Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 99.3%

Time: 8.4s

Alternatives: 15

Speedup: 1.0×

• 63.9% 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: 84.7% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cosh x}{x}\\ \mathbf{if}\;y \leq -4000000:\\ \;\;\;\;\frac{y}{z} \cdot t_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\frac{y \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (cosh x) x)))
   (if (<= y -4000000.0)
     (* (/ y z) t_0)
     (if (<= y 2e+125) (/ (* y t_0) z) (* (cosh x) (/ (/ y z) x))))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) / x;
	double tmp;
	if (y <= -4000000.0) {
		tmp = (y / z) * t_0;
	} else if (y <= 2e+125) {
		tmp = (y * t_0) / z;
	} else {
		tmp = cosh(x) * ((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) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) / x
    if (y <= (-4000000.0d0)) then
        tmp = (y / z) * t_0
    else if (y <= 2d+125) then
        tmp = (y * t_0) / z
    else
        tmp = cosh(x) * ((y / 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) / x;
	double tmp;
	if (y <= -4000000.0) {
		tmp = (y / z) * t_0;
	} else if (y <= 2e+125) {
		tmp = (y * t_0) / z;
	} else {
		tmp = Math.cosh(x) * ((y / z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) / x
	tmp = 0
	if y <= -4000000.0:
		tmp = (y / z) * t_0
	elif y <= 2e+125:
		tmp = (y * t_0) / z
	else:
		tmp = math.cosh(x) * ((y / z) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) / x)
	tmp = 0.0
	if (y <= -4000000.0)
		tmp = Float64(Float64(y / z) * t_0);
	elseif (y <= 2e+125)
		tmp = Float64(Float64(y * t_0) / z);
	else
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) / x;
	tmp = 0.0;
	if (y <= -4000000.0)
		tmp = (y / z) * t_0;
	elseif (y <= 2e+125)
		tmp = (y * t_0) / z;
	else
		tmp = cosh(x) * ((y / z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -4000000.0], N[(N[(y / z), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 2e+125], N[(N[(y * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4e6

   1. Initial program 88.7%

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

      1. associate-/l* 84.2%

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

      2. associate-/r/ 95.3%

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

   3. Simplified 95.3%

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

      1. *-un-lft-identity 95.3%

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

      2. times-frac 99.8%

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

      3. clear-num 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -4e6 < y < 1.9999999999999998e125

   1. Initial program 80.7%

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

      1. associate-*r/ 71.5%

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

      2. associate-/l/ 74.1%

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

      3. associate-/r* 73.8%

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

   3. Simplified 73.8%

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

      1. associate-/l/ 74.1%

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

      2. associate-*r/ 80.4%

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

      3. associate-*l/ 80.3%

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

      4. associate-/r* 95.2%

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

      5. associate-*l/ 99.7%

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

   5. Applied egg-rr 99.7%

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

   if 1.9999999999999998e125 < y

   1. Initial program 87.9%

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

      1. associate-*r/ 87.9%

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

      2. associate-/l/ 89.2%

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

      3. associate-/r* 100.0%

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

   3. Simplified 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4000000:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\cosh x}{x}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+125}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 2: 93.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\cosh x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 2e+223) (/ t_0 z) (* (/ y z) (/ (cosh x) x)))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 2e+223) {
		tmp = t_0 / z;
	} else {
		tmp = (y / z) * (cosh(x) / 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+223) then
        tmp = t_0 / z
    else
        tmp = (y / z) * (cosh(x) / 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+223) {
		tmp = t_0 / z;
	} else {
		tmp = (y / z) * (Math.cosh(x) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if t_0 <= 2e+223:
		tmp = t_0 / z
	else:
		tmp = (y / z) * (math.cosh(x) / x)
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= 2e+223)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(y / z) * Float64(cosh(x) / 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+223)
		tmp = t_0 / z;
	else
		tmp = (y / z) * (cosh(x) / 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+223], N[(t$95$0 / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2.00000000000000009e223

