cos2 (problem 3.4.1)

Percentage Accurate: 50.1% → 99.8%

Time: 14.3s

Alternatives: 10

Speedup: 107.0×

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: 50.1% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0002:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x\_m \cdot \left(\tan \left(\frac{x\_m}{2}\right) \cdot {x\_m}^{-2}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0002)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (* (sin x_m) (* (tan (/ x_m 2.0)) (pow x_m -2.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0002) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = sin(x_m) * (tan((x_m / 2.0)) * 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 <= 0.0002d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = sin(x_m) * (tan((x_m / 2.0d0)) * (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 <= 0.0002) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = Math.sin(x_m) * (Math.tan((x_m / 2.0)) * Math.pow(x_m, -2.0));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0002:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = math.sin(x_m) * (math.tan((x_m / 2.0)) * math.pow(x_m, -2.0))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0002)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(sin(x_m) * Float64(tan(Float64(x_m / 2.0)) * (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 <= 0.0002)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = sin(x_m) * (tan((x_m / 2.0)) * (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, 0.0002], 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[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision] * N[Power[x$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 34.4%

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

   3. Taylor expanded in x around 0 67.0%

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

   if 2.0000000000000001e-4 < x

   1. Initial program 97.0%

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

      1. flip-- 96.5%

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

      2. div-inv 96.5%

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

      3. metadata-eval 96.5%

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

      4. pow2 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. associate-*r/ 96.5%

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

      2. *-rgt-identity 96.5%

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

   6. Simplified 96.5%

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

      1. unpow2 96.5%

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

      2. 1-sub-cos 97.2%

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

   8. Applied egg-rr 97.2%

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

      1. div-inv 97.2%

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

      2. pow2 97.2%

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

      3. pow-flip 99.0%

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

      4. metadata-eval 99.0%

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

      5. div-inv 99.0%

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

      6. associate-*l* 98.9%

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

      7. pow2 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. associate-*r* 99.0%

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

       2. unpow2 99.0%

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

       3. associate-*r* 98.9%

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

       4. associate-*l* 98.8%

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

       5. associate-*r/ 98.8%

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

       6. *-rgt-identity 98.8%

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

       7. hang-0p-tan 99.5%

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

   12. Simplified 99.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(\tan \left(\frac{x}{2}\right) \cdot {x}^{-2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.002:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x\_m}^{2} + 0.001388888888888889 \cdot {x\_m}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x\_m \cdot \tan \left(\frac{x\_m}{2}\right)}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.002)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x_m 2.0))
     (* 0.001388888888888889 (pow x_m 4.0))))
   (/ (/ (* (sin x_m) (tan (/ x_m 2.0))) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.002) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x_m, 2.0)) + (0.001388888888888889 * pow(x_m, 4.0)));
	} else {
		tmp = ((sin(x_m) * tan((x_m / 2.0))) / 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 <= 0.002d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x_m ** 2.0d0)) + (0.001388888888888889d0 * (x_m ** 4.0d0)))
    else
        tmp = ((sin(x_m) * tan((x_m / 2.0d0))) / 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 <= 0.002) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x_m, 2.0)) + (0.001388888888888889 * Math.pow(x_m, 4.0)));
	} else {
		tmp = ((Math.sin(x_m) * Math.tan((x_m / 2.0))) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.002:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x_m, 2.0)) + (0.001388888888888889 * math.pow(x_m, 4.0)))
	else:
		tmp = ((math.sin(x_m) * math.tan((x_m / 2.0))) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.002)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x_m ^ 2.0)) + Float64(0.001388888888888889 * (x_m ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(sin(x_m) * tan(Float64(x_m / 2.0))) / 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 <= 0.002)
		tmp = 0.5 + ((-0.041666666666666664 * (x_m ^ 2.0)) + (0.001388888888888889 * (x_m ^ 4.0)));
	else
		tmp = ((sin(x_m) * tan((x_m / 2.0))) / 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, 0.002], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-3

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.1%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)} \]

   if 2e-3 < x

   1. Initial program 97.4%

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

      1. flip-- 96.9%

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

      2. div-inv 96.9%

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

      3. metadata-eval 96.9%

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

      4. pow2 96.9%

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

   4. Applied egg-rr 96.9%

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

      1. associate-*r/ 96.9%

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

      2. *-rgt-identity 96.9%

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

   6. Simplified 96.9%

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

      1. unpow2 96.9%

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

      2. 1-sub-cos 97.2%

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

   8. Applied egg-rr 97.2%

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

      1. div-inv 97.1%

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

      2. times-frac 98.9%

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

      3. pow2 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. associate-*l/ 98.9%

