cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.8%

Time: 11.7s

Alternatives: 6

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: 51.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-5}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x\_m \cdot \left({x\_m}^{-2} \cdot \tan \left(\frac{x\_m}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5e-5)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (* (sin x_m) (* (pow x_m -2.0) (tan (/ x_m 2.0))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5e-5) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = sin(x_m) * (pow(x_m, -2.0) * tan((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 <= 5d-5) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = sin(x_m) * ((x_m ** (-2.0d0)) * tan((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 <= 5e-5) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = Math.sin(x_m) * (Math.pow(x_m, -2.0) * Math.tan((x_m / 2.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 5e-5:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = math.sin(x_m) * (math.pow(x_m, -2.0) * math.tan((x_m / 2.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5e-5)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(sin(x_m) * Float64((x_m ^ -2.0) * tan(Float64(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 <= 5e-5)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = sin(x_m) * ((x_m ^ -2.0) * tan((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, 5e-5], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[x$95$m], $MachinePrecision] * N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000024e-5

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 65.0%

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

   if 5.00000000000000024e-5 < x

   1. Initial program 97.0%

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

      1. pow2 97.0%

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

      2. add-cube-cbrt 96.0%

         \[\leadsto \frac{1 - \cos x}{{\color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)}}^{2}} \]

      3. pow3 95.9%

         \[\leadsto \frac{1 - \cos x}{{\color{blue}{\left({\left(\sqrt[3]{x}\right)}^{3}\right)}}^{2}} \]

      4. pow-pow 96.0%

         \[\leadsto \frac{1 - \cos x}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{\left(3 \cdot 2\right)}}} \]

      5. metadata-eval 96.0%

         \[\leadsto \frac{1 - \cos x}{{\left(\sqrt[3]{x}\right)}^{\color{blue}{6}}} \]

   4. Applied egg-rr 96.0%

      \[\leadsto \frac{1 - \cos x}{\color{blue}{{\left(\sqrt[3]{x}\right)}^{6}}} \]
   5. Step-by-step derivation

      1. flip-- 95.9%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}{{\left(\sqrt[3]{x}\right)}^{6}} \]

      2. div-inv 95.8%

         \[\leadsto \frac{\color{blue}{\left(1 \cdot 1 - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}}{{\left(\sqrt[3]{x}\right)}^{6}} \]

      3. metadata-eval 95.8%

         \[\leadsto \frac{\left(\color{blue}{1} - \cos x \cdot \cos x\right) \cdot \frac{1}{1 + \cos x}}{{\left(\sqrt[3]{x}\right)}^{6}} \]

      4. pow2 95.8%

         \[\leadsto \frac{\left(1 - \color{blue}{{\cos x}^{2}}\right) \cdot \frac{1}{1 + \cos x}}{{\left(\sqrt[3]{x}\right)}^{6}} \]

   6. Applied egg-rr 95.8%

      \[\leadsto \frac{\color{blue}{\left(1 - {\cos x}^{2}\right) \cdot \frac{1}{1 + \cos x}}}{{\left(\sqrt[3]{x}\right)}^{6}} \]
   7. Step-by-step derivation

      1. unpow2 95.8%

         \[\leadsto \frac{\left(1 - \color{blue}{\cos x \cdot \cos x}\right) \cdot \frac{1}{1 + \cos x}}{{\left(\sqrt[3]{x}\right)}^{6}} \]

      2. 1-sub-cos 96.2%

         \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{{\left(\sqrt[3]{x}\right)}^{6}} \]

   8. Applied egg-rr 96.2%

      \[\leadsto \frac{\color{blue}{\left(\sin x \cdot \sin x\right)} \cdot \frac{1}{1 + \cos x}}{{\left(\sqrt[3]{x}\right)}^{6}} \]
   9. Step-by-step derivation

      1. associate-/l* 96.2%

         \[\leadsto \color{blue}{\left(\sin x \cdot \sin x\right) \cdot \frac{\frac{1}{1 + \cos x}}{{\left(\sqrt[3]{x}\right)}^{6}}} \]

      2. associate-*l* 96.2%

         \[\leadsto \color{blue}{\sin x \cdot \left(\sin x \cdot \frac{\frac{1}{1 + \cos x}}{{\left(\sqrt[3]{x}\right)}^{6}}\right)} \]

      3. pow1/3 92.1%

         \[\leadsto \sin x \cdot \left(\sin x \cdot \frac{\frac{1}{1 + \cos x}}{{\color{blue}{\left({x}^{0.3333333333333333}\right)}}^{6}}\right) \]

