cos2 (problem 3.4.1)

Percentage Accurate: 51.0% → 99.6%

Time: 10.9s

Alternatives: 8

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: 51.0% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.03:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x_m}^{2} + 0.001388888888888889 \cdot {x_m}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x_m}{x_m}}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.03)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x_m 2.0))
     (* 0.001388888888888889 (pow x_m 4.0))))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.03) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x_m, 2.0)) + (0.001388888888888889 * pow(x_m, 4.0)));
	} 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 <= 0.03d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x_m ** 2.0d0)) + (0.001388888888888889d0 * (x_m ** 4.0d0)))
    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 <= 0.03) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x_m, 2.0)) + (0.001388888888888889 * Math.pow(x_m, 4.0)));
	} 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 <= 0.03:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x_m, 2.0)) + (0.001388888888888889 * math.pow(x_m, 4.0)))
	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 <= 0.03)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x_m ^ 2.0)) + Float64(0.001388888888888889 * (x_m ^ 4.0))));
	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 <= 0.03)
		tmp = 0.5 + ((-0.041666666666666664 * (x_m ^ 2.0)) + (0.001388888888888889 * (x_m ^ 4.0)));
	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, 0.03], 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[(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 < 0.029999999999999999

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.5%

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

   if 0.029999999999999999 < x

   1. Initial program 98.9%

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

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 99.0%

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

      5. unpow-prod-down 99.3%

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

      6. inv-pow 99.3%

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

      7. clear-num 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. frac-2neg 99.3%

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

      2. div-inv 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. metadata-eval 99.2%

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

      6. add-sqr-sqrt 57.3%

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

      7. sqrt-unprod 82.2%

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

      8. sqr-neg 82.2%

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

      9. sqrt-unprod 24.9%

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

      10. add-sqr-sqrt 55.0%

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

      11. add-sqr-sqrt 30.1%

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

      12. sqrt-unprod 72.0%

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

      13. sqr-neg 72.0%

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

      14. sqrt-unprod 41.8%

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

      15. add-sqr-sqrt 99.2%

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

   6. Applied egg-rr 99.2%

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

   8. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. unpow-1 99.3%

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

      3. /-rgt-identity 99.3%

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

      4. un-div-inv 99.5%

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

      5. associate-/r/ 99.4%

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

      6. associate-*l/ 99.5%

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

      7. neg-mul-1 99.5%

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

      8. distribute-neg-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. sub-neg 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x_m}{x_m \cdot x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0042)
   (+ 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 <= 0.0042) {
		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 <= 0.0042d0) 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 <= 0.0042) {
		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 <= 0.0042:
		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 <= 0.0042)
		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 <= 0.0042)
		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, 0.0042], 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 < 0.00419999999999999974

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.4%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x_m}{x_m}}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0042)
   (+ 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 <= 0.0042) {
		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 <= 0.0042d0) 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 <= 0.0042) {
		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 <= 0.0042:
		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 <= 0.0042)
		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 <= 0.0042)
		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, 0.0042], 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 < 0.00419999999999999974

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.4%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.9%

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

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 99.0%

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

      5. unpow-prod-down 99.3%

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

      6. inv-pow 99.3%

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

      7. clear-num 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. frac-2neg 99.3%

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

      2. div-inv 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. metadata-eval 99.2%

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

      6. add-sqr-sqrt 57.3%

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

      7. sqrt-unprod 82.2%

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

      8. sqr-neg 82.2%

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

      9. sqrt-unprod 24.9%

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

      10. add-sqr-sqrt 55.0%

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

      11. add-sqr-sqrt 30.1%

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

      12. sqrt-unprod 72.0%

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

      13. sqr-neg 72.0%

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

      14. sqrt-unprod 41.8%

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

      15. add-sqr-sqrt 99.2%

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

   6. Applied egg-rr 99.2%

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

   8. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. unpow-1 99.3%

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

      3. /-rgt-identity 99.3%

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

      4. un-div-inv 99.5%

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

      5. associate-/r/ 99.4%

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

      6. associate-*l/ 99.5%

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

      7. neg-mul-1 99.5%

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

      8. distribute-neg-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. sub-neg 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.45:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x_m}^{2}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.45)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ 2.0 (pow x_m 2.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.45) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = 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 <= 2.45d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = 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 <= 2.45) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = 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 <= 2.45:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = 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 <= 2.45)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(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 <= 2.45)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = 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, 2.45], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4500000000000002

   1. Initial program 34.6%

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

   3. Taylor expanded in x around 0 67.4%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 67.4%

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

   5. Simplified 67.4%

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

   if 2.4500000000000002 < x

   1. Initial program 98.9%

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

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 99.0%

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

      5. unpow-prod-down 99.3%

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

      6. inv-pow 99.3%

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

      7. clear-num 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. unpow-1 99.3%

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

      2. /-rgt-identity 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in x around 0 56.9%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.45:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 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 <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 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 <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 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 <= 2.0:
		tmp = 0.5
	else:
		tmp = 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 <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(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 <= 2.0)
		tmp = 0.5;
	else
		tmp = 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, 2.0], 0.5, N[(2.0 * N[Power[x$95$m, -2.0], $MachinePrecision]), $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 67.6%

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

   if 2 < x

   1. Initial program 98.9%

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

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 99.0%

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

      5. unpow-prod-down 99.3%

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

      6. inv-pow 99.3%

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

      7. clear-num 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. unpow-1 99.3%

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

      2. /-rgt-identity 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in x around 0 56.9%

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

      1. expm1-log1p-u 56.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{x}}{x}\right)\right)} \]

      2. expm1-udef 48.6%

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

      3. associate-/l/ 48.6%

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

      4. div-inv 48.6%

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

      5. pow2 48.6%

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

      6. pow-flip 48.6%

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

      7. metadata-eval 48.6%

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

   9. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {x}^{-2}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 56.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {x}^{-2}\right)\right)} \]

       2. expm1-log1p 56.9%

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

   11. Simplified 56.9%

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

4. Final simplification 64.3%

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

Alternative 6: 79.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 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 <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 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 <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 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 <= 2.0:
		tmp = 0.5
	else:
		tmp = 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 <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(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 <= 2.0)
		tmp = 0.5;
	else
		tmp = 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, 2.0], 0.5, N[(2.0 / N[Power[x$95$m, 2.0], $MachinePrecision]), $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 67.6%

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

   if 2 < x

   1. Initial program 98.9%

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

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 99.0%

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

      5. unpow-prod-down 99.3%

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

      6. inv-pow 99.3%

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

      7. clear-num 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. unpow-1 99.3%

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

      2. /-rgt-identity 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in x around 0 56.9%

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

4. Final simplification 64.4%

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

Alternative 7: 79.0% 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 67.6%

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

   if 2 < x

   1. Initial program 98.9%

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

      1. clear-num 98.9%

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

      2. inv-pow 98.9%

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

      3. *-un-lft-identity 98.9%

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

      4. times-frac 99.0%

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

      5. unpow-prod-down 99.3%

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

      6. inv-pow 99.3%

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

      7. clear-num 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. unpow-1 99.3%

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

      2. /-rgt-identity 99.3%

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

      3. associate-*l/ 99.5%

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

      4. *-un-lft-identity 99.5%

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

   6. Applied egg-rr 55.0%

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

   7. Taylor expanded in x around 0 56.9%

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

4. Final simplification 64.3%

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

Alternative 8: 51.4% 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 54.2%

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

3. Taylor expanded in x around 0 48.3%

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

4. Final simplification 48.3%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))