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

       2. associate-*r/ 98.9%

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

       3. unpow2 98.9%

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

       4. associate-*r* 98.8%

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

       5. associate-*r/ 98.8%

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

       6. *-rgt-identity 98.8%

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

       7. hang-0p-tan 99.4%

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

   12. Simplified 99.4%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x\_m \cdot \tan \left(\frac{x\_m}{2}\right)}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5e-6)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (/ (* (sin x_m) (tan (/ x_m 2.0))) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5e-6) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = ((sin(x_m) * tan((x_m / 2.0))) / 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 <= 5d-6) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = ((sin(x_m) * tan((x_m / 2.0d0))) / 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 <= 5e-6) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = ((Math.sin(x_m) * Math.tan((x_m / 2.0))) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 5e-6:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = ((math.sin(x_m) * math.tan((x_m / 2.0))) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5e-6)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(sin(x_m) * tan(Float64(x_m / 2.0))) / 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 <= 5e-6)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = ((sin(x_m) * tan((x_m / 2.0))) / 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, 5e-6], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] * N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000041e-6

   1. Initial program 34.4%

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

   3. Taylor expanded in x around 0 67.0%

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

   if 5.00000000000000041e-6 < x

   1. Initial program 97.0%

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

      1. flip-- 96.5%

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

      2. div-inv 96.5%

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

      3. metadata-eval 96.5%

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

      4. pow2 96.5%

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

   4. Applied egg-rr 96.5%

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

      1. associate-*r/ 96.5%

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

      2. *-rgt-identity 96.5%

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

   6. Simplified 96.5%

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

      1. unpow2 96.5%

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

      2. 1-sub-cos 97.2%

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

   8. Applied egg-rr 97.2%

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

      1. div-inv 97.1%

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

      2. times-frac 98.9%

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

      3. pow2 98.9%

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

   10. Applied egg-rr 98.9%

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

       1. associate-*l/ 98.9%

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

       2. associate-*r/ 98.9%

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

       3. unpow2 98.9%

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

       4. associate-*r* 98.8%

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

       5. associate-*r/ 98.8%

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

       6. *-rgt-identity 98.8%

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

       7. hang-0p-tan 99.4%

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

   12. Simplified 99.4%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \tan \left(\frac{x}{2}\right)}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x\_m}^{-2} \cdot \left(1 - \cos x\_m\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (+ 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 <= 0.0056) {
		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 <= 0.0056d0) 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 <= 0.0056) {
		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 <= 0.0056:
		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 <= 0.0056)
		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 <= 0.0056)
		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, 0.0056], 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 < 0.00559999999999999994

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.1%

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

   if 0.00559999999999999994 < x

   1. Initial program 97.4%

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

      1. clear-num 97.3%

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

      2. associate-/r/ 97.4%

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

      3. pow2 97.4%

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

      4. pow-flip 99.3%

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

      5. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;{x}^{-2} \cdot \left(1 - \cos x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (+ 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 <= 0.0056) {
		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 <= 0.0056d0) 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 <= 0.0056) {
		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 <= 0.0056:
		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 <= 0.0056)
		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 <= 0.0056)
		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, 0.0056], 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 < 0.00559999999999999994

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.1%

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

   if 0.00559999999999999994 < x

   1. Initial program 97.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x\_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0056)
   (+ 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 <= 0.0056) {
		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 <= 0.0056d0) 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 <= 0.0056) {
		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 <= 0.0056:
		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 <= 0.0056)
		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 <= 0.0056)
		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, 0.0056], 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 < 0.00559999999999999994

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.1%

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

   if 0.00559999999999999994 < x

   1. Initial program 97.4%

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

      1. associate-/r* 99.3%

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

      2. div-inv 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0056:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.5% accurate, 8.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{x\_m \cdot \left(x\_m \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x\_m}\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (* x_m (+ (* x_m 0.16666666666666666) (* 2.0 (/ 1.0 x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 / (x_m * ((x_m * 0.16666666666666666) + (2.0 * (1.0 / x_m))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0 / (x_m * ((x_m * 0.16666666666666666d0) + (2.0d0 * (1.0d0 / x_m))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0 / (x_m * ((x_m * 0.16666666666666666) + (2.0 * (1.0 / x_m))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0 / (x_m * ((x_m * 0.16666666666666666) + (2.0 * (1.0 / x_m))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / Float64(x_m * Float64(Float64(x_m * 0.16666666666666666) + Float64(2.0 * Float64(1.0 / x_m)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0 / (x_m * ((x_m * 0.16666666666666666) + (2.0 * (1.0 / x_m))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(x$95$m * N[(N[(x$95$m * 0.16666666666666666), $MachinePrecision] + N[(2.0 * N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.8%

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

   1. associate-/r* 51.4%

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

   2. div-inv 51.4%

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

4. Applied egg-rr 51.4%

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

   1. clear-num 51.4%

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

   2. frac-times 50.6%

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

   3. metadata-eval 50.6%

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

6. Applied egg-rr 50.6%

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

7. Taylor expanded in x around 0 77.1%

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

8. Final simplification 77.1%

   \[\leadsto \frac{1}{x \cdot \left(x \cdot 0.16666666666666666 + 2 \cdot \frac{1}{x}\right)} \]
9. Add Preprocessing

