Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 96.4%

Time: 11.5s

Alternatives: 13

Speedup: 1.0×

• 64.0% 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: 84.5% 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: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. *-commutative 81.5%

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

   2. associate-*l/ 92.8%

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

   3. associate-/l* 92.8%

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

   4. associate-/l* 95.8%

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

   5. associate-/r* 82.9%

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

3. Simplified 82.9%

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

   1. associate-*r/ 83.5%

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

   2. *-commutative 83.5%

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

   3. *-commutative 83.5%

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

   4. associate-/r* 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

   \[\leadsto \frac{\frac{\cosh x \cdot y}{z}}{x} \]
8. Add Preprocessing

Alternative 2: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+237}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 1.25e-147)
   (/ (/ y z) x)
   (if (<= x 3.15e+237)
     (* y (/ (cosh x) (* x z)))
     (* y (+ (* 0.5 (/ x z)) (/ 1.0 (* x z)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.25e-147) {
		tmp = (y / z) / x;
	} else if (x <= 3.15e+237) {
		tmp = y * (cosh(x) / (x * z));
	} else {
		tmp = y * ((0.5 * (x / z)) + (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 (x <= 1.25d-147) then
        tmp = (y / z) / x
    else if (x <= 3.15d+237) then
        tmp = y * (cosh(x) / (x * z))
    else
        tmp = y * ((0.5d0 * (x / z)) + (1.0d0 / (x * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 1.25e-147) {
		tmp = (y / z) / x;
	} else if (x <= 3.15e+237) {
		tmp = y * (Math.cosh(x) / (x * z));
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 1.25e-147:
		tmp = (y / z) / x
	elif x <= 3.15e+237:
		tmp = y * (math.cosh(x) / (x * z))
	else:
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 1.25e-147)
		tmp = Float64(Float64(y / z) / x);
	elseif (x <= 3.15e+237)
		tmp = Float64(y * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(y * Float64(Float64(0.5 * Float64(x / z)) + Float64(1.0 / Float64(x * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 1.25e-147)
		tmp = (y / z) / x;
	elseif (x <= 3.15e+237)
		tmp = y * (cosh(x) / (x * z));
	else
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 1.25e-147], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 3.15e+237], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.25000000000000003e-147

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-*l/ 90.5%

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

      3. associate-/l* 90.5%

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

      4. associate-/l* 93.7%

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

      5. associate-/r* 84.6%

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

   3. Simplified 84.6%

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

      1. associate-*r/ 85.2%

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

      2. *-commutative 85.2%

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

      3. *-commutative 85.2%

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

      4. associate-/r* 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0 63.4%

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

   if 1.25000000000000003e-147 < x < 3.15000000000000004e237

   1. Initial program 86.7%

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

      1. *-commutative 86.7%

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

      2. associate-*l/ 95.6%

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

      3. associate-/l* 95.6%

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

      4. associate-/l* 98.7%

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

      5. associate-/r* 84.3%

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

   3. Simplified 84.3%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   if 3.15000000000000004e237 < x

   1. Initial program 61.5%

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

      1. *-commutative 61.5%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 53.8%

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

   3. Simplified 53.8%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 85.0%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.25 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 3.15 \cdot 10^{+237}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.5%

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

   1. associate-*r/ 92.8%

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

   2. associate-/r* 83.5%

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

   3. times-frac 92.5%

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

4. Applied egg-rr 92.5%

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

5. Final simplification 92.5%

   \[\leadsto \frac{\cosh x}{x} \cdot \frac{y}{z} \]
6. Add Preprocessing

Alternative 4: 66.0% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+187}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{z} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 7.2e+187)
   (+ (* 0.5 (/ (* x y) z)) (* (/ y z) (/ 1.0 x)))
   (* 0.5 (* y (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 7.2e+187) {
		tmp = (0.5 * ((x * y) / z)) + ((y / z) * (1.0 / 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 <= 7.2d+187) then
        tmp = (0.5d0 * ((x * y) / z)) + ((y / z) * (1.0d0 / 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 <= 7.2e+187) {
		tmp = (0.5 * ((x * y) / z)) + ((y / z) * (1.0 / x));
	} else {
		tmp = 0.5 * (y * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 7.2e+187:
		tmp = (0.5 * ((x * y) / z)) + ((y / z) * (1.0 / x))
	else:
		tmp = 0.5 * (y * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 7.2e+187)
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z)) + Float64(Float64(y / z) * Float64(1.0 / 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 <= 7.2e+187)
		tmp = (0.5 * ((x * y) / z)) + ((y / z) * (1.0 / x));
	else
		tmp = 0.5 * (y * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 7.2e+187], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(N[(y / z), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.20000000000000072e187

