Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 97.2%

Time: 11.0s

Alternatives: 18

Speedup: 1.0×

• 62.9% 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: 84.8% 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: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+113}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.7e+113) (/ (* y (/ (cosh x) x)) z) (/ (cosh x) (* x (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.7e+113) {
		tmp = (y * (cosh(x) / x)) / z;
	} else {
		tmp = cosh(x) / (x * (z / y));
	}
	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 <= 2.7d+113) then
        tmp = (y * (cosh(x) / x)) / z
    else
        tmp = cosh(x) / (x * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.7e+113) {
		tmp = (y * (Math.cosh(x) / x)) / z;
	} else {
		tmp = Math.cosh(x) / (x * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.7e+113:
		tmp = (y * (math.cosh(x) / x)) / z
	else:
		tmp = math.cosh(x) / (x * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.7e+113)
		tmp = Float64(Float64(y * Float64(cosh(x) / x)) / z);
	else
		tmp = Float64(cosh(x) / Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.7e+113)
		tmp = (y * (cosh(x) / x)) / z;
	else
		tmp = cosh(x) / (x * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.7e+113], N[(N[(y * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] / N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.70000000000000011e113

   1. Initial program 85.1%

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

      1. expm1-log1p-u 50.0%

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

      2. expm1-udef 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. expm1-def 50.0%

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

      2. expm1-log1p 85.1%

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

      3. associate-*r/ 97.6%

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

      4. associate-*l/ 97.5%

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

      5. *-commutative 97.5%

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

   5. Simplified 97.5%

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

   if 2.70000000000000011e113 < y

   1. Initial program 88.0%

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

      1. associate-/l* 88.0%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l/ 95.6%

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

      4. *-commutative 95.6%

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

   3. Simplified 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-/l* 88.0%

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

      3. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+113}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 2: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.2e+108) (* (/ (cosh x) z) (/ y x)) (/ (cosh x) (* x (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.2e+108) {
		tmp = (cosh(x) / z) * (y / x);
	} else {
		tmp = cosh(x) / (x * (z / y));
	}
	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 <= 2.2d+108) then
        tmp = (cosh(x) / z) * (y / x)
    else
        tmp = cosh(x) / (x * (z / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.2e+108) {
		tmp = (Math.cosh(x) / z) * (y / x);
	} else {
		tmp = Math.cosh(x) / (x * (z / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.2e+108:
		tmp = (math.cosh(x) / z) * (y / x)
	else:
		tmp = math.cosh(x) / (x * (z / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.2e+108)
		tmp = Float64(Float64(cosh(x) / z) * Float64(y / x));
	else
		tmp = Float64(cosh(x) / Float64(x * Float64(z / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.2e+108)
		tmp = (cosh(x) / z) * (y / x);
	else
		tmp = cosh(x) / (x * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.2e+108], N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] / N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.2000000000000001e108

   1. Initial program 85.1%

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

      1. associate-*l/ 85.1%

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

   3. Simplified 85.1%

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

   if 2.2000000000000001e108 < y

   1. Initial program 88.0%

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

      1. associate-/l* 88.0%

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

      2. associate-/r/ 99.8%

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

      3. associate-*l/ 95.6%

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

      4. *-commutative 95.6%

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

   3. Simplified 95.6%

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

      1. *-commutative 95.6%

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

      2. associate-/l* 88.0%

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

      3. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{+108}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 3: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. associate-*l/ 85.6%