Alternative 8: 76.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 3.7 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 3.7e+76) 0.5 (/ 0.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 3.7e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0 / (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 <= 3.7d+76) then
        tmp = 0.5d0
    else
        tmp = 0.0d0 / (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 <= 3.7e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0 / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 3.7e+76:
		tmp = 0.5
	else:
		tmp = 0.0 / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 3.7e+76)
		tmp = 0.5;
	else
		tmp = Float64(0.0 / 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 <= 3.7e+76)
		tmp = 0.5;
	else
		tmp = 0.0 / (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, 3.7e+76], 0.5, N[(0.0 / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.6999999999999999e76

   1. Initial program 40.1%

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

   3. Taylor expanded in x around 0 62.3%

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

   if 3.6999999999999999e76 < x

   1. Initial program 96.8%

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

      1. add-log-exp 96.8%

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

   4. Applied egg-rr 96.8%

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

      1. exp-diff 96.8%

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

      2. exp-1-e 96.8%

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

   6. Applied egg-rr 96.8%

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

   7. Taylor expanded in x around 0 54.7%

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

      1. exp-1-e 54.7%

         \[\leadsto \frac{\log \left(\frac{e}{\color{blue}{e}}\right)}{x \cdot x} \]

      2. *-inverses 54.7%

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

      3. metadata-eval 54.7%

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

   9. Simplified 54.7%

      \[\leadsto \frac{\color{blue}{0}}{x \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 2.0) 0.5 (/ (/ 2.0 x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / 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 <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = (2.0d0 / 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 <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = (2.0 / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.0:
		tmp = 0.5
	else:
		tmp = (2.0 / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(Float64(2.0 / 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 <= 2.0)
		tmp = 0.5;
	else
		tmp = (2.0 / 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, 2.0], 0.5, N[(N[(2.0 / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.5%

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

   if 2 < x

   1. Initial program 97.4%

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

      1. div-sub 97.3%

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

      2. pow2 97.3%

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

      3. pow-flip 97.5%

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

      4. metadata-eval 97.5%

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

      5. div-inv 97.5%

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

      6. pow2 97.5%

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

      7. pow-flip 99.3%

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

      8. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{{x}^{-2} - \cos x \cdot {x}^{-2}} \]
   5. Step-by-step derivation

      1. *-un-lft-identity 99.3%

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

      2. distribute-rgt-out-- 99.3%

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

      3. add-exp-log 99.2%

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

      4. sub-neg 99.2%

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

      5. log1p-udef 99.2%

         \[\leadsto {x}^{-2} \cdot e^{\color{blue}{\mathsf{log1p}\left(-\cos x\right)}} \]

      6. *-un-lft-identity 99.2%

         \[\leadsto {x}^{-2} \cdot e^{\color{blue}{1 \cdot \mathsf{log1p}\left(-\cos x\right)}} \]

      7. metadata-eval 99.2%

         \[\leadsto {x}^{-2} \cdot e^{\color{blue}{\left(0.3333333333333333 \cdot 3\right)} \cdot \mathsf{log1p}\left(-\cos x\right)} \]

      8. associate-*r* 99.3%

         \[\leadsto {x}^{-2} \cdot e^{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \mathsf{log1p}\left(-\cos x\right)\right)}} \]

      9. *-commutative 99.3%

         \[\leadsto {x}^{-2} \cdot e^{\color{blue}{\left(3 \cdot \mathsf{log1p}\left(-\cos x\right)\right) \cdot 0.3333333333333333}} \]

      10. metadata-eval 99.3%

          \[\leadsto {x}^{\color{blue}{\left(-2\right)}} \cdot e^{\left(3 \cdot \mathsf{log1p}\left(-\cos x\right)\right) \cdot 0.3333333333333333} \]

      11. pow-flip 97.4%

          \[\leadsto \color{blue}{\frac{1}{{x}^{2}}} \cdot e^{\left(3 \cdot \mathsf{log1p}\left(-\cos x\right)\right) \cdot 0.3333333333333333} \]

      12. pow2 97.4%

          \[\leadsto \frac{1}{\color{blue}{x \cdot x}} \cdot e^{\left(3 \cdot \mathsf{log1p}\left(-\cos x\right)\right) \cdot 0.3333333333333333} \]

      13. associate-/r/ 97.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x}{e^{\left(3 \cdot \mathsf{log1p}\left(-\cos x\right)\right) \cdot 0.3333333333333333}}}} \]

      14. associate-/l* 97.3%

          \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{e^{\left(3 \cdot \mathsf{log1p}\left(-\cos x\right)\right) \cdot 0.3333333333333333}}{x}}}} \]

      15. clear-num 99.3%

          \[\leadsto \color{blue}{\frac{\frac{e^{\left(3 \cdot \mathsf{log1p}\left(-\cos x\right)\right) \cdot 0.3333333333333333}}{x}}{x}} \]

   6. Applied egg-rr 47.6%

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

   7. Taylor expanded in x around 0 49.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 52.3% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.5d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.5;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 49.8%

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

3. Taylor expanded in x around 0 52.2%

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

4. Final simplification 52.2%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))