Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.6% → 98.0%

Time: 9.0s

Alternatives: 18

Speedup: 1.0×

• 65.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \frac{y}{x}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (cosh(x) * (y / x)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (Math.cosh(x) * (y / x)) / z;
}

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (cosh(x) * (y / x)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \frac{y}{x}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (cosh(x) * (y / x)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (Math.cosh(x) * (y / x)) / z;
}

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (cosh(x) * (y / x)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+210}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 2e+210) (/ t_0 z) (* y (/ (/ (cosh x) z) x)))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 2e+210) {
		tmp = t_0 / z;
	} else {
		tmp = y * ((cosh(x) / z) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) * (y / x)
    if (t_0 <= 2d+210) then
        tmp = t_0 / z
    else
        tmp = y * ((cosh(x) / z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 2e+210) {
		tmp = t_0 / z;
	} else {
		tmp = y * ((Math.cosh(x) / z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if t_0 <= 2e+210:
		tmp = t_0 / z
	else:
		tmp = y * ((math.cosh(x) / z) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= 2e+210)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if (t_0 <= 2e+210)
		tmp = t_0 / z;
	else
		tmp = y * ((cosh(x) / z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+210], N[(t$95$0 / z), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.99999999999999985e210

   1. Initial program 98.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   if 1.99999999999999985e210 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 68.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 60.2%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 69.5%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 69.5%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 84.7%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 84.7%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 68.8%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 32.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 29.6%

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

      6. frac-times 40.7%

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

      7. *-commutative 40.7%

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

      8. associate-*r/ 35.0%

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

      9. associate-/r* 27.7%

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

   5. Applied egg-rr 27.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 60.2%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 68.7%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 68.8%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 68.8%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 99.9%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 99.9%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+210}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 2: 92.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{\frac{x \cdot 0.5 + \frac{-1}{x}}{x \cdot \left(x \cdot 0.25\right) + \frac{-1}{x \cdot x}}}}{z}\\ t_1 := y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{if}\;x \leq -1.45 \cdot 10^{+191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{z}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (/
          (/
           y
           (/ (+ (* x 0.5) (/ -1.0 x)) (+ (* x (* x 0.25)) (/ -1.0 (* x x)))))
          z))
        (t_1 (* y (/ (cosh x) (* x z)))))
   (if (<= x -1.45e+191)
     t_0
     (if (<= x -1.65e-76)
       t_1
       (if (<= x -1.8e-222)
         (* (/ y x) (/ 1.0 z))
         (if (<= x 2.7e+154) t_1 t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (y / (((x * 0.5) + (-1.0 / x)) / ((x * (x * 0.25)) + (-1.0 / (x * x))))) / z;
	double t_1 = y * (cosh(x) / (x * z));
	double tmp;
	if (x <= -1.45e+191) {
		tmp = t_0;
	} else if (x <= -1.65e-76) {
		tmp = t_1;
	} else if (x <= -1.8e-222) {
		tmp = (y / x) * (1.0 / z);
	} else if (x <= 2.7e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y / (((x * 0.5d0) + ((-1.0d0) / x)) / ((x * (x * 0.25d0)) + ((-1.0d0) / (x * x))))) / z
    t_1 = y * (cosh(x) / (x * z))
    if (x <= (-1.45d+191)) then
        tmp = t_0
    else if (x <= (-1.65d-76)) then
        tmp = t_1
    else if (x <= (-1.8d-222)) then
        tmp = (y / x) * (1.0d0 / z)
    else if (x <= 2.7d+154) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / (((x * 0.5) + (-1.0 / x)) / ((x * (x * 0.25)) + (-1.0 / (x * x))))) / z;
	double t_1 = y * (Math.cosh(x) / (x * z));
	double tmp;
	if (x <= -1.45e+191) {
		tmp = t_0;
	} else if (x <= -1.65e-76) {
		tmp = t_1;
	} else if (x <= -1.8e-222) {
		tmp = (y / x) * (1.0 / z);
	} else if (x <= 2.7e+154) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / (((x * 0.5) + (-1.0 / x)) / ((x * (x * 0.25)) + (-1.0 / (x * x))))) / z
	t_1 = y * (math.cosh(x) / (x * z))
	tmp = 0
	if x <= -1.45e+191:
		tmp = t_0
	elif x <= -1.65e-76:
		tmp = t_1
	elif x <= -1.8e-222:
		tmp = (y / x) * (1.0 / z)
	elif x <= 2.7e+154:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / Float64(Float64(Float64(x * 0.5) + Float64(-1.0 / x)) / Float64(Float64(x * Float64(x * 0.25)) + Float64(-1.0 / Float64(x * x))))) / z)
	t_1 = Float64(y * Float64(cosh(x) / Float64(x * z)))
	tmp = 0.0
	if (x <= -1.45e+191)
		tmp = t_0;
	elseif (x <= -1.65e-76)
		tmp = t_1;
	elseif (x <= -1.8e-222)
		tmp = Float64(Float64(y / x) * Float64(1.0 / z));
	elseif (x <= 2.7e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / (((x * 0.5) + (-1.0 / x)) / ((x * (x * 0.25)) + (-1.0 / (x * x))))) / z;
	t_1 = y * (cosh(x) / (x * z));
	tmp = 0.0;
	if (x <= -1.45e+191)
		tmp = t_0;
	elseif (x <= -1.65e-76)
		tmp = t_1;
	elseif (x <= -1.8e-222)
		tmp = (y / x) * (1.0 / z);
	elseif (x <= 2.7e+154)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / N[(N[(N[(x * 0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45e+191], t$95$0, If[LessEqual[x, -1.65e-76], t$95$1, If[LessEqual[x, -1.8e-222], N[(N[(y / x), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e+154], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4500000000000001e191 or 2.70000000000000006e154 < x

   1. Initial program 63.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 48.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in y around 0 48.3%

      \[\leadsto \color{blue}{\frac{y \cdot \left(0.5 \cdot x + \frac{1}{x}\right)}{z}} \]
   4. Step-by-step derivation

      1. distribute-lft-in 48.3%

         \[\leadsto \frac{\color{blue}{y \cdot \left(0.5 \cdot x\right) + y \cdot \frac{1}{x}}}{z} \]

      2. flip-+ 45.3%

         \[\leadsto \frac{\color{blue}{\frac{\left(y \cdot \left(0.5 \cdot x\right)\right) \cdot \left(y \cdot \left(0.5 \cdot x\right)\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}}{z} \]

      3. *-commutative 45.3%

         \[\leadsto \frac{\frac{\left(y \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) \cdot \left(y \cdot \left(0.5 \cdot x\right)\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      4. *-commutative 45.3%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      5. un-div-inv 45.3%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \color{blue}{\frac{y}{x}} \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      6. un-div-inv 45.3%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      7. *-commutative 45.3%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \frac{y}{x} \cdot \frac{y}{x}}{y \cdot \color{blue}{\left(x \cdot 0.5\right)} - y \cdot \frac{1}{x}}}{z} \]

      8. un-div-inv 45.3%

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

   5. Applied egg-rr 45.3%

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

   6. Taylor expanded in y around 0 100.0%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(0.25 \cdot {x}^{2} - \frac{1}{{x}^{2}}\right)}{0.5 \cdot x - \frac{1}{x}}}}{z} \]
   7. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{0.5 \cdot x - \frac{1}{x}}{0.25 \cdot {x}^{2} - \frac{1}{{x}^{2}}}}}}{z} \]