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

3. Simplified 85.6%

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

4. Final simplification 85.6%

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

Alternative 4: 71.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -560 \lor \neg \left(x \leq 380\right):\\ \;\;\;\;\frac{y \cdot \left(z + x \cdot \left(0.5 \cdot \left(x \cdot z\right)\right)\right)}{z} \cdot \frac{1}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -560.0) (not (<= x 380.0)))
   (* (/ (* y (+ z (* x (* 0.5 (* x z))))) z) (/ 1.0 (* x z)))
   (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -560.0) || !(x <= 380.0)) {
		tmp = ((y * (z + (x * (0.5 * (x * z))))) / z) * (1.0 / (x * z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-560.0d0)) .or. (.not. (x <= 380.0d0))) then
        tmp = ((y * (z + (x * (0.5d0 * (x * z))))) / z) * (1.0d0 / (x * z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -560.0) || !(x <= 380.0)) {
		tmp = ((y * (z + (x * (0.5 * (x * z))))) / z) * (1.0 / (x * z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -560.0) or not (x <= 380.0):
		tmp = ((y * (z + (x * (0.5 * (x * z))))) / z) * (1.0 / (x * z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -560.0) || !(x <= 380.0))
		tmp = Float64(Float64(Float64(y * Float64(z + Float64(x * Float64(0.5 * Float64(x * z))))) / z) * Float64(1.0 / Float64(x * z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -560.0) || ~((x <= 380.0)))
		tmp = ((y * (z + (x * (0.5 * (x * z))))) / z) * (1.0 / (x * z));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -560.0], N[Not[LessEqual[x, 380.0]], $MachinePrecision]], N[(N[(N[(y * N[(z + N[(x * N[(0.5 * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -560 or 380 < x

   1. Initial program 79.4%

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

      1. associate-*l/ 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in x around 0 42.5%

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

      1. associate-*r/ 42.5%

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

      2. frac-add 39.9%

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

      3. *-commutative 39.9%

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

      4. *-commutative 39.9%

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

      5. associate-*l* 39.9%

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

   6. Applied egg-rr 39.9%

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

      1. associate-/r* 47.0%

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

      2. div-inv 47.0%

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

      3. associate-*l* 55.5%

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

      4. *-commutative 55.5%

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

      5. distribute-lft-out 55.5%

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

      6. associate-*l* 55.5%

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

   8. Applied egg-rr 55.5%

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

   if -560 < x < 380

   1. Initial program 91.8%

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

      1. associate-*l/ 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around 0 89.7%

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

      1. *-un-lft-identity 89.7%

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

      2. times-frac 92.8%

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

   6. Applied egg-rr 92.8%

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

      1. associate-*l/ 92.8%

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

      2. *-un-lft-identity 92.8%

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

   8. Applied egg-rr 92.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -560 \lor \neg \left(x \leq 380\right):\\ \;\;\;\;\frac{y \cdot \left(z + x \cdot \left(0.5 \cdot \left(x \cdot z\right)\right)\right)}{z} \cdot \frac{1}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 5: 70.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{y \cdot x}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot t_0 + z \cdot 0.5}{z \cdot t_0}\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z + x \cdot \left(0.5 \cdot \left(x \cdot z\right)\right)\right)}{z} \cdot \frac{1}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (* y x))))
   (if (<= x -2.7e-7)
     (/ (+ (* (/ y x) t_0) (* z 0.5)) (* z t_0))
     (if (<= x 370.0)
       (/ (/ y z) x)
       (* (/ (* y (+ z (* x (* 0.5 (* x z))))) z) (/ 1.0 (* x z)))))))