   1. Initial program 96.7%

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

   if 2.00000000000000009e223 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 64.6%

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

      1. associate-/l* 56.5%

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

      2. associate-/r/ 77.0%

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

   3. Simplified 77.0%

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

      1. *-un-lft-identity 77.0%

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

      2. times-frac 89.9%

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

      3. clear-num 90.9%

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

   5. Applied egg-rr 90.9%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{\cosh x}{x}\\ \end{array} \]

Alternative 3: 89.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{if}\;y \leq -1.9 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ (/ y z) x))))
   (if (<= y -1.9e+51)
     t_0
     (if (<= y -1.35e-142)
       (* (/ y x) (/ (cosh x) z))
       (if (<= y 2.05e-76) (* y (/ (cosh x) (* z x))) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * ((y / z) / x);
	double tmp;
	if (y <= -1.9e+51) {
		tmp = t_0;
	} else if (y <= -1.35e-142) {
		tmp = (y / x) * (cosh(x) / z);
	} else if (y <= 2.05e-76) {
		tmp = y * (cosh(x) / (z * x));
	} 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 = cosh(x) * ((y / z) / x)
    if (y <= (-1.9d+51)) then
        tmp = t_0
    else if (y <= (-1.35d-142)) then
        tmp = (y / x) * (cosh(x) / z)
    else if (y <= 2.05d-76) then
        tmp = y * (cosh(x) / (z * x))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cosh(x) * ((y / z) / x);
	double tmp;
	if (y <= -1.9e+51) {
		tmp = t_0;
	} else if (y <= -1.35e-142) {
		tmp = (y / x) * (Math.cosh(x) / z);
	} else if (y <= 2.05e-76) {
		tmp = y * (Math.cosh(x) / (z * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * ((y / z) / x)
	tmp = 0
	if y <= -1.9e+51:
		tmp = t_0
	elif y <= -1.35e-142:
		tmp = (y / x) * (math.cosh(x) / z)
	elif y <= 2.05e-76:
		tmp = y * (math.cosh(x) / (z * x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(Float64(y / z) / x))
	tmp = 0.0
	if (y <= -1.9e+51)
		tmp = t_0;
	elseif (y <= -1.35e-142)
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	elseif (y <= 2.05e-76)
		tmp = Float64(y * Float64(cosh(x) / Float64(z * x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) * ((y / z) / x);
	tmp = 0.0;
	if (y <= -1.9e+51)
		tmp = t_0;
	elseif (y <= -1.35e-142)
		tmp = (y / x) * (cosh(x) / z);
	elseif (y <= 2.05e-76)
		tmp = y * (cosh(x) / (z * x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.9e+51], t$95$0, If[LessEqual[y, -1.35e-142], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.05e-76], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8999999999999999e51 or 2.0499999999999999e-76 < y