   1. Initial program 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-*l/ 92.0%

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

      3. associate-/l* 92.0%

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

      4. associate-/l* 95.3%

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

      5. associate-/r* 85.0%

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

   3. Simplified 85.0%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.4%

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

      1. *-rgt-identity 67.4%

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

      2. *-commutative 67.4%

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

      3. times-frac 69.5%

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

   7. Applied egg-rr 69.5%

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

   if 7.20000000000000072e187 < x

   1. Initial program 64.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.3%

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

   4. Taylor expanded in x around inf 54.3%

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

      1. associate-*r/ 54.3%

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

      2. *-commutative 54.3%

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

      3. associate-*l/ 54.3%

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

      4. associate-*r/ 65.3%

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

   6. Simplified 65.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+187}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{z} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 9.5e-149) (/ (/ y z) x) (* y (+ (* 0.5 (/ x z)) (/ 1.0 (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.5e-149) {
		tmp = (y / z) / x;
	} else {
		tmp = y * ((0.5 * (x / z)) + (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 (x <= 9.5d-149) then
        tmp = (y / z) / x
    else
        tmp = y * ((0.5d0 * (x / z)) + (1.0d0 / (x * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 9.5e-149) {
		tmp = (y / z) / x;
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 9.5e-149:
		tmp = (y / z) / x
	else:
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 9.5e-149)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(y * Float64(Float64(0.5 * Float64(x / z)) + Float64(1.0 / Float64(x * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 9.5e-149)
		tmp = (y / z) / x;
	else
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 9.5e-149], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.50000000000000034e-149

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-*l/ 90.5%

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

      3. associate-/l* 90.5%

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

      4. associate-/l* 93.7%

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

      5. associate-/r* 84.6%

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

   3. Simplified 84.6%

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

      1. associate-*r/ 85.2%

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

      2. *-commutative 85.2%

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

      3. *-commutative 85.2%

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

      4. associate-/r* 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0 63.4%

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

   if 9.50000000000000034e-149 < x

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l/ 96.1%

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

      3. associate-/l* 96.2%

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

      4. associate-/l* 98.9%

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

      5. associate-/r* 80.4%

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.3%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 59.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 3e-148) (/ (/ y z) x) (+ (* 0.5 (/ y (/ z x))) (/ y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e-148) {
		tmp = (y / z) / x;
	} else {
		tmp = (0.5 * (y / (z / x))) + (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 <= 3d-148) then
        tmp = (y / z) / x
    else
        tmp = (0.5d0 * (y / (z / x))) + (y / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 3e-148) {
		tmp = (y / z) / x;
	} else {
		tmp = (0.5 * (y / (z / x))) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 3e-148:
		tmp = (y / z) / x
	else:
		tmp = (0.5 * (y / (z / x))) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 3e-148)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(0.5 * Float64(y / Float64(z / x))) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 3e-148)
		tmp = (y / z) / x;
	else
		tmp = (0.5 * (y / (z / x))) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 3e-148], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999998e-148

