28)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 (* (/ 1.0 x) x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 7.6e+28) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - ((1.0 / x) * x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.6e+28)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - Float64(Float64(1.0 / x) * x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 7.6e+28], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - N[(N[(1.0 / x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.5999999999999998e28

   1. Initial program 29.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   if 7.5999999999999998e28 < x

   1. Initial program 98.2%

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

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

      5. div-sub N/A

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

      6. sub-div N/A

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

      7. frac-sub N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 98.2

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 67.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{1}{x} \cdot x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+77) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+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 <= 1.4d+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 <= 1.4e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4e+77:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+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 <= 1.4e+77)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+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 < 1.4e77

   1. Initial program 31.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.2%

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

   if 1.4e77 < x

   1. Initial program 98.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.8%

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

Alternative 7: 50.7% 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 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 58.5%

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

Reproduce

herbie shell --seed 2024270 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))