      2. *-commutative 100.0%

         \[\leadsto \frac{\frac{y}{\frac{\color{blue}{x \cdot 0.5} - \frac{1}{x}}{0.25 \cdot {x}^{2} - \frac{1}{{x}^{2}}}}}{z} \]

      3. *-commutative 100.0%

         \[\leadsto \frac{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{\color{blue}{{x}^{2} \cdot 0.25} - \frac{1}{{x}^{2}}}}}{z} \]

      4. unpow2 100.0%

         \[\leadsto \frac{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{\color{blue}{\left(x \cdot x\right)} \cdot 0.25 - \frac{1}{{x}^{2}}}}}{z} \]

      5. associate-*l* 100.0%

         \[\leadsto \frac{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{\color{blue}{x \cdot \left(x \cdot 0.25\right)} - \frac{1}{{x}^{2}}}}}{z} \]

      6. unpow2 100.0%

         \[\leadsto \frac{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{x \cdot \left(x \cdot 0.25\right) - \frac{1}{\color{blue}{x \cdot x}}}}}{z} \]

   8. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{x \cdot \left(x \cdot 0.25\right) - \frac{1}{x \cdot x}}}}}{z} \]

   if -1.4500000000000001e191 < x < -1.64999999999999992e-76 or -1.79999999999999987e-222 < x < 2.70000000000000006e154

   1. Initial program 91.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 85.1%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 86.3%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 86.3%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 91.4%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 91.4%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 91.4%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 55.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 39.8%

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

      6. frac-times 41.4%

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

      7. *-commutative 41.4%

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

      8. associate-*r/ 38.5%

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

      9. associate-/r* 36.3%

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

   5. Applied egg-rr 36.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 51.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 85.1%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 91.4%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 91.4%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 91.4%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. *-rgt-identity 91.4%

         \[\leadsto \frac{\color{blue}{y \cdot 1}}{x} \cdot \frac{\cosh x}{z} \]

      7. associate-*r/ 91.3%

         \[\leadsto \color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \frac{\cosh x}{z} \]

      8. associate-*r* 97.6%

         \[\leadsto \color{blue}{y \cdot \left(\frac{1}{x} \cdot \frac{\cosh x}{z}\right)} \]

      9. associate-*r/ 97.7%

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      10. associate-*r/ 97.6%

          \[\leadsto y \cdot \color{blue}{\left(\frac{1}{x} \cdot \frac{\cosh x}{z}\right)} \]

      11. associate-*l/ 97.7%

          \[\leadsto y \cdot \color{blue}{\frac{1 \cdot \frac{\cosh x}{z}}{x}} \]

      12. *-lft-identity 97.7%

          \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{z}}}{x} \]

      13. associate-/l/ 91.4%

          \[\leadsto y \cdot \color{blue}{\frac{\cosh x}{x \cdot z}} \]

   7. Simplified 91.4%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]

   if -1.64999999999999992e-76 < x < -1.79999999999999987e-222

   1. Initial program 96.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 96.6%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   3. Simplified 96.6%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   4. Taylor expanded in x around 0 96.6%

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \frac{y}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{y}{\frac{x \cdot 0.5 + \frac{-1}{x}}{x \cdot \left(x \cdot 0.25\right) + \frac{-1}{x \cdot x}}}}{z}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-222}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{z}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{+154}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{x \cdot 0.5 + \frac{-1}{x}}{x \cdot \left(x \cdot 0.25\right) + \frac{-1}{x \cdot x}}}}{z}\\ \end{array} \]

Alternative 3: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{\frac{\cosh x}{z}}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* y (/ (/ (cosh x) z) x)))

C

double code(double x, double y, double z) {
	return y * ((cosh(x) / z) / x);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y * ((cosh(x) / z) / x)
end function

Java

public static double code(double x, double y, double z) {
	return y * ((Math.cosh(x) / z) / x);
}

Python

def code(x, y, z):
	return y * ((math.cosh(x) / z) / x)

Julia

function code(x, y, z)
	return Float64(y * Float64(Float64(cosh(x) / z) / x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = y * ((cosh(x) / z) / x);
end

Wolfram

code[x_, y_, z_] := N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Step-by-step derivation

   1. associate-*r/ 80.1%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

   2. associate-/r* 78.7%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

3. Simplified 78.7%

   \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
4. Step-by-step derivation

   1. associate-*r/ 85.4%

      \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

   2. *-commutative 85.4%

      \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

   3. frac-times 86.3%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   4. expm1-log1p-u 48.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

   5. expm1-udef 36.7%

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

   6. frac-times 38.0%

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

   7. *-commutative 38.0%

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

   8. associate-*r/ 35.2%

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

   9. associate-/r* 33.9%

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

5. Applied egg-rr 33.9%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 46.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

   2. expm1-log1p 80.1%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

   3. associate-*r/ 86.4%

      \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

   4. associate-*l/ 86.3%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   5. *-commutative 86.3%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

   6. associate-*l/ 97.3%

      \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

   7. associate-*r/ 96.3%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

7. Simplified 96.3%

   \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

8. Final simplification 96.3%

   \[\leadsto y \cdot \frac{\frac{\cosh x}{z}}{x} \]

Alternative 4: 75.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{z} \cdot \frac{x \cdot \left(x \cdot 0.25\right) + \frac{-1}{x \cdot x}}{x \cdot 0.5 + \frac{-1}{x}}\\ \mathbf{if}\;x \leq -7 \cdot 10^{+163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+184}:\\ \;\;\;\;y \cdot \frac{z \cdot \left(\left(x \cdot 0.5\right) \cdot \left(-x\right)\right) - z}{x \cdot \left(z \cdot \left(-z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0
         (*
          (/ y z)
          (/ (+ (* x (* x 0.25)) (/ -1.0 (* x x))) (+ (* x 0.5) (/ -1.0 x))))))
   (if (<= x -7e+163)
     t_0
     (if (<= x 2.8e+25)
       (/ (+ (/ y x) (* 0.5 (* x y))) z)
       (if (<= x 7.4e+184)
         (* y (/ (- (* z (* (* x 0.5) (- x))) z) (* x (* z (- z)))))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (y / z) * (((x * (x * 0.25)) + (-1.0 / (x * x))) / ((x * 0.5) + (-1.0 / x)));
	double tmp;
	if (x <= -7e+163) {
		tmp = t_0;
	} else if (x <= 2.8e+25) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else if (x <= 7.4e+184) {
		tmp = y * (((z * ((x * 0.5) * -x)) - z) / (x * (z * -z)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / z) * (((x * (x * 0.25d0)) + ((-1.0d0) / (x * x))) / ((x * 0.5d0) + ((-1.0d0) / x)))
    if (x <= (-7d+163)) then
        tmp = t_0
    else if (x <= 2.8d+25) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else if (x <= 7.4d+184) then
        tmp = y * (((z * ((x * 0.5d0) * -x)) - z) / (x * (z * -z)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / z) * (((x * (x * 0.25)) + (-1.0 / (x * x))) / ((x * 0.5) + (-1.0 / x)));
	double tmp;
	if (x <= -7e+163) {
		tmp = t_0;
	} else if (x <= 2.8e+25) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else if (x <= 7.4e+184) {
		tmp = y * (((z * ((x * 0.5) * -x)) - z) / (x * (z * -z)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / z) * (((x * (x * 0.25)) + (-1.0 / (x * x))) / ((x * 0.5) + (-1.0 / x)))
	tmp = 0
	if x <= -7e+163:
		tmp = t_0
	elif x <= 2.8e+25:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	elif x <= 7.4e+184:
		tmp = y * (((z * ((x * 0.5) * -x)) - z) / (x * (z * -z)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / z) * Float64(Float64(Float64(x * Float64(x * 0.25)) + Float64(-1.0 / Float64(x * x))) / Float64(Float64(x * 0.5) + Float64(-1.0 / x))))
	tmp = 0.0
	if (x <= -7e+163)
		tmp = t_0;
	elseif (x <= 2.8e+25)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	elseif (x <= 7.4e+184)
		tmp = Float64(y * Float64(Float64(Float64(z * Float64(Float64(x * 0.5) * Float64(-x))) - z) / Float64(x * Float64(z * Float64(-z)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / z) * (((x * (x * 0.25)) + (-1.0 / (x * x))) / ((x * 0.5) + (-1.0 / x)));
	tmp = 0.0;
	if (x <= -7e+163)
		tmp = t_0;
	elseif (x <= 2.8e+25)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	elseif (x <= 7.4e+184)
		tmp = y * (((z * ((x * 0.5) * -x)) - z) / (x * (z * -z)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / z), $MachinePrecision] * N[(N[(N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+163], t$95$0, If[LessEqual[x, 2.8e+25], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 7.4e+184], N[(y * N[(N[(N[(z * N[(N[(x * 0.5), $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / N[(x * N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.0000000000000005e163 or 7.3999999999999995e184 < x