   1. Initial program 90.3%

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

      1. associate-*r/ 87.4%

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

      2. associate-/l/ 87.5%

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

      3. associate-/r* 97.0%

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

   3. Simplified 97.0%

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

   if -1.8999999999999999e51 < y < -1.3499999999999999e-142

   1. Initial program 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-*r/ 95.3%

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

   3. Simplified 95.3%

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

   if -1.3499999999999999e-142 < y < 2.0499999999999999e-76

   1. Initial program 65.5%

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

      1. associate-*r/ 99.7%

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

      2. associate-/l/ 81.0%

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

      3. associate-*l/ 80.9%

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

      4. *-commutative 80.9%

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

      5. *-commutative 80.9%

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

   3. Simplified 80.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+51}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;y \leq -1.35 \cdot 10^{-142}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-76}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 4: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-53} \lor \neg \left(y \leq 2 \cdot 10^{-76}\right):\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2e-53) (not (<= y 2e-76)))
   (* (cosh x) (/ (/ y z) x))
   (* y (/ (cosh x) (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2e-53) || !(y <= 2e-76)) {
		tmp = cosh(x) * ((y / z) / x);
	} 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) :: tmp
    if ((y <= (-2d-53)) .or. (.not. (y <= 2d-76))) then
        tmp = cosh(x) * ((y / z) / x)
    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 tmp;
	if ((y <= -2e-53) || !(y <= 2e-76)) {
		tmp = Math.cosh(x) * ((y / z) / x);
	} else {
		tmp = y * (Math.cosh(x) / (z * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2e-53) or not (y <= 2e-76):
		tmp = math.cosh(x) * ((y / z) / x)
	else:
		tmp = y * (math.cosh(x) / (z * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2e-53) || !(y <= 2e-76))
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / x));
	else
		tmp = Float64(y * Float64(cosh(x) / Float64(z * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2e-53) || ~((y <= 2e-76)))
		tmp = cosh(x) * ((y / z) / x);
	else
		tmp = y * (cosh(x) / (z * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2e-53], N[Not[LessEqual[y, 2e-76]], $MachinePrecision]], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.00000000000000006e-53 or 1.99999999999999985e-76 < y

   1. Initial program 91.9%

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

      1. associate-*r/ 86.4%

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

      2. associate-/l/ 86.5%

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

      3. associate-/r* 94.4%

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

   3. Simplified 94.4%

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

   if -2.00000000000000006e-53 < y < 1.99999999999999985e-76

   1. Initial program 70.3%

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

      1. associate-*r/ 99.7%

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

      2. associate-/l/ 80.7%

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

      3. associate-*l/ 80.6%

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

      4. *-commutative 80.6%

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

      5. *-commutative 80.6%

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

   3. Simplified 80.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-53} \lor \neg \left(y \leq 2 \cdot 10^{-76}\right):\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \end{array} \]

Alternative 5: 83.5% accurate, 1.0× speedup

Math

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

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(cosh(x) / Float64(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[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

   1. associate-*r/ 94.7%

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

   2. associate-/l/ 84.4%

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

   3. associate-*l/ 84.3%

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

   4. *-commutative 84.3%

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

   5. *-commutative 84.3%

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

3. Simplified 84.3%

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

4. Final simplification 84.3%

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

Alternative 6: 90.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. associate-/l* 77.4%

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

   2. associate-/r/ 83.3%

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

3. Simplified 83.3%

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

   1. *-un-lft-identity 83.3%

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

   2. times-frac 91.4%

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

   3. clear-num 92.4%

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

5. Applied egg-rr 92.4%

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

6. Final simplification 92.4%

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

Alternative 7: 68.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{0.5}{\frac{z}{x \cdot x}}\\ \mathbf{if}\;x \leq -5 \cdot 10^{+91}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+169}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+241}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ y x) (/ 0.5 (/ z (* x x))))))
   (if (<= x -5e+91)
     t_0
     (if (<= x 2.75e+169)
       (* y (/ (+ (* x 0.5) (/ 1.0 x)) z))
       (if (<= x 1.95e+241) t_0 (* 0.5 (/ y (/ z x))))))))