   1. Initial program 80.1%

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

      1. *-commutative 80.1%

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

      2. associate-*l/ 90.5%

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

      3. associate-/l* 90.5%

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

      4. associate-/l* 93.7%

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

      5. associate-/r* 84.6%

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

   3. Simplified 84.6%

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

      1. associate-*r/ 85.2%

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

      2. *-commutative 85.2%

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

      3. *-commutative 85.2%

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

      4. associate-/r* 97.9%

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

   6. Applied egg-rr 97.9%

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

   7. Taylor expanded in x around 0 63.4%

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

   if 2.99999999999999998e-148 < x

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l/ 96.1%

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

      3. associate-/l* 96.2%

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

      4. associate-/l* 98.9%

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

      5. associate-/r* 80.4%

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

   3. Simplified 80.4%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 59.3%

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

      1. div-inv 59.3%

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

      2. *-commutative 59.3%

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

      3. associate-*l* 61.1%

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

   7. Applied egg-rr 61.1%

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

      1. un-div-inv 61.1%

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

      2. clear-num 61.1%

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

   9. Applied egg-rr 61.1%

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

       1. un-div-inv 61.1%

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

   11. Applied egg-rr 61.1%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq 2 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + t\_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (* x z))))
   (if (<= z 2e-72)
     (+ (* 0.5 (/ y (/ z x))) t_0)
     (+ (* 0.5 (/ (* x y) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y / (x * z);
	double tmp;
	if (z <= 2e-72) {
		tmp = (0.5 * (y / (z / x))) + t_0;
	} else {
		tmp = (0.5 * ((x * y) / z)) + 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 / (x * z)
    if (z <= 2d-72) then
        tmp = (0.5d0 * (y / (z / x))) + t_0
    else
        tmp = (0.5d0 * ((x * y) / z)) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (x * z);
	double tmp;
	if (z <= 2e-72) {
		tmp = (0.5 * (y / (z / x))) + t_0;
	} else {
		tmp = (0.5 * ((x * y) / z)) + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (x * z)
	tmp = 0
	if z <= 2e-72:
		tmp = (0.5 * (y / (z / x))) + t_0
	else:
		tmp = (0.5 * ((x * y) / z)) + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(x * z))
	tmp = 0.0
	if (z <= 2e-72)
		tmp = Float64(Float64(0.5 * Float64(y / Float64(z / x))) + t_0);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (x * z);
	tmp = 0.0;
	if (z <= 2e-72)
		tmp = (0.5 * (y / (z / x))) + t_0;
	else
		tmp = (0.5 * ((x * y) / z)) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 2e-72], N[(N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < 1.9999999999999999e-72

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. associate-*l/ 94.1%

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

      3. associate-/l* 94.1%

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

      4. associate-/l* 94.1%

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

      5. associate-/r* 86.4%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 68.8%

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

      1. div-inv 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-*l* 73.3%

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

   7. Applied egg-rr 73.3%

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

      1. un-div-inv 73.3%

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

      2. clear-num 73.3%

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

   9. Applied egg-rr 73.3%

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

       1. un-div-inv 73.3%

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

   11. Applied egg-rr 73.3%

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

   if 1.9999999999999999e-72 < z

   1. Initial program 81.6%

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

      1. *-commutative 81.6%

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

      2. associate-*l/ 90.4%

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

      3. associate-/l* 90.5%

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

      4. associate-/l* 98.7%

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

      5. associate-/r* 76.6%

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

   3. Simplified 76.6%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 520000:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 520000.0)
   (* (/ y z) (+ (/ 1.0 x) (* x 0.5)))
   (* y (* x (/ 0.5 z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 520000.0) {
		tmp = (y / z) * ((1.0 / x) + (x * 0.5));
	} else {
		tmp = y * (x * (0.5 / 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 <= 520000.0d0) then
        tmp = (y / z) * ((1.0d0 / x) + (x * 0.5d0))
    else
        tmp = y * (x * (0.5d0 / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 520000.0) {
		tmp = (y / z) * ((1.0 / x) + (x * 0.5));
	} else {
		tmp = y * (x * (0.5 / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 520000.0:
		tmp = (y / z) * ((1.0 / x) + (x * 0.5))
	else:
		tmp = y * (x * (0.5 / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 520000.0)
		tmp = Float64(Float64(y / z) * Float64(Float64(1.0 / x) + Float64(x * 0.5)));
	else
		tmp = Float64(y * Float64(x * Float64(0.5 / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 520000.0)
		tmp = (y / z) * ((1.0 / x) + (x * 0.5));
	else
		tmp = y * (x * (0.5 / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 520000.0], N[(N[(y / z), $MachinePrecision] * N[(N[(1.0 / x), $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.2e5