   1. Initial program 66.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 52.1%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in y around 0 52.1%

      \[\leadsto \color{blue}{\frac{y \cdot \left(0.5 \cdot x + \frac{1}{x}\right)}{z}} \]
   4. Step-by-step derivation

      1. distribute-lft-in 52.1%

         \[\leadsto \frac{\color{blue}{y \cdot \left(0.5 \cdot x\right) + y \cdot \frac{1}{x}}}{z} \]

      2. flip-+ 46.9%

         \[\leadsto \frac{\color{blue}{\frac{\left(y \cdot \left(0.5 \cdot x\right)\right) \cdot \left(y \cdot \left(0.5 \cdot x\right)\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}}{z} \]

      3. *-commutative 46.9%

         \[\leadsto \frac{\frac{\left(y \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) \cdot \left(y \cdot \left(0.5 \cdot x\right)\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      4. *-commutative 46.9%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      5. un-div-inv 46.9%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \color{blue}{\frac{y}{x}} \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      6. un-div-inv 46.9%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      7. *-commutative 46.9%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \frac{y}{x} \cdot \frac{y}{x}}{y \cdot \color{blue}{\left(x \cdot 0.5\right)} - y \cdot \frac{1}{x}}}{z} \]

      8. un-div-inv 46.9%

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

   5. Applied egg-rr 46.9%

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

   6. Taylor expanded in y around 0 66.0%

      \[\leadsto \color{blue}{\frac{y \cdot \left(0.25 \cdot {x}^{2} - \frac{1}{{x}^{2}}\right)}{z \cdot \left(0.5 \cdot x - \frac{1}{x}\right)}} \]
   7. Step-by-step derivation

      1. times-frac 90.0%

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{0.25 \cdot {x}^{2} - \frac{1}{{x}^{2}}}{0.5 \cdot x - \frac{1}{x}}} \]

      2. *-commutative 90.0%

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{{x}^{2} \cdot 0.25} - \frac{1}{{x}^{2}}}{0.5 \cdot x - \frac{1}{x}} \]

      3. unpow2 90.0%

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot 0.25 - \frac{1}{{x}^{2}}}{0.5 \cdot x - \frac{1}{x}} \]

      4. associate-*l* 90.0%

         \[\leadsto \frac{y}{z} \cdot \frac{\color{blue}{x \cdot \left(x \cdot 0.25\right)} - \frac{1}{{x}^{2}}}{0.5 \cdot x - \frac{1}{x}} \]

      5. unpow2 90.0%

         \[\leadsto \frac{y}{z} \cdot \frac{x \cdot \left(x \cdot 0.25\right) - \frac{1}{\color{blue}{x \cdot x}}}{0.5 \cdot x - \frac{1}{x}} \]

      6. *-commutative 90.0%

         \[\leadsto \frac{y}{z} \cdot \frac{x \cdot \left(x \cdot 0.25\right) - \frac{1}{x \cdot x}}{\color{blue}{x \cdot 0.5} - \frac{1}{x}} \]

   8. Simplified 90.0%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x \cdot \left(x \cdot 0.25\right) - \frac{1}{x \cdot x}}{x \cdot 0.5 - \frac{1}{x}}} \]

   if -7.0000000000000005e163 < x < 2.8000000000000002e25

   1. Initial program 94.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 73.6%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   if 2.8000000000000002e25 < x < 7.3999999999999995e184

   1. Initial program 75.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 61.1%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 75.0%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 75.0%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 86.1%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 86.1%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 75.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 47.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 47.2%

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

      6. frac-times 55.6%

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

      7. *-commutative 55.6%

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

      8. associate-*r/ 47.2%

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

      9. associate-/r* 36.1%

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

   5. Applied egg-rr 36.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 36.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 61.1%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 75.0%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 75.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 75.0%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 100.0%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   8. Taylor expanded in x around 0 30.4%

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

      1. associate-*r/ 30.4%

         \[\leadsto y \cdot \left(\color{blue}{\frac{0.5 \cdot x}{z}} + \frac{1}{x \cdot z}\right) \]

      2. *-commutative 30.4%

         \[\leadsto y \cdot \left(\frac{\color{blue}{x \cdot 0.5}}{z} + \frac{1}{x \cdot z}\right) \]

      3. frac-2neg 30.4%

         \[\leadsto y \cdot \left(\frac{x \cdot 0.5}{z} + \color{blue}{\frac{-1}{-x \cdot z}}\right) \]

      4. metadata-eval 30.4%

         \[\leadsto y \cdot \left(\frac{x \cdot 0.5}{z} + \frac{\color{blue}{-1}}{-x \cdot z}\right) \]

      5. frac-add 40.0%

         \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot \left(-x \cdot z\right) + z \cdot -1}{z \cdot \left(-x \cdot z\right)}} \]

      6. distribute-rgt-neg-in 40.0%

         \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \color{blue}{\left(x \cdot \left(-z\right)\right)} + z \cdot -1}{z \cdot \left(-x \cdot z\right)} \]

      7. distribute-rgt-neg-in 40.0%

         \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + z \cdot -1}{z \cdot \color{blue}{\left(x \cdot \left(-z\right)\right)}} \]

   10. Applied egg-rr 40.0%

       \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + z \cdot -1}{z \cdot \left(x \cdot \left(-z\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 40.0%

          \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + z \cdot -1}{\color{blue}{\left(x \cdot \left(-z\right)\right) \cdot z}} \]

       2. associate-*l* 40.0%

          \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + z \cdot -1}{\color{blue}{x \cdot \left(\left(-z\right) \cdot z\right)}} \]

       3. *-commutative 40.0%

          \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + \color{blue}{-1 \cdot z}}{x \cdot \left(\left(-z\right) \cdot z\right)} \]

       4. neg-mul-1 40.0%

          \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + \color{blue}{\left(-z\right)}}{x \cdot \left(\left(-z\right) \cdot z\right)} \]