C

double code(double x, double y, double z) {
	double t_0 = (y / x) * (0.5 / (z / (x * x)));
	double tmp;
	if (x <= -5e+91) {
		tmp = t_0;
	} else if (x <= 2.75e+169) {
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	} else if (x <= 1.95e+241) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (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) :: t_0
    real(8) :: tmp
    t_0 = (y / x) * (0.5d0 / (z / (x * x)))
    if (x <= (-5d+91)) then
        tmp = t_0
    else if (x <= 2.75d+169) then
        tmp = y * (((x * 0.5d0) + (1.0d0 / x)) / z)
    else if (x <= 1.95d+241) then
        tmp = t_0
    else
        tmp = 0.5d0 * (y / (z / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / x) * (0.5 / (z / (x * x)));
	double tmp;
	if (x <= -5e+91) {
		tmp = t_0;
	} else if (x <= 2.75e+169) {
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	} else if (x <= 1.95e+241) {
		tmp = t_0;
	} else {
		tmp = 0.5 * (y / (z / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / x) * (0.5 / (z / (x * x)))
	tmp = 0
	if x <= -5e+91:
		tmp = t_0
	elif x <= 2.75e+169:
		tmp = y * (((x * 0.5) + (1.0 / x)) / z)
	elif x <= 1.95e+241:
		tmp = t_0
	else:
		tmp = 0.5 * (y / (z / x))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / x) * Float64(0.5 / Float64(z / Float64(x * x))))
	tmp = 0.0
	if (x <= -5e+91)
		tmp = t_0;
	elseif (x <= 2.75e+169)
		tmp = Float64(y * Float64(Float64(Float64(x * 0.5) + Float64(1.0 / x)) / z));
	elseif (x <= 1.95e+241)
		tmp = t_0;
	else
		tmp = Float64(0.5 * Float64(y / Float64(z / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / x) * (0.5 / (z / (x * x)));
	tmp = 0.0;
	if (x <= -5e+91)
		tmp = t_0;
	elseif (x <= 2.75e+169)
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	elseif (x <= 1.95e+241)
		tmp = t_0;
	else
		tmp = 0.5 * (y / (z / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] * N[(0.5 / N[(z / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5e+91], t$95$0, If[LessEqual[x, 2.75e+169], N[(y * N[(N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e+241], t$95$0, N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.0000000000000002e91 or 2.74999999999999986e169 < x < 1.95000000000000013e241

   1. Initial program 80.6%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 64.2%

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

      3. associate-*l/ 64.2%

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

      4. *-commutative 64.2%

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

      5. *-commutative 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in x around 0 54.2%

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

      1. associate-/r* 54.2%

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

      2. associate-*r/ 54.2%

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

      3. frac-add 57.1%

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

      4. *-commutative 57.1%

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

   6. Applied egg-rr 57.1%

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

   7. Taylor expanded in y around 0 57.1%

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

      1. times-frac 73.4%

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

      2. unpow2 73.4%

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

   9. Simplified 73.4%

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

   10. Taylor expanded in x around inf 73.4%

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

       1. associate-*r/ 73.4%

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

       2. associate-/l* 73.4%

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

       3. unpow2 73.4%

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

   12. Simplified 73.4%

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

   if -5.0000000000000002e91 < x < 2.74999999999999986e169

   1. Initial program 88.7%

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

      1. associate-*r/ 92.1%

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

      2. associate-/l/ 93.7%

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

      3. associate-*l/ 93.6%

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

      4. *-commutative 93.6%

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

      5. *-commutative 93.6%

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

   3. Simplified 93.6%

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

   4. Taylor expanded in x around 0 73.7%

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

   5. Taylor expanded in z around 0 73.7%

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

   if 1.95000000000000013e241 < x

   1. Initial program 46.7%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 66.7%

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

      3. associate-*l/ 66.7%

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

      4. *-commutative 66.7%

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

      5. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x around 0 61.5%

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

   5. Taylor expanded in x around inf 55.7%

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

      1. associate-/l* 61.5%

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

   7. Simplified 61.5%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+91}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{0.5}{\frac{z}{x \cdot x}}\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+169}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+241}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{0.5}{\frac{z}{x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 8: 69.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+183} \lor \neg \left(z \leq 3.2 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -9.5e+183) (not (<= z 3.2e+104)))
   (* (/ y x) (/ (+ 1.0 (* 0.5 (* x x))) z))
   (* y (/ (+ 1.0 (* x (* x 0.5))) (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.5e+183) || !(z <= 3.2e+104)) {
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	} else {
		tmp = y * ((1.0 + (x * (x * 0.5))) / (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 ((z <= (-9.5d+183)) .or. (.not. (z <= 3.2d+104))) then
        tmp = (y / x) * ((1.0d0 + (0.5d0 * (x * x))) / z)
    else
        tmp = y * ((1.0d0 + (x * (x * 0.5d0))) / (z * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -9.5e+183) || !(z <= 3.2e+104)) {
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	} else {
		tmp = y * ((1.0 + (x * (x * 0.5))) / (z * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -9.5e+183) or not (z <= 3.2e+104):
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z)
	else:
		tmp = y * ((1.0 + (x * (x * 0.5))) / (z * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -9.5e+183) || !(z <= 3.2e+104))
		tmp = Float64(Float64(y / x) * Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / z));
	else
		tmp = Float64(y * Float64(Float64(1.0 + Float64(x * Float64(x * 0.5))) / Float64(z * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -9.5e+183) || ~((z <= 3.2e+104)))
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	else
		tmp = y * ((1.0 + (x * (x * 0.5))) / (z * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -9.5e+183], N[Not[LessEqual[z, 3.2e+104]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.5000000000000003e183 or 3.2e104 < z