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-*l/ 90.3%

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

      3. associate-/l* 90.3%

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

      4. associate-/l* 94.3%

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

      5. associate-/r* 87.0%

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

   3. Simplified 87.0%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.7%

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

      1. *-rgt-identity 75.7%

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

      2. *-commutative 75.7%

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

      3. times-frac 78.3%

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

   7. Applied egg-rr 78.3%

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

   8. Taylor expanded in x around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. +-commutative 75.7%

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

      3. *-rgt-identity 75.7%

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

      4. associate-*l/ 75.7%

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

      5. times-frac 78.3%

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

      6. metadata-eval 78.3%

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

      7. associate-/r* 78.3%

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

      8. neg-mul-1 78.3%

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

      9. associate-/l* 76.3%

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

      10. associate-*r* 76.3%

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

      11. *-commutative 76.3%

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

      12. *-commutative 76.3%

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

      13. distribute-lft-out 76.3%

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

      14. neg-mul-1 76.3%

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

      15. associate-/r* 76.3%

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

      16. metadata-eval 76.3%

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

   10. Simplified 76.3%

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

   if 5.2e5 < x

   1. Initial program 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 71.2%

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

   3. Simplified 71.2%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 38.6%

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

      1. *-rgt-identity 38.6%

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

      2. *-commutative 38.6%

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

      3. times-frac 38.6%

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

   7. Applied egg-rr 38.6%

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

   8. Taylor expanded in x around inf 38.6%

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

      1. associate-*r/ 38.6%

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

      2. *-commutative 38.6%

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

      3. *-commutative 38.6%

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

      4. associate-*r* 38.6%

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

      5. associate-*r/ 41.3%

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

      6. associate-/l* 41.3%

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

   10. Simplified 41.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 520000:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.3% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

      2. associate-*l/ 90.2%

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

      3. associate-/l* 90.2%

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

      4. associate-/l* 94.2%

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

      5. associate-/r* 86.9%

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

   3. Simplified 86.9%

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

      1. associate-*r/ 87.7%

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

      2. *-commutative 87.7%

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

      3. *-commutative 87.7%

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

      4. associate-/r* 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in x around 0 69.1%

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

   if 1.3999999999999999 < x

   1. Initial program 80.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 39.0%

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

   4. Taylor expanded in x around inf 39.0%

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

      1. associate-*r/ 34.8%

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

      2. *-commutative 34.8%

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

      3. associate-*l/ 39.0%

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

      4. associate-*r/ 41.7%

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

   6. Simplified 41.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.3% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 81.7%

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

      1. *-commutative 81.7%

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

      2. associate-*l/ 90.2%

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

      3. associate-/l* 90.2%

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

      4. associate-/l* 94.2%

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

      5. associate-/r* 86.9%

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

   3. Simplified 86.9%

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

      1. associate-*r/ 87.7%

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

      2. *-commutative 87.7%

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

      3. *-commutative 87.7%

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

      4. associate-/r* 97.8%

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

   6. Applied egg-rr 97.8%

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

   7. Taylor expanded in x around 0 69.1%

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

   if 1.3999999999999999 < x

   1. Initial program 80.9%

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

      1. *-commutative 80.9%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 100.0%

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

      4. associate-/l* 100.0%

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

      5. associate-/r* 72.1%

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

   3. Simplified 72.1%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 39.0%

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

      1. *-rgt-identity 39.0%

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

      2. *-commutative 39.0%

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

      3. times-frac 39.0%

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

   7. Applied egg-rr 39.0%

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

   8. Taylor expanded in x around inf 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-commutative 39.0%

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

      3. *-commutative 39.0%

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

      4. associate-*r* 39.0%

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

      5. associate-*r/ 41.7%

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

      6. associate-/l* 41.7%

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

   10. Simplified 41.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 5e-20) (/ (/ y x) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 5e-20) {
		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 <= 5d-20) 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 <= 5e-20) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 5e-20:
		tmp = (y / x) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 5e-20)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 5e-20)
		tmp = (y / x) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 5e-20], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.9999999999999999e-20

   1. Initial program 82.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 50.9%

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

   if 4.9999999999999999e-20 < z

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*l/ 87.9%

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

      3. associate-/l* 88.0%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 71.7%

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

   3. Simplified 71.7%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 43.6%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.1% 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 81.5%

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

   1. *-commutative 81.5%

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

   2. associate-*l/ 92.8%

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

   3. associate-/l* 92.8%

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

   4. associate-/l* 95.8%

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

   5. associate-/r* 82.9%

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

3. Simplified 82.9%

   \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 49.2%

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

6. Final simplification 49.2%

   \[\leadsto \frac{y}{x \cdot z} \]
7. Add Preprocessing

Alternative 13: 53.2% 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 81.5%

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

   1. *-commutative 81.5%

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

   2. associate-*l/ 92.8%

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

   3. associate-/l* 92.8%

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

   4. associate-/l* 95.8%

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

   5. associate-/r* 82.9%

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

3. Simplified 82.9%

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

   1. associate-*r/ 83.5%

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

   2. *-commutative 83.5%

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

   3. *-commutative 83.5%

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

   4. associate-/r* 98.4%

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

6. Applied egg-rr 98.4%

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

7. Taylor expanded in x around 0 56.0%

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

8. Final simplification 56.0%

   \[\leadsto \frac{\frac{y}{z}}{x} \]
9. Add Preprocessing

Developer target: 96.9% accurate, 0.9× 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 2024041 
(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))