cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.8%

Time: 9.2s

Alternatives: 10

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.6%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. 1-sub-cos N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. hang-0p-tan N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-/.f64 75.8

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

4. Applied rewrites 75.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. clear-num N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 99.7

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

   11. lift-/.f64 N/A

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

   12. clear-num N/A

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

   13. associate-/r/ N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 99.7

       \[\leadsto \frac{1}{\frac{x}{\frac{\sin x}{x} \cdot \tan \color{blue}{\left(0.5 \cdot x\right)}}} \]

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{\sin x}{x} \cdot \tan \left(0.5 \cdot x\right)}}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 99.7

      \[\leadsto \frac{1}{\frac{x}{\sin x \cdot \color{blue}{\frac{\tan \left(0.5 \cdot x\right)}{x}}}} \]

8. Applied rewrites 99.7%

   \[\leadsto \frac{1}{\frac{x}{\color{blue}{\sin x \cdot \frac{\tan \left(0.5 \cdot x\right)}{x}}}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. frac-2neg N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-neg.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-155}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan \left(0.5 \cdot x\right) \cdot \sin x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 5e-155) 0.5 (/ (* (tan (* 0.5 x)) (sin x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 5e-155) {
		tmp = 0.5;
	} else {
		tmp = (tan((0.5 * x)) * sin(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 5d-155) then
        tmp = 0.5d0
    else
        tmp = (tan((0.5d0 * x)) * sin(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 5e-155) {
		tmp = 0.5;
	} else {
		tmp = (Math.tan((0.5 * x)) * Math.sin(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 5e-155:
		tmp = 0.5
	else:
		tmp = (math.tan((0.5 * x)) * math.sin(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 5e-155)
		tmp = 0.5;
	else
		tmp = Float64(Float64(tan(Float64(0.5 * x)) * sin(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 5e-155)
		tmp = 0.5;
	else
		tmp = (tan((0.5 * x)) * sin(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 5e-155], 0.5, N[(N[(N[Tan[N[(0.5 * x), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999999e-155

   1. Initial program 40.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 61.7%

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

   if 4.9999999999999999e-155 < x

   1. Initial program 64.5%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. metadata-eval N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. 1-sub-cos N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. hang-0p-tan N/A

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

      12. lower-tan.f64 N/A

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

      13. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.5

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f64 99.5

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

   6. Applied rewrites 99.5%

      \[\leadsto \frac{\color{blue}{\tan \left(0.5 \cdot x\right) \cdot \sin x}}{x \cdot x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos x - 1}{x} \cdot \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00016) 0.5 (* (/ (- (cos x) 1.0) x) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = ((cos(x) - 1.0) / x) * (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00016d0) then
        tmp = 0.5d0
    else
        tmp = ((cos(x) - 1.0d0) / x) * ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = ((Math.cos(x) - 1.0) / x) * (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00016:
		tmp = 0.5
	else:
		tmp = ((math.cos(x) - 1.0) / x) * (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00016)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(cos(x) - 1.0) / x) * Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00016)
		tmp = 0.5;
	else
		tmp = ((cos(x) - 1.0) / x) * (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00016], 0.5, N[(N[(N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.60000000000000013e-4

   1. Initial program 33.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.6%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 99.1%

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

Alternative 4: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00016) 0.5 (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		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 <= 0.00016d0) 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 <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00016:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00016)
		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 <= 0.00016)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00016], 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 < 1.60000000000000013e-4

   1. Initial program 33.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.6%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 98.8%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00016:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00016) 0.5 (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00016) {
		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 <= 0.00016d0) 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 <= 0.00016) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00016:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00016)
		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 <= 0.00016)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00016], 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 < 1.60000000000000013e-4

   1. Initial program 33.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 68.6%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 98.8%

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

Alternative 6: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.8)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (pow (* (* 0.16666666666666666 x) x) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 4.8) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = pow(((0.16666666666666666 * x) * x), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.8)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(0.16666666666666666 * x) * x) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.8], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[Power[N[(N[(0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.79999999999999982

   1. Initial program 33.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 68.1

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

   5. Applied rewrites 68.1%

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

   if 4.79999999999999982 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 53.7

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

   7. Applied rewrites 53.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.8:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.3)
   (fma -0.041666666666666664 (* x x) 0.5)
   (pow (* (* 0.16666666666666666 x) x) -1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 3.3) {
		tmp = fma(-0.041666666666666664, (x * x), 0.5);
	} else {
		tmp = pow(((0.16666666666666666 * x) * x), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.3)
		tmp = fma(-0.041666666666666664, Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(0.16666666666666666 * x) * x) ^ -1.0;
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 3.3], N[(-0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[Power[N[(N[(0.16666666666666666 * x), $MachinePrecision] * x), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2999999999999998

   1. Initial program 33.9%

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

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

      3. unpow2 N/A

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

      4. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   if 3.2999999999999998 < x

   1. Initial program 98.8%

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

   4. Applied rewrites 98.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 53.7

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

   7. Applied rewrites 53.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 53.7%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(0.16666666666666666 \cdot x\right) \cdot x\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.6%

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

4. Applied rewrites 50.3%

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

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 75.7

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

7. Applied rewrites 75.7%

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

8. Final simplification 75.7%

   \[\leadsto {\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 2\right)\right)}^{-1} \]
9. Add Preprocessing

Alternative 9: 64.2% 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 40.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.7%

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

   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 70.3%

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

Alternative 10: 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 49.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 53.2%

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

Reproduce

herbie shell --seed 2024309 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))