C

double code(double x, double y, double z) {
	double t_0 = z / (y * x);
	double tmp;
	if (x <= -2.7e-7) {
		tmp = (((y / x) * t_0) + (z * 0.5)) / (z * t_0);
	} else if (x <= 370.0) {
		tmp = (y / z) / x;
	} else {
		tmp = ((y * (z + (x * (0.5 * (x * z))))) / 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) :: t_0
    real(8) :: tmp
    t_0 = z / (y * x)
    if (x <= (-2.7d-7)) then
        tmp = (((y / x) * t_0) + (z * 0.5d0)) / (z * t_0)
    else if (x <= 370.0d0) then
        tmp = (y / z) / x
    else
        tmp = ((y * (z + (x * (0.5d0 * (x * z))))) / z) * (1.0d0 / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z / (y * x);
	double tmp;
	if (x <= -2.7e-7) {
		tmp = (((y / x) * t_0) + (z * 0.5)) / (z * t_0);
	} else if (x <= 370.0) {
		tmp = (y / z) / x;
	} else {
		tmp = ((y * (z + (x * (0.5 * (x * z))))) / z) * (1.0 / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z / (y * x)
	tmp = 0
	if x <= -2.7e-7:
		tmp = (((y / x) * t_0) + (z * 0.5)) / (z * t_0)
	elif x <= 370.0:
		tmp = (y / z) / x
	else:
		tmp = ((y * (z + (x * (0.5 * (x * z))))) / z) * (1.0 / (x * z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z / Float64(y * x))
	tmp = 0.0
	if (x <= -2.7e-7)
		tmp = Float64(Float64(Float64(Float64(y / x) * t_0) + Float64(z * 0.5)) / Float64(z * t_0));
	elseif (x <= 370.0)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(Float64(y * Float64(z + Float64(x * Float64(0.5 * Float64(x * z))))) / z) * Float64(1.0 / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z / (y * x);
	tmp = 0.0;
	if (x <= -2.7e-7)
		tmp = (((y / x) * t_0) + (z * 0.5)) / (z * t_0);
	elseif (x <= 370.0)
		tmp = (y / z) / x;
	else
		tmp = ((y * (z + (x * (0.5 * (x * z))))) / z) * (1.0 / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z / N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.7e-7], N[(N[(N[(N[(y / x), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 370.0], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y * N[(z + N[(x * N[(0.5 * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.70000000000000009e-7

   1. Initial program 79.7%

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

      1. associate-*l/ 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in x around 0 47.1%

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

      1. associate-*r/ 47.1%

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

      2. clear-num 47.1%

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

      3. *-commutative 47.1%

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

      4. *-commutative 47.1%

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

      5. associate-*l* 47.1%

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

   6. Applied egg-rr 47.1%

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

      1. +-commutative 47.1%

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

      2. associate-/r* 47.1%

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

      3. associate-*r* 47.1%

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

      4. *-commutative 47.1%

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

      5. clear-num 47.1%

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

      6. associate-/l* 47.1%

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

      7. frac-add 54.4%

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

   8. Applied egg-rr 54.4%

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

   if -2.70000000000000009e-7 < x < 370

   1. Initial program 91.7%

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

      1. associate-*l/ 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in x around 0 89.7%

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

      1. *-un-lft-identity 89.7%

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

      2. times-frac 92.7%

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

   6. Applied egg-rr 92.7%

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

      1. associate-*l/ 92.8%

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

      2. *-un-lft-identity 92.8%

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

   8. Applied egg-rr 92.8%

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

   if 370 < x

   1. Initial program 79.7%

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

      1. associate-*l/ 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in x around 0 39.8%

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

      1. associate-*r/ 39.8%

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

      2. frac-add 41.7%

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

      3. *-commutative 41.7%

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

      4. *-commutative 41.7%

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

      5. associate-*l* 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. associate-/r* 49.4%

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

      2. div-inv 49.4%

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

      3. associate-*l* 58.5%

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

      4. *-commutative 58.5%

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

      5. distribute-lft-out 58.5%

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

      6. associate-*l* 58.5%

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

   8. Applied egg-rr 58.5%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \frac{z}{y \cdot x} + z \cdot 0.5}{z \cdot \frac{z}{y \cdot x}}\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(z + x \cdot \left(0.5 \cdot \left(x \cdot z\right)\right)\right)}{z} \cdot \frac{1}{x \cdot z}\\ \end{array} \]