   1. Initial program 77.9%

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

      1. associate-*r/ 88.5%

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

      2. associate-/l/ 60.4%

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

      3. associate-*l/ 60.3%

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

      4. *-commutative 60.3%

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

      5. *-commutative 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in x around 0 48.1%

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

      1. associate-/r* 48.1%

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

      2. associate-*r/ 48.1%

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

      3. frac-add 45.2%

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

      4. *-commutative 45.2%

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

   6. Applied egg-rr 45.2%

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

   7. Taylor expanded in y around 0 46.7%

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

      1. times-frac 59.7%

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

      2. unpow2 59.7%

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

   9. Simplified 59.7%

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

   if -9.5000000000000003e183 < z < 3.2e104

   1. Initial program 86.3%

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

      1. associate-*r/ 96.8%

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

      2. associate-/l/ 92.7%

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

      3. associate-*l/ 92.7%

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

      4. *-commutative 92.7%

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

      5. *-commutative 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in x around 0 74.8%

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

      1. associate-/r* 74.8%

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

      2. associate-*r/ 74.8%

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

      3. frac-add 77.1%

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

      4. *-commutative 77.1%

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

   6. Applied egg-rr 77.1%

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

   7. Taylor expanded in z around 0 77.1%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+183} \lor \neg \left(z \leq 3.2 \cdot 10^{+104}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{z \cdot x}\\ \end{array} \]

Alternative 9: 68.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e+183)
   (/ (+ (/ y x) (* 0.5 (* y x))) z)
   (* y (/ (+ 1.0 (* x (* x 0.5))) (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+183) {
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	} else {
		tmp = y * ((1.0 + (x * (x * 0.5))) / (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 (z <= (-1.9d+183)) then
        tmp = ((y / x) + (0.5d0 * (y * x))) / z
    else
        tmp = y * ((1.0d0 + (x * (x * 0.5d0))) / (z * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+183) {
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	} else {
		tmp = y * ((1.0 + (x * (x * 0.5))) / (z * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.9e+183:
		tmp = ((y / x) + (0.5 * (y * x))) / z
	else:
		tmp = y * ((1.0 + (x * (x * 0.5))) / (z * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e+183)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(y * x))) / z);
	else
		tmp = Float64(y * Float64(Float64(1.0 + Float64(x * Float64(x * 0.5))) / Float64(z * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.9e+183)
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	else
		tmp = y * ((1.0 + (x * (x * 0.5))) / (z * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e+183], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9e183

   1. Initial program 83.2%

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

   2. Taylor expanded in x around 0 56.3%

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

   if -1.9e183 < z

   1. Initial program 84.2%

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

      1. associate-*r/ 94.9%

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

      2. associate-/l/ 88.0%

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

      3. associate-*l/ 87.9%

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

      4. *-commutative 87.9%

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

      5. *-commutative 87.9%

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

   3. Simplified 87.9%

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

   4. Taylor expanded in x around 0 70.7%

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

      1. associate-/r* 70.8%

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

      2. associate-*r/ 70.8%

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

      3. frac-add 72.0%

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

      4. *-commutative 72.0%

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

   6. Applied egg-rr 72.0%

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

   7. Taylor expanded in z around 0 72.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+183}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{z \cdot x}\\ \end{array} \]