       5. unsub-neg 40.0%

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

       6. associate-*r* 47.9%

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

       7. *-commutative 47.9%

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

       8. *-commutative 47.9%

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

       9. *-commutative 47.9%

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

   12. Simplified 47.9%

       \[\leadsto y \cdot \color{blue}{\frac{\left(-z\right) \cdot \left(x \cdot \left(x \cdot 0.5\right)\right) - z}{x \cdot \left(z \cdot \left(-z\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x \cdot \left(x \cdot 0.25\right) + \frac{-1}{x \cdot x}}{x \cdot 0.5 + \frac{-1}{x}}\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;x \leq 7.4 \cdot 10^{+184}:\\ \;\;\;\;y \cdot \frac{z \cdot \left(\left(x \cdot 0.5\right) \cdot \left(-x\right)\right) - z}{x \cdot \left(z \cdot \left(-z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x \cdot \left(x \cdot 0.25\right) + \frac{-1}{x \cdot x}}{x \cdot 0.5 + \frac{-1}{x}}\\ \end{array} \]

Alternative 5: 78.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-133} \lor \neg \left(x \leq 2.1 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{\frac{y}{\frac{x \cdot 0.5 + \frac{-1}{x}}{x \cdot \left(x \cdot 0.25\right) + \frac{-1}{x \cdot x}}}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot z} + y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.7e-133) (not (<= x 2.1e+154)))
   (/
    (/ y (/ (+ (* x 0.5) (/ -1.0 x)) (+ (* x (* x 0.25)) (/ -1.0 (* x x)))))
    z)
   (+ (* y (/ 1.0 (* x z))) (* y (* 0.5 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.7e-133) || !(x <= 2.1e+154)) {
		tmp = (y / (((x * 0.5) + (-1.0 / x)) / ((x * (x * 0.25)) + (-1.0 / (x * x))))) / z;
	} else {
		tmp = (y * (1.0 / (x * z))) + (y * (0.5 * (x / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.7d-133)) .or. (.not. (x <= 2.1d+154))) then
        tmp = (y / (((x * 0.5d0) + ((-1.0d0) / x)) / ((x * (x * 0.25d0)) + ((-1.0d0) / (x * x))))) / z
    else
        tmp = (y * (1.0d0 / (x * z))) + (y * (0.5d0 * (x / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.7e-133) || !(x <= 2.1e+154)) {
		tmp = (y / (((x * 0.5) + (-1.0 / x)) / ((x * (x * 0.25)) + (-1.0 / (x * x))))) / z;
	} else {
		tmp = (y * (1.0 / (x * z))) + (y * (0.5 * (x / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.7e-133) or not (x <= 2.1e+154):
		tmp = (y / (((x * 0.5) + (-1.0 / x)) / ((x * (x * 0.25)) + (-1.0 / (x * x))))) / z
	else:
		tmp = (y * (1.0 / (x * z))) + (y * (0.5 * (x / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.7e-133) || !(x <= 2.1e+154))
		tmp = Float64(Float64(y / Float64(Float64(Float64(x * 0.5) + Float64(-1.0 / x)) / Float64(Float64(x * Float64(x * 0.25)) + Float64(-1.0 / Float64(x * x))))) / z);
	else
		tmp = Float64(Float64(y * Float64(1.0 / Float64(x * z))) + Float64(y * Float64(0.5 * Float64(x / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.7e-133) || ~((x <= 2.1e+154)))
		tmp = (y / (((x * 0.5) + (-1.0 / x)) / ((x * (x * 0.25)) + (-1.0 / (x * x))))) / z;
	else
		tmp = (y * (1.0 / (x * z))) + (y * (0.5 * (x / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.7e-133], N[Not[LessEqual[x, 2.1e+154]], $MachinePrecision]], N[(N[(y / N[(N[(N[(x * 0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y * N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6999999999999999e-133 or 2.09999999999999994e154 < x

   1. Initial program 83.3%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 56.3%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in y around 0 56.3%

      \[\leadsto \color{blue}{\frac{y \cdot \left(0.5 \cdot x + \frac{1}{x}\right)}{z}} \]
   4. Step-by-step derivation

      1. distribute-lft-in 56.3%

         \[\leadsto \frac{\color{blue}{y \cdot \left(0.5 \cdot x\right) + y \cdot \frac{1}{x}}}{z} \]

      2. flip-+ 41.6%

         \[\leadsto \frac{\color{blue}{\frac{\left(y \cdot \left(0.5 \cdot x\right)\right) \cdot \left(y \cdot \left(0.5 \cdot x\right)\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}}{z} \]

      3. *-commutative 41.6%

         \[\leadsto \frac{\frac{\left(y \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) \cdot \left(y \cdot \left(0.5 \cdot x\right)\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      4. *-commutative 41.6%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \color{blue}{\left(x \cdot 0.5\right)}\right) - \left(y \cdot \frac{1}{x}\right) \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      5. un-div-inv 41.7%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \color{blue}{\frac{y}{x}} \cdot \left(y \cdot \frac{1}{x}\right)}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      6. un-div-inv 41.6%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \frac{y}{x} \cdot \color{blue}{\frac{y}{x}}}{y \cdot \left(0.5 \cdot x\right) - y \cdot \frac{1}{x}}}{z} \]

      7. *-commutative 41.6%

         \[\leadsto \frac{\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(y \cdot \left(x \cdot 0.5\right)\right) - \frac{y}{x} \cdot \frac{y}{x}}{y \cdot \color{blue}{\left(x \cdot 0.5\right)} - y \cdot \frac{1}{x}}}{z} \]

      8. un-div-inv 41.7%

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

   5. Applied egg-rr 41.7%

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

   6. Taylor expanded in y around 0 78.1%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(0.25 \cdot {x}^{2} - \frac{1}{{x}^{2}}\right)}{0.5 \cdot x - \frac{1}{x}}}}{z} \]
   7. Step-by-step derivation

      1. associate-/l* 78.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{0.5 \cdot x - \frac{1}{x}}{0.25 \cdot {x}^{2} - \frac{1}{{x}^{2}}}}}}{z} \]

      2. *-commutative 78.1%

         \[\leadsto \frac{\frac{y}{\frac{\color{blue}{x \cdot 0.5} - \frac{1}{x}}{0.25 \cdot {x}^{2} - \frac{1}{{x}^{2}}}}}{z} \]

      3. *-commutative 78.1%

         \[\leadsto \frac{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{\color{blue}{{x}^{2} \cdot 0.25} - \frac{1}{{x}^{2}}}}}{z} \]

      4. unpow2 78.1%

         \[\leadsto \frac{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{\color{blue}{\left(x \cdot x\right)} \cdot 0.25 - \frac{1}{{x}^{2}}}}}{z} \]

      5. associate-*l* 78.1%

         \[\leadsto \frac{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{\color{blue}{x \cdot \left(x \cdot 0.25\right)} - \frac{1}{{x}^{2}}}}}{z} \]

      6. unpow2 78.1%

         \[\leadsto \frac{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{x \cdot \left(x \cdot 0.25\right) - \frac{1}{\color{blue}{x \cdot x}}}}}{z} \]

   8. Simplified 78.1%

      \[\leadsto \frac{\color{blue}{\frac{y}{\frac{x \cdot 0.5 - \frac{1}{x}}{x \cdot \left(x \cdot 0.25\right) - \frac{1}{x \cdot x}}}}}{z} \]

   if -2.6999999999999999e-133 < x < 2.09999999999999994e154

   1. Initial program 89.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 85.6%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 87.5%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 87.5%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 89.9%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 89.9%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 89.6%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 58.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 41.1%