      4. pow-pow 97.1%

         \[\leadsto \sin x \cdot \left(\sin x \cdot \frac{\frac{1}{1 + \cos x}}{\color{blue}{{x}^{\left(0.3333333333333333 \cdot 6\right)}}}\right) \]

      5. metadata-eval 97.1%

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

      6. unpow2 97.1%

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

      7. sqr-neg 97.1%

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

      8. div-inv 97.1%

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

      9. sqr-neg 97.1%

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

      10. unpow2 97.1%

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

      11. pow-flip 99.2%

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

      12. metadata-eval 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. associate-*r* 99.3%

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

       2. *-commutative 99.3%

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

       3. associate-*l/ 99.4%

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

       4. *-lft-identity 99.4%

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

       5. *-commutative 99.4%

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

       6. hang-0p-tan 99.7%

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

   12. Simplified 99.7%

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

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0055:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0055)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (* (pow x_m -2.0) (- 1.0 (cos x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0055) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = pow(x_m, -2.0) * (1.0 - cos(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 <= 0.0055d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = (x_m ** (-2.0d0)) * (1.0d0 - cos(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 <= 0.0055) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = Math.pow(x_m, -2.0) * (1.0 - Math.cos(x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0055:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = math.pow(x_m, -2.0) * (1.0 - math.cos(x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0055)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64((x_m ^ -2.0) * Float64(1.0 - cos(x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0055)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = (x_m ^ -2.0) * (1.0 - cos(x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0055], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[x$95$m, -2.0], $MachinePrecision] * N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054999999999999997

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 65.0%

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

   if 0.0054999999999999997 < x

   1. Initial program 97.0%

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

      1. clear-num 97.0%

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

      2. associate-/r/ 97.0%

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

      3. pow2 97.0%

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

      4. pow-flip 99.1%

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

      5. metadata-eval 99.1%

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

   4. Applied egg-rr 99.1%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0055:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0055)
   (+ 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 <= 0.0055) {
		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 <= 0.0055d0) 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 <= 0.0055) {
		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 <= 0.0055:
		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 <= 0.0055)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / 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 <= 0.0055)
		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, 0.0055], 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 < 0.0054999999999999997

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 65.0%

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

   if 0.0054999999999999997 < x

   1. Initial program 97.0%

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

      1. clear-num 97.0%

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

      2. inv-pow 97.0%

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

      3. pow2 97.0%

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

   4. Applied egg-rr 97.0%

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

      1. unpow-1 97.0%

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

      2. clear-num 97.0%

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

      3. remove-double-div 97.0%

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

      4. unpow2 97.0%

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

      5. associate-/r* 99.0%

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

   6. Applied egg-rr 99.0%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0055:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0055)
   (+ 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 <= 0.0055) {
		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 <= 0.0055d0) 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 <= 0.0055) {
		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 <= 0.0055:
		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 <= 0.0055)
		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 <= 0.0055)
		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, 0.0055], 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 < 0.0054999999999999997

   1. Initial program 36.7%

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

   3. Taylor expanded in x around 0 65.0%

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

   if 0.0054999999999999997 < x

   1. Initial program 97.0%

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

Alternative 5: 75.6% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.04 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.04e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.04e+77) {
		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 <= 1.04d+77) 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 <= 1.04e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.04e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.04e+77)
		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 <= 1.04e+77)
		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, 1.04e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.04e77

   1. Initial program 43.1%

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

   3. Taylor expanded in x around 0 59.6%

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

   if 1.04e77 < x

   1. Initial program 96.3%

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

      1. clear-num 96.3%

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

      2. inv-pow 96.3%

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

      3. pow2 96.3%

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

   4. Applied egg-rr 96.3%

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

      1. unpow-1 96.3%

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

      2. clear-num 96.3%

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

      3. remove-double-div 96.3%

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

      4. unpow2 96.3%

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

      5. associate-/r* 99.4%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in x around 0 63.3%

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

   8. Taylor expanded in x around 0 63.3%

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

Alternative 6: 27.8% 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 50.6%

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

   1. clear-num 50.6%

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

   2. inv-pow 50.6%

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

   3. pow2 50.6%

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

4. Applied egg-rr 50.6%

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

   1. unpow-1 50.6%

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

   2. clear-num 50.6%

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

   3. remove-double-div 50.6%

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

   4. unpow2 50.6%

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

   5. associate-/r* 51.9%

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

6. Applied egg-rr 51.9%

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

7. Taylor expanded in x around 0 25.1%

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

8. Taylor expanded in x around 0 25.1%

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

Reproduce

herbie shell --seed 2024186 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))