Alternative 10: 61.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.42\right):\\ \;\;\;\;0.5 \cdot \left(\frac{y}{z} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.42))) (* 0.5 (* (/ y z) x)) (/ y (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * ((y / z) * x);
	} 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 <= (-1.4d0)) .or. (.not. (x <= 1.42d0))) then
        tmp = 0.5d0 * ((y / z) * x)
    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 <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * ((y / z) * x);
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.42):
		tmp = 0.5 * ((y / z) * x)
	else:
		tmp = y / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.42))
		tmp = Float64(0.5 * Float64(Float64(y / z) * x));
	else
		tmp = Float64(y / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.42)))
		tmp = 0.5 * ((y / z) * x);
	else
		tmp = y / (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4], N[Not[LessEqual[x, 1.42]], $MachinePrecision]], N[(0.5 * N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.4199999999999999 < x

   1. Initial program 79.4%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 74.8%

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

      3. associate-*l/ 74.8%

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

      4. *-commutative 74.8%

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

      5. *-commutative 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in x around 0 43.5%

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

      1. associate-/r* 43.5%

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

      2. associate-*r/ 43.5%

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

      3. frac-add 45.4%

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

      4. *-commutative 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in x around inf 44.3%

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

      1. associate-/l* 42.8%

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

      2. associate-/r/ 36.3%

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

   9. Simplified 36.3%

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

   if -1.3999999999999999 < x < 1.4199999999999999

   1. Initial program 89.1%

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

      1. associate-*r/ 89.1%

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

      2. associate-/l/ 94.4%

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

      3. associate-/r* 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in x around 0 92.5%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.42\right):\\ \;\;\;\;0.5 \cdot \left(\frac{y}{z} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 11: 65.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.42\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.42))) (* 0.5 (/ y (/ z x))) (/ y (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * (y / (z / x));
	} 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 <= (-1.4d0)) .or. (.not. (x <= 1.42d0))) then
        tmp = 0.5d0 * (y / (z / x))
    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 <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * (y / (z / x));
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.42):
		tmp = 0.5 * (y / (z / x))
	else:
		tmp = y / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.42))
		tmp = Float64(0.5 * Float64(y / Float64(z / x)));
	else
		tmp = Float64(y / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.42)))
		tmp = 0.5 * (y / (z / x));
	else
		tmp = y / (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4], N[Not[LessEqual[x, 1.42]], $MachinePrecision]], N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.4199999999999999 < x

   1. Initial program 79.4%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 74.8%

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

      3. associate-*l/ 74.8%

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

      4. *-commutative 74.8%

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

      5. *-commutative 74.8%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in x around 0 43.5%

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

   5. Taylor expanded in x around inf 44.3%

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

      1. associate-/l* 42.8%

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

   7. Simplified 42.8%

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

   if -1.3999999999999999 < x < 1.4199999999999999

   1. Initial program 89.1%

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

      1. associate-*r/ 89.1%

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

      2. associate-/l/ 94.4%

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

      3. associate-/r* 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in x around 0 92.5%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.42\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 12: 65.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4)
   (* 0.5 (/ y (/ z x)))
   (if (<= x 1.42) (/ y (* z x)) (* 0.5 (/ (* y x) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4) {
		tmp = 0.5 * (y / (z / x));
	} else if (x <= 1.42) {
		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 <= (-1.4d0)) then
        tmp = 0.5d0 * (y / (z / x))
    else if (x <= 1.42d0) 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 <= -1.4) {
		tmp = 0.5 * (y / (z / x));
	} else if (x <= 1.42) {
		tmp = y / (z * x);
	} else {
		tmp = 0.5 * ((y * x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4:
		tmp = 0.5 * (y / (z / x))
	elif x <= 1.42:
		tmp = y / (z * x)
	else:
		tmp = 0.5 * ((y * x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(0.5 * Float64(y / Float64(z / x)));
	elseif (x <= 1.42)
		tmp = Float64(y / Float64(z * x));
	else
		tmp = Float64(0.5 * Float64(Float64(y * x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4)
		tmp = 0.5 * (y / (z / x));
	elseif (x <= 1.42)
		tmp = y / (z * x);
	else
		tmp = 0.5 * ((y * x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4], N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.42], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 83.8%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 73.5%