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

      6. frac-times 42.1%

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

      7. *-commutative 42.1%

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

      8. associate-*r/ 39.6%

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

      9. associate-/r* 37.9%

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

   5. Applied egg-rr 37.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 55.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 85.6%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 89.7%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 89.6%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 89.6%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 94.5%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 93.9%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 93.9%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   8. Taylor expanded in x around 0 70.3%

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

      1. +-commutative 70.3%

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)} \]

      2. distribute-rgt-in 70.3%

         \[\leadsto \color{blue}{\frac{1}{x \cdot z} \cdot y + \left(0.5 \cdot \frac{x}{z}\right) \cdot y} \]

      3. *-commutative 70.3%

         \[\leadsto \frac{1}{x \cdot z} \cdot y + \color{blue}{\left(\frac{x}{z} \cdot 0.5\right)} \cdot y \]

   10. Applied egg-rr 70.3%

       \[\leadsto \color{blue}{\frac{1}{x \cdot z} \cdot y + \left(\frac{x}{z} \cdot 0.5\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-133} \lor \neg \left(x \leq 2.1 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{\frac{y}{\frac{x \cdot 0.5 + \frac{-1}{x}}{x \cdot \left(x \cdot 0.25\right) + \frac{-1}{x \cdot x}}}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot z} + y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \]

Alternative 6: 68.1% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+15} \lor \neg \left(x \leq 2.8 \cdot 10^{+25}\right):\\ \;\;\;\;y \cdot \frac{z \cdot \left(\left(x \cdot 0.5\right) \cdot \left(-x\right)\right) - z}{x \cdot \left(z \cdot \left(-z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.5e+15) (not (<= x 2.8e+25)))
   (* y (/ (- (* z (* (* x 0.5) (- x))) z) (* x (* z (- z)))))
   (/ (+ (/ y x) (* 0.5 (* x y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.5e+15) || !(x <= 2.8e+25)) {
		tmp = y * (((z * ((x * 0.5) * -x)) - z) / (x * (z * -z)));
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-4.5d+15)) .or. (.not. (x <= 2.8d+25))) then
        tmp = y * (((z * ((x * 0.5d0) * -x)) - z) / (x * (z * -z)))
    else
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.5e+15) || !(x <= 2.8e+25)) {
		tmp = y * (((z * ((x * 0.5) * -x)) - z) / (x * (z * -z)));
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.5e+15) or not (x <= 2.8e+25):
		tmp = y * (((z * ((x * 0.5) * -x)) - z) / (x * (z * -z)))
	else:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.5e+15) || !(x <= 2.8e+25))
		tmp = Float64(y * Float64(Float64(Float64(z * Float64(Float64(x * 0.5) * Float64(-x))) - z) / Float64(x * Float64(z * Float64(-z)))));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.5e+15) || ~((x <= 2.8e+25)))
		tmp = y * (((z * ((x * 0.5) * -x)) - z) / (x * (z * -z)));
	else
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.5e+15], N[Not[LessEqual[x, 2.8e+25]], $MachinePrecision]], N[(y * N[(N[(N[(z * N[(N[(x * 0.5), $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision] / N[(x * N[(z * (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5e15 or 2.8000000000000002e25 < x

   1. Initial program 77.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 65.0%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 65.0%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 65.0%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 78.0%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 78.0%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 77.2%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 36.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 36.6%

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

      6. frac-times 39.8%

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

      7. *-commutative 39.8%

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

      8. associate-*r/ 34.1%

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

      9. associate-/r* 30.9%

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

   5. Applied egg-rr 30.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 30.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 65.0%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 77.2%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 77.2%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 77.2%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 100.0%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   8. Taylor expanded in x around 0 39.9%

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

      1. associate-*r/ 39.9%

         \[\leadsto y \cdot \left(\color{blue}{\frac{0.5 \cdot x}{z}} + \frac{1}{x \cdot z}\right) \]

      2. *-commutative 39.9%

         \[\leadsto y \cdot \left(\frac{\color{blue}{x \cdot 0.5}}{z} + \frac{1}{x \cdot z}\right) \]

      3. frac-2neg 39.9%

         \[\leadsto y \cdot \left(\frac{x \cdot 0.5}{z} + \color{blue}{\frac{-1}{-x \cdot z}}\right) \]

      4. metadata-eval 39.9%

         \[\leadsto y \cdot \left(\frac{x \cdot 0.5}{z} + \frac{\color{blue}{-1}}{-x \cdot z}\right) \]

      5. frac-add 47.9%

         \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot \left(-x \cdot z\right) + z \cdot -1}{z \cdot \left(-x \cdot z\right)}} \]

      6. distribute-rgt-neg-in 47.9%

         \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \color{blue}{\left(x \cdot \left(-z\right)\right)} + z \cdot -1}{z \cdot \left(-x \cdot z\right)} \]

      7. distribute-rgt-neg-in 47.9%

         \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + z \cdot -1}{z \cdot \color{blue}{\left(x \cdot \left(-z\right)\right)}} \]

   10. Applied egg-rr 47.9%

       \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + z \cdot -1}{z \cdot \left(x \cdot \left(-z\right)\right)}} \]
   11. Step-by-step derivation

       1. *-commutative 47.9%

          \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + z \cdot -1}{\color{blue}{\left(x \cdot \left(-z\right)\right) \cdot z}} \]

       2. associate-*l* 48.7%

          \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + z \cdot -1}{\color{blue}{x \cdot \left(\left(-z\right) \cdot z\right)}} \]

       3. *-commutative 48.7%

          \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + \color{blue}{-1 \cdot z}}{x \cdot \left(\left(-z\right) \cdot z\right)} \]

       4. neg-mul-1 48.7%

          \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot \left(x \cdot \left(-z\right)\right) + \color{blue}{\left(-z\right)}}{x \cdot \left(\left(-z\right) \cdot z\right)} \]