Alternative 6: 68.9% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.85 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{y \cdot \left(z + x \cdot \left(0.5 \cdot \left(x \cdot z\right)\right)\right)}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.85e-74)
   (/ (+ (/ y x) (* (* y x) 0.5)) z)
   (if (<= z 2.9e+62)
     (/ (* y (+ z (* x (* 0.5 (* x z))))) (* z (* x z)))
     (+ (* 0.5 (/ (* y x) z)) (/ y (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.85e-74) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else if (z <= 2.9e+62) {
		tmp = (y * (z + (x * (0.5 * (x * z))))) / (z * (x * z));
	} else {
		tmp = (0.5 * ((y * x) / z)) + (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 <= 2.85d-74) then
        tmp = ((y / x) + ((y * x) * 0.5d0)) / z
    else if (z <= 2.9d+62) then
        tmp = (y * (z + (x * (0.5d0 * (x * z))))) / (z * (x * z))
    else
        tmp = (0.5d0 * ((y * x) / z)) + (y / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.85e-74) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else if (z <= 2.9e+62) {
		tmp = (y * (z + (x * (0.5 * (x * z))))) / (z * (x * z));
	} else {
		tmp = (0.5 * ((y * x) / z)) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.85e-74:
		tmp = ((y / x) + ((y * x) * 0.5)) / z
	elif z <= 2.9e+62:
		tmp = (y * (z + (x * (0.5 * (x * z))))) / (z * (x * z))
	else:
		tmp = (0.5 * ((y * x) / z)) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.85e-74)
		tmp = Float64(Float64(Float64(y / x) + Float64(Float64(y * x) * 0.5)) / z);
	elseif (z <= 2.9e+62)
		tmp = Float64(Float64(y * Float64(z + Float64(x * Float64(0.5 * Float64(x * z))))) / Float64(z * Float64(x * z)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / z)) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.85e-74)
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	elseif (z <= 2.9e+62)
		tmp = (y * (z + (x * (0.5 * (x * z))))) / (z * (x * z));
	else
		tmp = (0.5 * ((y * x) / z)) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.85e-74], N[(N[(N[(y / x), $MachinePrecision] + N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 2.9e+62], N[(N[(y * N[(z + N[(x * N[(0.5 * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < 2.85000000000000012e-74

   1. Initial program 88.1%

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

   2. Taylor expanded in x around 0 75.5%

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

   if 2.85000000000000012e-74 < z < 2.89999999999999984e62

   1. Initial program 90.8%

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

      1. associate-*l/ 90.8%

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

   3. Simplified 90.8%

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

   4. Taylor expanded in x around 0 56.1%

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

      1. associate-*r/ 56.1%

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

      2. frac-add 64.8%

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

      3. *-commutative 64.8%

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

      4. *-commutative 64.8%

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

      5. associate-*l* 64.8%

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

   6. Applied egg-rr 64.8%

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

      1. associate-*l* 70.6%

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

      2. *-commutative 70.6%

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

      3. distribute-lft-out 70.6%

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

      4. associate-*l* 70.6%

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

   8. Applied egg-rr 70.6%

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

   if 2.89999999999999984e62 < z

   1. Initial program 72.5%

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

      1. associate-*l/ 72.4%

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

   3. Simplified 72.4%

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

   4. Taylor expanded in x around 0 57.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.85 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+62}:\\ \;\;\;\;\frac{y \cdot \left(z + x \cdot \left(0.5 \cdot \left(x \cdot z\right)\right)\right)}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 7: 65.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{y}{z} - x \cdot \left(y \cdot \left(x \cdot -0.5\right)\right)}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 7.2e-116)
   (/ (+ (/ y x) (* (* y x) 0.5)) z)
   (/ (- (* z (/ y z)) (* x (* y (* x -0.5)))) (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7.2e-116) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = ((z * (y / z)) - (x * (y * (x * -0.5)))) / (x * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 7.2d-116) then
        tmp = ((y / x) + ((y * x) * 0.5d0)) / z
    else
        tmp = ((z * (y / z)) - (x * (y * (x * (-0.5d0))))) / (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 7.2e-116) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = ((z * (y / z)) - (x * (y * (x * -0.5)))) / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7.2e-116:
		tmp = ((y / x) + ((y * x) * 0.5)) / z
	else:
		tmp = ((z * (y / z)) - (x * (y * (x * -0.5)))) / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7.2e-116)
		tmp = Float64(Float64(Float64(y / x) + Float64(Float64(y * x) * 0.5)) / z);
	else
		tmp = Float64(Float64(Float64(z * Float64(y / z)) - Float64(x * Float64(y * Float64(x * -0.5)))) / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 7.2e-116)
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	else
		tmp = ((z * (y / z)) - (x * (y * (x * -0.5)))) / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7.2e-116], N[(N[(N[(y / x), $MachinePrecision] + N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(z * N[(y / z), $MachinePrecision]), $MachinePrecision] - N[(x * N[(y * N[(x * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.19999999999999951e-116