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

      3. associate-*l/ 73.5%

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

      4. *-commutative 73.5%

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

      5. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in x around 0 46.3%

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

   5. Taylor expanded in x around inf 44.9%

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

      1. associate-/l* 46.3%

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

   7. Simplified 46.3%

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

   if -1.3999999999999999 < x < 1.4199999999999999

   1. Initial program 89.1%

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

      1. associate-*r/ 89.1%

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

      2. associate-/l/ 94.4%

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

      3. associate-/r* 92.5%

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

   3. Simplified 92.5%

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

   4. Taylor expanded in x around 0 92.5%

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

   if 1.4199999999999999 < x

   1. Initial program 74.6%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 76.2%

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

      3. associate-*l/ 76.2%

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

      4. *-commutative 76.2%

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

      5. *-commutative 76.2%

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

   3. Simplified 76.2%

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

   4. Taylor expanded in x around 0 40.4%

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

   5. Taylor expanded in x around inf 43.6%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z}\\ \end{array} \]

Alternative 13: 65.5% accurate, 9.7× speedup

Math

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

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(y * Float64(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[(y * N[(N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

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

   1. associate-*r/ 94.7%

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

   2. associate-/l/ 84.4%

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

   3. associate-*l/ 84.3%

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

   4. *-commutative 84.3%

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

   5. *-commutative 84.3%

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

3. Simplified 84.3%

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

4. Taylor expanded in x around 0 67.9%

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

5. Taylor expanded in z around 0 67.9%

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

6. Final simplification 67.9%

   \[\leadsto y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z} \]

Alternative 14: 56.9% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-21} \lor \neg \left(z \leq 6.5 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -5e-21) (not (<= z 6.5e-25))) (/ y (* z x)) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -5e-21) || !(z <= 6.5e-25)) {
		tmp = y / (z * x);
	} 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 ((z <= (-5d-21)) .or. (.not. (z <= 6.5d-25))) then
        tmp = y / (z * x)
    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 ((z <= -5e-21) || !(z <= 6.5e-25)) {
		tmp = y / (z * x);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -5e-21) or not (z <= 6.5e-25):
		tmp = y / (z * x)
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -5e-21) || !(z <= 6.5e-25))
		tmp = Float64(y / Float64(z * x));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -5e-21) || ~((z <= 6.5e-25)))
		tmp = y / (z * x);
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -5e-21], N[Not[LessEqual[z, 6.5e-25]], $MachinePrecision]], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.99999999999999973e-21 or 6.5e-25 < z

   1. Initial program 80.1%

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

      1. associate-*r/ 68.8%

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

      2. associate-/l/ 69.7%

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

      3. associate-/r* 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in x around 0 52.0%

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

   if -4.99999999999999973e-21 < z < 6.5e-25

   1. Initial program 88.4%

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

      1. associate-*r/ 99.8%

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

      2. associate-/l/ 94.6%

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

      3. associate-*l/ 94.5%

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

      4. *-commutative 94.5%

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

      5. *-commutative 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in x around 0 45.5%

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

      1. div-inv 45.5%

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

      2. associate-/r* 63.9%

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

   6. Applied egg-rr 63.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-21} \lor \neg \left(z \leq 6.5 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 15: 49.4% accurate, 21.4× speedup

Math

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

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(y / Float64(z * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. associate-*r/ 78.3%

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

   2. associate-/l/ 80.9%

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

   3. associate-/r* 84.6%

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

3. Simplified 84.6%

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

4. Taylor expanded in x around 0 48.9%

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

5. Final simplification 48.9%

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

Developer target: 97.1% 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 2023240 
(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))