       5. unsub-neg 48.7%

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

       6. associate-*r* 51.8%

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

       7. *-commutative 51.8%

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

       8. *-commutative 51.8%

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

       9. *-commutative 51.8%

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

   12. Simplified 51.8%

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

   if -4.5e15 < x < 2.8000000000000002e25

   1. Initial program 94.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 83.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+15} \lor \neg \left(x \leq 2.8 \cdot 10^{+25}\right):\\ \;\;\;\;y \cdot \frac{z \cdot \left(\left(x \cdot 0.5\right) \cdot \left(-x\right)\right) - z}{x \cdot \left(z \cdot \left(-z\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 7: 67.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+232}:\\ \;\;\;\;y \cdot \frac{z \cdot \left(x \cdot 0.5\right) + \frac{z}{x}}{z \cdot z}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z + \left(x \cdot 0.5\right) \cdot \left(x \cdot z\right)}{z \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.3e+232)
   (* y (/ (+ (* z (* x 0.5)) (/ z x)) (* z z)))
   (if (<= x 1.05e+26)
     (/ (+ (/ y x) (* 0.5 (* x y))) z)
     (* y (/ (+ z (* (* x 0.5) (* x z))) (* z (* x z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+232) {
		tmp = y * (((z * (x * 0.5)) + (z / x)) / (z * z));
	} else if (x <= 1.05e+26) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = y * ((z + ((x * 0.5) * (x * z))) / (z * (x * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.3d+232)) then
        tmp = y * (((z * (x * 0.5d0)) + (z / x)) / (z * z))
    else if (x <= 1.05d+26) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else
        tmp = y * ((z + ((x * 0.5d0) * (x * z))) / (z * (x * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.3e+232) {
		tmp = y * (((z * (x * 0.5)) + (z / x)) / (z * z));
	} else if (x <= 1.05e+26) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = y * ((z + ((x * 0.5) * (x * z))) / (z * (x * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.3e+232:
		tmp = y * (((z * (x * 0.5)) + (z / x)) / (z * z))
	elif x <= 1.05e+26:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	else:
		tmp = y * ((z + ((x * 0.5) * (x * z))) / (z * (x * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.3e+232)
		tmp = Float64(y * Float64(Float64(Float64(z * Float64(x * 0.5)) + Float64(z / x)) / Float64(z * z)));
	elseif (x <= 1.05e+26)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	else
		tmp = Float64(y * Float64(Float64(z + Float64(Float64(x * 0.5) * Float64(x * z))) / Float64(z * Float64(x * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.3e+232)
		tmp = y * (((z * (x * 0.5)) + (z / x)) / (z * z));
	elseif (x <= 1.05e+26)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	else
		tmp = y * ((z + ((x * 0.5) * (x * z))) / (z * (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.3e+232], N[(y * N[(N[(N[(z * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(z / x), $MachinePrecision]), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+26], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(z + N[(N[(x * 0.5), $MachinePrecision] * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.29999999999999987e232

   1. Initial program 47.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 35.3%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 35.3%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 35.3%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 58.8%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 58.8%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 47.1%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 29.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 29.4%

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

      6. frac-times 35.3%

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

      7. *-commutative 35.3%

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

      8. associate-*r/ 29.4%

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

      9. associate-/r* 23.5%

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

   5. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 35.3%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 47.1%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 47.1%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 47.1%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 100.0%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   8. Taylor expanded in x around 0 65.7%

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

      1. associate-*r/ 65.7%

         \[\leadsto y \cdot \left(\color{blue}{\frac{0.5 \cdot x}{z}} + \frac{1}{x \cdot z}\right) \]

      2. *-commutative 65.7%

         \[\leadsto y \cdot \left(\frac{\color{blue}{x \cdot 0.5}}{z} + \frac{1}{x \cdot z}\right) \]

      3. associate-/r* 65.7%

         \[\leadsto y \cdot \left(\frac{x \cdot 0.5}{z} + \color{blue}{\frac{\frac{1}{x}}{z}}\right) \]

      4. frac-add 76.5%

         \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot z + z \cdot \frac{1}{x}}{z \cdot z}} \]

      5. div-inv 76.5%

         \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot z + \color{blue}{\frac{z}{x}}}{z \cdot z} \]

   10. Applied egg-rr 76.5%

       \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot z + \frac{z}{x}}{z \cdot z}} \]

   if -1.29999999999999987e232 < x < 1.05e26

   1. Initial program 94.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 71.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   if 1.05e26 < x

   1. Initial program 73.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 62.5%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 69.6%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 69.6%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 78.6%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 78.6%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 73.2%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 39.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 39.3%

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

      6. frac-times 46.4%

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

      7. *-commutative 46.4%

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

      8. associate-*r/ 39.3%

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

      9. associate-/r* 32.1%

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

   5. Applied egg-rr 32.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 32.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 62.5%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 73.2%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 73.2%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 73.2%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 100.0%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   8. Taylor expanded in x around 0 38.2%

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

      1. +-commutative 38.2%

         \[\leadsto y \cdot \color{blue}{\left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)} \]

      2. associate-*r/ 38.2%

         \[\leadsto y \cdot \left(\frac{1}{x \cdot z} + \color{blue}{\frac{0.5 \cdot x}{z}}\right) \]

      3. *-commutative 38.2%

         \[\leadsto y \cdot \left(\frac{1}{x \cdot z} + \frac{\color{blue}{x \cdot 0.5}}{z}\right) \]

      4. frac-add 47.1%

         \[\leadsto y \cdot \color{blue}{\frac{1 \cdot z + \left(x \cdot z\right) \cdot \left(x \cdot 0.5\right)}{\left(x \cdot z\right) \cdot z}} \]

      5. *-un-lft-identity 47.1%

         \[\leadsto y \cdot \frac{\color{blue}{z} + \left(x \cdot z\right) \cdot \left(x \cdot 0.5\right)}{\left(x \cdot z\right) \cdot z} \]

   10. Applied egg-rr 47.1%

       \[\leadsto y \cdot \color{blue}{\frac{z + \left(x \cdot z\right) \cdot \left(x \cdot 0.5\right)}{\left(x \cdot z\right) \cdot z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+232}:\\ \;\;\;\;y \cdot \frac{z \cdot \left(x \cdot 0.5\right) + \frac{z}{x}}{z \cdot z}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z + \left(x \cdot 0.5\right) \cdot \left(x \cdot z\right)}{z \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 8: 67.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+232} \lor \neg \left(x \leq 1.3 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \frac{z \cdot \left(x \cdot 0.5\right) + \frac{z}{x}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5.5e+232) (not (<= x 1.3e+17)))
   (* y (/ (+ (* z (* x 0.5)) (/ z x)) (* z z)))
   (/ (+ (/ y x) (* 0.5 (* x y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5e+232) || !(x <= 1.3e+17)) {
		tmp = y * (((z * (x * 0.5)) + (z / x)) / (z * z));
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-5.5d+232)) .or. (.not. (x <= 1.3d+17))) then
        tmp = y * (((z * (x * 0.5d0)) + (z / x)) / (z * z))
    else
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5.5e+232) || !(x <= 1.3e+17)) {
		tmp = y * (((z * (x * 0.5)) + (z / x)) / (z * z));
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5.5e+232) or not (x <= 1.3e+17):
		tmp = y * (((z * (x * 0.5)) + (z / x)) / (z * z))
	else:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5.5e+232) || !(x <= 1.3e+17))
		tmp = Float64(y * Float64(Float64(Float64(z * Float64(x * 0.5)) + Float64(z / x)) / Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -5.5e+232) || ~((x <= 1.3e+17)))
		tmp = y * (((z * (x * 0.5)) + (z / x)) / (z * z));
	else
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5.5e+232], N[Not[LessEqual[x, 1.3e+17]], $MachinePrecision]], N[(y * N[(N[(N[(z * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(z / x), $MachinePrecision]), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.49999999999999997e232 or 1.3e17 < x

   1. Initial program 68.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 57.9%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 63.2%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 63.2%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 75.0%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 75.0%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 68.4%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 38.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 38.2%

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

      6. frac-times 44.7%

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

      7. *-commutative 44.7%

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

      8. associate-*r/ 38.2%

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

      9. associate-/r* 31.6%

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

   5. Applied egg-rr 31.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 31.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 57.9%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 68.4%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 68.4%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 68.4%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 100.0%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   8. Taylor expanded in x around 0 43.0%

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

      1. associate-*r/ 43.0%

         \[\leadsto y \cdot \left(\color{blue}{\frac{0.5 \cdot x}{z}} + \frac{1}{x \cdot z}\right) \]

      2. *-commutative 43.0%

         \[\leadsto y \cdot \left(\frac{\color{blue}{x \cdot 0.5}}{z} + \frac{1}{x \cdot z}\right) \]

      3. associate-/r* 43.0%

         \[\leadsto y \cdot \left(\frac{x \cdot 0.5}{z} + \color{blue}{\frac{\frac{1}{x}}{z}}\right) \]