   1. Initial program 88.0%

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

   2. Taylor expanded in x around 0 75.1%

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

   if 7.19999999999999951e-116 < z

   1. Initial program 81.3%

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

      1. associate-*l/ 81.2%

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

   3. Simplified 81.2%

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

   4. Taylor expanded in x around 0 59.6%

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

      1. associate-*r/ 59.6%

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

      2. clear-num 59.6%

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

      3. *-commutative 59.6%

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

      4. *-commutative 59.6%

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

      5. associate-*l* 59.6%

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

   6. Applied egg-rr 59.6%

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

      1. +-commutative 59.6%

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

      2. associate-/l/ 58.6%

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

      3. frac-2neg 58.6%

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

      4. frac-2neg 58.6%

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

      5. clear-num 58.6%

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

      6. frac-add 62.4%

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

      7. distribute-neg-frac 62.4%

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

      8. associate-*r* 62.4%

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

      9. distribute-rgt-neg-in 62.4%

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

      10. metadata-eval 62.4%

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

   8. Applied egg-rr 62.4%

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

      1. *-commutative 62.4%

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

      2. *-commutative 62.4%

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

      3. distribute-rgt-neg-out 62.4%

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

      4. unsub-neg 62.4%

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

      5. distribute-frac-neg 62.4%

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

      6. distribute-rgt-neg-in 62.4%

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

      7. distribute-lft-neg-out 62.4%

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

      8. remove-double-neg 62.4%

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

      9. *-commutative 62.4%

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

      10. associate-*l* 62.4%

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

      11. distribute-lft-neg-out 62.4%

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

      12. distribute-rgt-neg-out 62.4%

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

      13. remove-double-neg 62.4%

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

   10. Simplified 62.4%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \frac{y}{z} - x \cdot \left(y \cdot \left(x \cdot -0.5\right)\right)}{x \cdot z}\\ \end{array} \]

Alternative 8: 66.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 6.8e+152)
   (/ (+ (/ y x) (* (* y x) 0.5)) z)
   (+ (/ y (* x z)) (* 0.5 (* x (/ y z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.8e+152) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = (y / (x * z)) + (0.5 * (x * (y / z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 6.8d+152) then
        tmp = ((y / x) + ((y * x) * 0.5d0)) / z
    else
        tmp = (y / (x * z)) + (0.5d0 * (x * (y / z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 6.8e+152) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = (y / (x * z)) + (0.5 * (x * (y / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 6.8e+152:
		tmp = ((y / x) + ((y * x) * 0.5)) / z
	else:
		tmp = (y / (x * z)) + (0.5 * (x * (y / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 6.8e+152)
		tmp = Float64(Float64(Float64(y / x) + Float64(Float64(y * x) * 0.5)) / z);
	else
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(0.5 * Float64(x * Float64(y / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 6.8e+152)
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	else
		tmp = (y / (x * z)) + (0.5 * (x * (y / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 6.8e+152], N[(N[(N[(y / x), $MachinePrecision] + N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.80000000000000041e152

