cos2 (problem 3.4.1)

Percentage Accurate: 50.8% → 99.5%

Time: 10.1s

Alternatives: 7

Speedup: 17.8×

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: 50.8% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x\_m \cdot \frac{-1}{1 - \cos x\_m}\right)}^{-1}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.004)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (pow (* x_m (/ -1.0 (- 1.0 (cos x_m)))) -1.0) (- x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = pow((x_m * (-1.0 / (1.0 - cos(x_m)))), -1.0) / -x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.004d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = ((x_m * ((-1.0d0) / (1.0d0 - cos(x_m)))) ** (-1.0d0)) / -x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = Math.pow((x_m * (-1.0 / (1.0 - Math.cos(x_m)))), -1.0) / -x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.004:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = math.pow((x_m * (-1.0 / (1.0 - math.cos(x_m)))), -1.0) / -x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.004)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64((Float64(x_m * Float64(-1.0 / Float64(1.0 - cos(x_m)))) ^ -1.0) / Float64(-x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.004)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = ((x_m * (-1.0 / (1.0 - cos(x_m)))) ^ -1.0) / -x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.004], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(x$95$m * N[(-1.0 / N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0040000000000000001

   1. Initial program 35.6%

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

   3. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0040000000000000001 < x

   1. Initial program 98.9%

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

      1. frac-2neg 98.9%

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

      2. div-inv 98.9%

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

      3. pow2 98.9%

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

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. clear-num 99.1%

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

      2. associate-/r/ 99.2%

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

      3. frac-2neg 99.2%

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

      4. metadata-eval 99.2%

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

      5. distribute-neg-in 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

      8. associate-+r+ 99.1%

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

      9. sub-neg 99.1%

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

      10. associate-+r- 99.2%

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

      11. metadata-eval 99.2%

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

   10. Applied egg-rr 99.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(x \cdot \frac{-1}{1 - \cos x}\right)}^{-1}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\_m\right) \cdot {x\_m}^{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.004)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (* (- 1.0 (cos x_m)) (pow x_m -2.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = (1.0 - cos(x_m)) * pow(x_m, -2.0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.004d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = (1.0d0 - cos(x_m)) * (x_m ** (-2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = (1.0 - Math.cos(x_m)) * Math.pow(x_m, -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.004:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = (1.0 - math.cos(x_m)) * math.pow(x_m, -2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.004)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) * (x_m ^ -2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.004)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = (1.0 - cos(x_m)) * (x_m ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.004], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[x$95$m, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0040000000000000001

   1. Initial program 35.6%

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

   3. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0040000000000000001 < x

   1. Initial program 98.9%

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

      1. clear-num 98.8%

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

      2. associate-/r/ 98.9%

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

      3. pow2 98.9%

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

      4. pow-flip 99.2%

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

      5. metadata-eval 99.2%

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

   4. Applied egg-rr 99.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{x\_m}{-1 + \cos x\_m}}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.004)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (/ -1.0 (/ x_m (+ -1.0 (cos x_m)))) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = (-1.0 / (x_m / (-1.0 + cos(x_m)))) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.004d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = ((-1.0d0) / (x_m / ((-1.0d0) + cos(x_m)))) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = (-1.0 / (x_m / (-1.0 + Math.cos(x_m)))) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.004:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = (-1.0 / (x_m / (-1.0 + math.cos(x_m)))) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.004)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(-1.0 / Float64(x_m / Float64(-1.0 + cos(x_m)))) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.004)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = (-1.0 / (x_m / (-1.0 + cos(x_m)))) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.004], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(x$95$m / N[(-1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0040000000000000001

   1. Initial program 35.6%

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

   3. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0040000000000000001 < x

   1. Initial program 98.9%

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

      1. frac-2neg 98.9%

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

      2. div-inv 98.9%

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

      3. pow2 98.9%

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

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   8. Applied egg-rr 99.2%

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

      1. unpow-1 99.2%

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

   10. Applied egg-rr 99.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{x}{-1 + \cos x}}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1 + \cos x\_m}{x\_m}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.004)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (/ (+ -1.0 (cos x_m)) x_m) (- x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = ((-1.0 + cos(x_m)) / x_m) / -x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.004d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = (((-1.0d0) + cos(x_m)) / x_m) / -x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = ((-1.0 + Math.cos(x_m)) / x_m) / -x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.004:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = ((-1.0 + math.cos(x_m)) / x_m) / -x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.004)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(-1.0 + cos(x_m)) / x_m) / Float64(-x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.004)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = ((-1.0 + cos(x_m)) / x_m) / -x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.004], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 + N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0040000000000000001

   1. Initial program 35.6%

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

   3. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0040000000000000001 < x

   1. Initial program 98.9%

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

      1. frac-2neg 98.9%

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

      2. div-inv 98.9%

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

      3. pow2 98.9%

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

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 99.2%

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

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.004:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.004)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.004d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = (1.0d0 - cos(x_m)) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.004) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = (1.0 - Math.cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.004:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = (1.0 - math.cos(x_m)) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.004)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.004)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.004], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0040000000000000001

   1. Initial program 35.6%

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

   3. Taylor expanded in x around 0 66.2%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0040000000000000001 < x

   1. Initial program 98.9%

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

Alternative 6: 75.7% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 4.4 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 4.4e+76) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 4.4e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 4.4d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 4.4e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 4.4e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 4.4e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 4.4e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 4.4e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.4000000000000001e76

   1. Initial program 41.0%

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

   3. Taylor expanded in x around 0 61.7%

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

   if 4.4000000000000001e76 < x

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0 73.2%

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

   4. Taylor expanded in x around 0 73.2%

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

Alternative 7: 27.4% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 51.4%

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

3. Taylor expanded in x around 0 29.1%

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

4. Taylor expanded in x around 0 29.8%

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

Reproduce

herbie shell --seed 2024177 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))