      4. frac-add 48.7%

         \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot z + z \cdot \frac{1}{x}}{z \cdot z}} \]

      5. div-inv 48.7%

         \[\leadsto y \cdot \frac{\left(x \cdot 0.5\right) \cdot z + \color{blue}{\frac{z}{x}}}{z \cdot z} \]

   10. Applied egg-rr 48.7%

       \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot 0.5\right) \cdot z + \frac{z}{x}}{z \cdot z}} \]

   if -5.49999999999999997e232 < x < 1.3e17

   1. Initial program 93.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 73.0%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+232} \lor \neg \left(x \leq 1.3 \cdot 10^{+17}\right):\\ \;\;\;\;y \cdot \frac{z \cdot \left(x \cdot 0.5\right) + \frac{z}{x}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 9: 65.4% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+233}:\\ \;\;\;\;\frac{-y}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.15e-290)
   (/ (+ (/ y x) (* 0.5 (* x y))) z)
   (if (<= y 2.8e+233)
     (/ (- y) (/ z (+ (* x -0.5) (/ -1.0 x))))
     (/ (* y (+ (* x 0.5) (/ 1.0 x))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e-290) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else if (y <= 2.8e+233) {
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)));
	} else {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.15d-290)) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else if (y <= 2.8d+233) then
        tmp = -y / (z / ((x * (-0.5d0)) + ((-1.0d0) / x)))
    else
        tmp = (y * ((x * 0.5d0) + (1.0d0 / x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.15e-290) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else if (y <= 2.8e+233) {
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)));
	} else {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.15e-290:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	elif y <= 2.8e+233:
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)))
	else:
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.15e-290)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	elseif (y <= 2.8e+233)
		tmp = Float64(Float64(-y) / Float64(z / Float64(Float64(x * -0.5) + Float64(-1.0 / x))));
	else
		tmp = Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.15e-290)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	elseif (y <= 2.8e+233)
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)));
	else
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.15e-290], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[y, 2.8e+233], N[((-y) / N[(z / N[(N[(x * -0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15e-290

   1. Initial program 88.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 55.9%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   if -1.15e-290 < y < 2.8000000000000001e233

   1. Initial program 83.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 80.6%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 84.1%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 92.2%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 92.2%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 83.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 50.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 35.2%

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

      6. frac-times 39.5%

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

      7. *-commutative 39.5%

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

      8. associate-*r/ 36.3%

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

      9. associate-/r* 32.8%

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

   5. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 48.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 80.6%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 83.0%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 83.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 83.0%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 97.5%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 96.1%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 96.1%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   8. Taylor expanded in x around 0 66.5%

      \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)} \]

   9. Taylor expanded in z around -inf 61.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot \left(-0.5 \cdot x - \frac{1}{x}\right)}{z}} \]
   10. Step-by-step derivation

       1. mul-1-neg 61.7%

          \[\leadsto \color{blue}{-\frac{y \cdot \left(-0.5 \cdot x - \frac{1}{x}\right)}{z}} \]

       2. associate-/l* 65.7%

          \[\leadsto -\color{blue}{\frac{y}{\frac{z}{-0.5 \cdot x - \frac{1}{x}}}} \]

       3. *-commutative 65.7%

          \[\leadsto -\frac{y}{\frac{z}{\color{blue}{x \cdot -0.5} - \frac{1}{x}}} \]

   11. Simplified 65.7%

       \[\leadsto \color{blue}{-\frac{y}{\frac{z}{x \cdot -0.5 - \frac{1}{x}}}} \]

   if 2.8000000000000001e233 < y

   1. Initial program 95.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 95.6%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in y around 0 95.6%

      \[\leadsto \color{blue}{\frac{y \cdot \left(0.5 \cdot x + \frac{1}{x}\right)}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+233}:\\ \;\;\;\;\frac{-y}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \end{array} \]

Alternative 10: 66.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-222}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.7e-222)
   (/ (+ (/ y x) (* 0.5 (* x y))) z)
   (* y (+ (/ 1.0 (* x z)) (* 0.5 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.7e-222) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = y * ((1.0 / (x * z)) + (0.5 * (x / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3.7d-222)) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else
        tmp = y * ((1.0d0 / (x * z)) + (0.5d0 * (x / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.7e-222) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = y * ((1.0 / (x * z)) + (0.5 * (x / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.7e-222:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	else:
		tmp = y * ((1.0 / (x * z)) + (0.5 * (x / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.7e-222)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	else
		tmp = Float64(y * Float64(Float64(1.0 / Float64(x * z)) + Float64(0.5 * Float64(x / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.7e-222)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	else
		tmp = y * ((1.0 / (x * z)) + (0.5 * (x / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.7e-222], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6999999999999999e-222

   1. Initial program 88.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 64.6%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   if -3.6999999999999999e-222 < x

   1. Initial program 83.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 79.1%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 83.8%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 83.8%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. associate-*r/ 87.7%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x \cdot z}} \]

      2. *-commutative 87.7%

         \[\leadsto \frac{\cosh x \cdot y}{\color{blue}{z \cdot x}} \]

      3. frac-times 83.7%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      4. expm1-log1p-u 51.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\cosh x}{z} \cdot \frac{y}{x}\right)\right)} \]

      5. expm1-udef 37.7%

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

      6. frac-times 42.2%

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

      7. *-commutative 42.2%

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

      8. associate-*r/ 39.1%

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

      9. associate-/r* 34.6%

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

   5. Applied egg-rr 34.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 48.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x \cdot \frac{\frac{y}{x}}{z}\right)\right)} \]

      2. expm1-log1p 79.1%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      3. associate-*r/ 83.8%

         \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      4. associate-*l/ 83.7%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

      5. *-commutative 83.7%

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\cosh x}{z}} \]

      6. associate-*l/ 96.2%

         \[\leadsto \color{blue}{\frac{y \cdot \frac{\cosh x}{z}}{x}} \]

      7. associate-*r/ 97.0%

         \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   7. Simplified 97.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{z}}{x}} \]

   8. Taylor expanded in x around 0 62.9%

      \[\leadsto y \cdot \color{blue}{\left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{-222}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)\\ \end{array} \]

Alternative 11: 62.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.6) (not (<= x 1.4))) (* 0.5 (* x (/ y z))) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6) || !(x <= 1.4)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.6d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = 0.5d0 * (x * (y / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6) || !(x <= 1.4)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.6) or not (x <= 1.4):
		tmp = 0.5 * (x * (y / z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.6) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(x * Float64(y / z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.6) || ~((x <= 1.4)))
		tmp = 0.5 * (x * (y / z));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.6], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 1.3999999999999999 < x

   1. Initial program 79.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 35.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 35.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-/l* 29.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-/r/ 37.2%

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

   5. Simplified 37.2%

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

   6. Taylor expanded in x around 0 35.7%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-commutative 35.7%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{y \cdot x}}{z} \]

      2. associate-*l/ 29.0%

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

      3. *-commutative 29.0%

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

   8. Simplified 29.0%

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

   if -1.6000000000000001 < x < 1.3999999999999999

   1. Initial program 94.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 94.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   4. Taylor expanded in x around 0 91.7%

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

      1. associate-*r/ 91.8%

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

      2. associate-*l/ 91.9%

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

      3. *-un-lft-identity 91.9%

         \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   6. Applied egg-rr 91.9%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 12: 66.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.6) (not (<= x 1.4))) (* 0.5 (* y (/ x z))) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6) || !(x <= 1.4)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.6d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = 0.5d0 * (y * (x / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.6) || !(x <= 1.4)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.6) or not (x <= 1.4):
		tmp = 0.5 * (y * (x / z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.6) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(y * Float64(x / z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.6) || ~((x <= 1.4)))
		tmp = 0.5 * (y * (x / z));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.6], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001 or 1.3999999999999999 < x