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0 70.7%

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

   if 6.80000000000000041e152 < z

   1. Initial program 66.5%

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

      1. associate-*l/ 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in x around 0 60.0%

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

      1. div-inv 60.0%

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

      2. associate-*l* 56.9%

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

   6. Applied egg-rr 56.9%

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

   7. Taylor expanded in y around 0 56.9%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.8 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \end{array} \]

Alternative 9: 66.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.2e-191)
   (/ (+ (/ y x) (* (* y x) 0.5)) z)
   (+ (/ y (* x z)) (* 0.5 (* y (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.2e-191) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = (y / (x * z)) + (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 (z <= 1.2d-191) then
        tmp = ((y / x) + ((y * x) * 0.5d0)) / z
    else
        tmp = (y / (x * z)) + (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 (z <= 1.2e-191) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = (y / (x * z)) + (0.5 * (y * (x / z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.2e-191:
		tmp = ((y / x) + ((y * x) * 0.5)) / z
	else:
		tmp = (y / (x * z)) + (0.5 * (y * (x / z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.2e-191)
		tmp = Float64(Float64(Float64(y / x) + Float64(Float64(y * x) * 0.5)) / z);
	else
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(0.5 * Float64(y * Float64(x / z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.2e-191)
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	else
		tmp = (y / (x * z)) + (0.5 * (y * (x / z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.2e-191], N[(N[(N[(y / x), $MachinePrecision] + N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e-191

   1. Initial program 87.8%

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

   2. Taylor expanded in x around 0 74.5%

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

   if 1.2e-191 < z

   1. Initial program 82.5%

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

      1. associate-*l/ 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in x around 0 62.5%

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

      1. associate-/l* 60.5%

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

      2. associate-/r/ 61.5%

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

   6. Applied egg-rr 61.5%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{-191}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \]

Alternative 10: 68.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2e+32)
   (/ (+ (/ y x) (* (* y x) 0.5)) z)
   (+ (* 0.5 (/ (* y x) z)) (/ y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e+32) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = (0.5 * ((y * x) / z)) + (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 <= 2d+32) then
        tmp = ((y / x) + ((y * x) * 0.5d0)) / z
    else
        tmp = (0.5d0 * ((y * x) / z)) + (y / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e+32) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = (0.5 * ((y * x) / z)) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2e+32:
		tmp = ((y / x) + ((y * x) * 0.5)) / z
	else:
		tmp = (0.5 * ((y * x) / z)) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2e+32)
		tmp = Float64(Float64(Float64(y / x) + Float64(Float64(y * x) * 0.5)) / z);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / z)) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2e+32)
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	else
		tmp = (0.5 * ((y * x) / z)) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2e+32], N[(N[(N[(y / x), $MachinePrecision] + N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.00000000000000011e32

   1. Initial program 88.6%

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

   2. Taylor expanded in x around 0 73.8%

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

   if 2.00000000000000011e32 < z

   1. Initial program 74.7%

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

      1. associate-*l/ 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in x around 0 54.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 11: 65.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.45e+153) (/ (* y (+ (* x 0.5) (/ 1.0 x))) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.45e+153) {
		tmp = (y * ((x * 0.5) + (1.0 / 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 <= 1.45d+153) then
        tmp = (y * ((x * 0.5d0) + (1.0d0 / 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 <= 1.45e+153) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.45e+153:
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.45e+153)
		tmp = Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / 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 <= 1.45e+153)
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.45e+153], N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.45000000000000001e153

   1. Initial program 88.4%

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

      1. expm1-log1p-u 51.4%

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

      2. expm1-udef 41.5%

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

   3. Applied egg-rr 41.5%

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

      1. expm1-def 51.4%

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

      2. expm1-log1p 88.4%

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

      3. associate-*r/ 97.8%

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

      4. associate-*l/ 97.7%

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

      5. *-commutative 97.7%

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

   5. Simplified 97.7%

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

   6. Taylor expanded in x around 0 70.7%

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

   if 1.45000000000000001e153 < z

   1. Initial program 66.5%

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

      1. associate-*l/ 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in x around 0 56.7%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{+153}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 12: 66.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.65e+151) (/ (+ (/ y x) (* (* y x) 0.5)) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.65e+151) {
		tmp = ((y / x) + ((y * x) * 0.5)) / 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 <= 1.65d+151) then
        tmp = ((y / x) + ((y * x) * 0.5d0)) / 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 <= 1.65e+151) {
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.65e+151:
		tmp = ((y / x) + ((y * x) * 0.5)) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.65e+151)
		tmp = Float64(Float64(Float64(y / x) + Float64(Float64(y * x) * 0.5)) / 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 <= 1.65e+151)
		tmp = ((y / x) + ((y * x) * 0.5)) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.65e+151], N[(N[(N[(y / x), $MachinePrecision] + N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.65000000000000012e151

   1. Initial program 88.4%

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

   2. Taylor expanded in x around 0 70.7%

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

   if 1.65000000000000012e151 < z

   1. Initial program 66.5%

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

      1. associate-*l/ 66.5%

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

   3. Simplified 66.5%

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

   4. Taylor expanded in x around 0 56.7%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{+151}:\\ \;\;\;\;\frac{\frac{y}{x} + \left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 13: 62.4% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.45))) (* x (* 0.5 (/ y z))) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.45)) {
		tmp = x * (0.5 * (y / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d0)) .or. (.not. (x <= 1.45d0))) then
        tmp = x * (0.5d0 * (y / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.45)) {
		tmp = x * (0.5 * (y / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.45):
		tmp = x * (0.5 * (y / z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.45))
		tmp = Float64(x * Float64(0.5 * Float64(y / z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.45)))
		tmp = x * (0.5 * (y / z));
	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, 1.45]], $MachinePrecision]], N[(x * N[(0.5 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.44999999999999996 < x

   1. Initial program 79.5%

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

      1. expm1-log1p-u 36.2%

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

      2. expm1-udef 35.5%

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

   3. Applied egg-rr 35.5%

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

      1. expm1-def 36.2%

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

      2. expm1-log1p 79.5%

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

      3. associate-*r/ 100.0%

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

      4. associate-*l/ 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 42.3%

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

   7. Taylor expanded in x around inf 42.3%

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

      1. *-commutative 42.3%

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

      2. associate-/l* 37.8%

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

      3. associate-*r/ 37.8%

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

      4. associate-/l* 42.3%

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

      5. associate-*r/ 39.3%

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

      6. *-commutative 39.3%

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

      7. associate-*l/ 42.3%

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

      8. associate-*r/ 34.0%

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

      9. *-commutative 34.0%

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

      10. associate-*l/ 34.0%

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

   9. Simplified 34.0%

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

   if -1.3999999999999999 < x < 1.44999999999999996

   1. Initial program 91.7%

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

      1. associate-*l/ 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around 0 90.4%

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

      1. *-un-lft-identity 90.4%

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

      2. times-frac 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. associate-*l/ 93.5%

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

      2. *-un-lft-identity 93.5%

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

   8. Applied egg-rr 93.5%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;x \cdot \left(0.5 \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 14: 66.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.45))) (* y (* 0.5 (/ x z))) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.45)) {
		tmp = y * (0.5 * (x / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d0)) .or. (.not. (x <= 1.45d0))) then
        tmp = y * (0.5d0 * (x / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.45)) {
		tmp = y * (0.5 * (x / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.45):
		tmp = y * (0.5 * (x / z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.45))
		tmp = Float64(y * Float64(0.5 * Float64(x / z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.45)))
		tmp = y * (0.5 * (x / z));
	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, 1.45]], $MachinePrecision]], N[(y * N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.44999999999999996 < x