   1. Initial program 79.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 35.7%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 35.7%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-/l* 29.0%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-/r/ 37.2%

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

   5. Simplified 37.2%

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

   if -1.6000000000000001 < x < 1.3999999999999999

   1. Initial program 94.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 94.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   4. Taylor expanded in x around 0 91.7%

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

      1. associate-*r/ 91.8%

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

      2. associate-*l/ 91.9%

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

      3. *-un-lft-identity 91.9%

         \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   6. Applied egg-rr 91.9%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 13: 66.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.6)
   (/ (* 0.5 (* x y)) z)
   (if (<= x 1.4) (/ (/ y z) x) (* 0.5 (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6) {
		tmp = (0.5 * (x * y)) / z;
	} else if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = 0.5 * (y * (x / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.6d0)) then
        tmp = (0.5d0 * (x * y)) / z
    else if (x <= 1.4d0) then
        tmp = (y / z) / x
    else
        tmp = 0.5d0 * (y * (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.6) {
		tmp = (0.5 * (x * y)) / z;
	} else if (x <= 1.4) {
		tmp = (y / z) / x;
	} else {
		tmp = 0.5 * (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.6:
		tmp = (0.5 * (x * y)) / z
	elif x <= 1.4:
		tmp = (y / z) / x
	else:
		tmp = 0.5 * (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.6)
		tmp = Float64(Float64(0.5 * Float64(x * y)) / z);
	elseif (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(0.5 * Float64(y * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.6)
		tmp = (0.5 * (x * y)) / z;
	elseif (x <= 1.4)
		tmp = (y / z) / x;
	else
		tmp = 0.5 * (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.6], N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6000000000000001

   1. Initial program 82.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 40.5%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 40.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. *-commutative 40.5%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{y \cdot x}}{z} \]

      2. associate-*r/ 40.5%

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

   5. Simplified 40.5%

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

   if -1.6000000000000001 < x < 1.3999999999999999

   1. Initial program 94.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ 94.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

   4. Taylor expanded in x around 0 91.7%

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

      1. associate-*r/ 91.8%

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

      2. associate-*l/ 91.9%

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

      3. *-un-lft-identity 91.9%

         \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

   6. Applied egg-rr 91.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 1.3999999999999999 < x

   1. Initial program 77.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

   2. Taylor expanded in x around 0 30.5%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

   3. Taylor expanded in x around inf 30.5%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-/l* 24.8%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{x}{\frac{z}{y}}} \]

      2. associate-/r/ 34.8%

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

   5. Simplified 34.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(\frac{x}{z} \cdot y\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \]

Alternative 14: 65.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* y (+ (* x 0.5) (/ 1.0 x))) z))

C

double code(double x, double y, double z) {
	return (y * ((x * 0.5) + (1.0 / x))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * ((x * 0.5d0) + (1.0d0 / x))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (y * ((x * 0.5) + (1.0 / x))) / z;
}

Python

def code(x, y, z):
	return (y * ((x * 0.5) + (1.0 / x))) / z

Julia

function code(x, y, z)
	return Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

2. Taylor expanded in x around 0 61.9%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

3. Taylor expanded in y around 0 61.8%

   \[\leadsto \color{blue}{\frac{y \cdot \left(0.5 \cdot x + \frac{1}{x}\right)}{z}} \]

4. Final simplification 61.8%

   \[\leadsto \frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z} \]

Alternative 15: 65.7% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ (/ y x) (* 0.5 (* x y))) z))

C

double code(double x, double y, double z) {
	return ((y / x) + (0.5 * (x * y))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((y / x) + (0.5d0 * (x * y))) / z
end function

Java

public static double code(double x, double y, double z) {
	return ((y / x) + (0.5 * (x * y))) / z;
}

Python

def code(x, y, z):
	return ((y / x) + (0.5 * (x * y))) / z

Julia

function code(x, y, z)
	return Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((y / x) + (0.5 * (x * y))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]

2. Taylor expanded in x around 0 61.9%

   \[\leadsto \frac{\color{blue}{0.5 \cdot \left(x \cdot y\right) + \frac{y}{x}}}{z} \]

3. Final simplification 61.9%

   \[\leadsto \frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z} \]

Alternative 16: 49.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e-167) (/ y (* x z)) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-167) {
		tmp = y / (x * z);
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1d-167)) then
        tmp = y / (x * z)
    else
        tmp = (y / x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e-167) {
		tmp = y / (x * z);
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e-167:
		tmp = y / (x * z)
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e-167)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e-167)
		tmp = y / (x * z);
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e-167], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1e-167

   1. Initial program 88.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 77.5%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 76.4%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 76.4%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]

   4. Taylor expanded in x around 0 46.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   if -1e-167 < z

   1. Initial program 84.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Step-by-step derivation

      1. associate-*r/ 81.9%

         \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* 80.4%

         \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

   3. Simplified 80.4%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]

   4. Taylor expanded in x around 0 44.7%

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

      1. associate-/r* 49.5%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{z}} \]

   6. Simplified 49.5%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 17: 48.4% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{y}{x \cdot z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ y (* x z)))

C

double code(double x, double y, double z) {
	return y / (x * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = y / (x * z)
end function

Java

public static double code(double x, double y, double z) {
	return y / (x * z);
}

Python

def code(x, y, z):
	return y / (x * z)

Julia

function code(x, y, z)
	return Float64(y / Float64(x * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = y / (x * z);
end

Wolfram

code[x_, y_, z_] := N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Step-by-step derivation

   1. associate-*r/ 80.1%

      \[\leadsto \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

   2. associate-/r* 78.7%

      \[\leadsto \cosh x \cdot \color{blue}{\frac{y}{x \cdot z}} \]

3. Simplified 78.7%

   \[\leadsto \color{blue}{\cosh x \cdot \frac{y}{x \cdot z}} \]

4. Taylor expanded in x around 0 45.5%

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

5. Final simplification 45.5%

   \[\leadsto \frac{y}{x \cdot z} \]

Alternative 18: 53.1% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{y}{z}}{x} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (/ y z) x))

C

double code(double x, double y, double z) {
	return (y / z) / x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y / z) / x
end function

Java

public static double code(double x, double y, double z) {
	return (y / z) / x;
}

Python

def code(x, y, z):
	return (y / z) / x

Julia

function code(x, y, z)
	return Float64(Float64(y / z) / x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y / z) / x;
end

Wolfram

code[x_, y_, z_] := N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 86.4%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Step-by-step derivation

   1. associate-*l/ 86.3%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

3. Simplified 86.3%

   \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]

4. Taylor expanded in x around 0 46.4%

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

   1. associate-*r/ 48.7%

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

   2. associate-*l/ 48.7%

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

   3. *-un-lft-identity 48.7%

      \[\leadsto \frac{\frac{\color{blue}{y}}{z}}{x} \]

6. Applied egg-rr 48.7%

   \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

7. Final simplification 48.7%

   \[\leadsto \frac{\frac{y}{z}}{x} \]

Developer target: 96.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2023293 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< y -4.618902267687042e-52) (* (/ (/ y z) x) (cosh x)) (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x))))

  (/ (* (cosh x) (/ y x)) z))