   1. Initial program 79.5%

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

      1. expm1-log1p-u 36.2%

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

      2. expm1-udef 35.5%

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

   3. Applied egg-rr 35.5%

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

      1. expm1-def 36.2%

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

      2. expm1-log1p 79.5%

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

      3. associate-*r/ 100.0%

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

      4. associate-*l/ 100.0%

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

      5. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 42.3%

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

   7. Taylor expanded in x around inf 42.3%

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

      1. associate-*r/ 42.3%

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

      2. associate-*r* 42.3%

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

      3. associate-*l/ 39.3%

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

      4. associate-*r/ 39.3%

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

      5. *-commutative 39.3%

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

   9. Simplified 39.3%

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

   if -1.3999999999999999 < x < 1.44999999999999996

   1. Initial program 91.7%

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

      1. associate-*l/ 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around 0 90.4%

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

      1. *-un-lft-identity 90.4%

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

      2. times-frac 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. associate-*l/ 93.5%

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

      2. *-un-lft-identity 93.5%

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

   8. Applied egg-rr 93.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 15: 66.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;\frac{\left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.45))) (/ (* (* y x) 0.5) z) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.45)) {
		tmp = ((y * x) * 0.5) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d0)) .or. (.not. (x <= 1.45d0))) then
        tmp = ((y * x) * 0.5d0) / z
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.45)) {
		tmp = ((y * x) * 0.5) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.45):
		tmp = ((y * x) * 0.5) / z
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.45))
		tmp = Float64(Float64(Float64(y * x) * 0.5) / z);
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.45)))
		tmp = ((y * x) * 0.5) / z;
	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, 1.45]], $MachinePrecision]], N[(N[(N[(y * x), $MachinePrecision] * 0.5), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.44999999999999996 < x

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 42.3%

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

   3. Taylor expanded in x around inf 42.3%

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

      1. *-commutative 42.3%

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

   5. Simplified 42.3%

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

   if -1.3999999999999999 < x < 1.44999999999999996

   1. Initial program 91.7%

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

      1. associate-*l/ 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around 0 90.4%

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

      1. *-un-lft-identity 90.4%

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

      2. times-frac 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. associate-*l/ 93.5%

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

      2. *-un-lft-identity 93.5%

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

   8. Applied egg-rr 93.5%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;\frac{\left(y \cdot x\right) \cdot 0.5}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 16: 51.6% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.95e-124) (/ (/ y x) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.95e-124) {
		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 (y <= 2.95d-124) 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 (y <= 2.95e-124) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.95e-124:
		tmp = (y / x) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.95e-124)
		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 (y <= 2.95e-124)
		tmp = (y / x) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.95e-124], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9500000000000001e-124

   1. Initial program 81.9%

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

   2. Taylor expanded in x around 0 49.9%

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

   if 2.9500000000000001e-124 < y

   1. Initial program 91.9%

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

      1. associate-*l/ 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in x around 0 55.3%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.95 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 17: 56.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -20000.0) (/ y (* x z)) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -20000.0) {
		tmp = y / (x * z);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-20000.0d0)) then
        tmp = y / (x * z)
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -20000.0) {
		tmp = y / (x * z);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -20000.0:
		tmp = y / (x * z)
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -20000.0)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -20000.0)
		tmp = y / (x * z);
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -20000.0], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2e4

   1. Initial program 83.3%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around 0 45.6%

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

   if -2e4 < z

   1. Initial program 86.5%

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

      1. associate-*l/ 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around 0 49.9%

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

      1. *-un-lft-identity 49.9%

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

      2. times-frac 61.8%

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

   6. Applied egg-rr 61.8%

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

      1. associate-*l/ 61.9%

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

      2. *-un-lft-identity 61.9%

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

   8. Applied egg-rr 61.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -20000:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 18: 50.2% 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 85.7%

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

   1. associate-*l/ 85.6%

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

3. Simplified 85.6%

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

4. Taylor expanded in x around 0 48.8%

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

5. Final simplification 48.8%

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

Developer target: 97.2% 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 2023332 
(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))