Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.1% → 97.6%

Time: 11.2s

Alternatives: 16

Speedup: 1.0×

• 63.5% 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: 85.1% 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: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -5e-41) (/ (/ (* y (cosh x)) z) x) (/ (* y (/ (cosh x) x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-41) {
		tmp = ((y * cosh(x)) / z) / x;
	} else {
		tmp = (y * (cosh(x) / 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 <= (-5d-41)) then
        tmp = ((y * cosh(x)) / z) / x
    else
        tmp = (y * (cosh(x) / x)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -5e-41) {
		tmp = ((y * Math.cosh(x)) / z) / x;
	} else {
		tmp = (y * (Math.cosh(x) / x)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -5e-41:
		tmp = ((y * math.cosh(x)) / z) / x
	else:
		tmp = (y * (math.cosh(x) / x)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -5e-41)
		tmp = Float64(Float64(Float64(y * cosh(x)) / z) / x);
	else
		tmp = Float64(Float64(y * Float64(cosh(x) / x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -5e-41)
		tmp = ((y * cosh(x)) / z) / x;
	else
		tmp = (y * (cosh(x) / x)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -5e-41], N[(N[(N[(y * N[Cosh[x], $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.9999999999999996e-41

   1. Initial program 89.1%

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

      1. associate-*l/ 89.1%

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

   3. Simplified 89.1%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -4.9999999999999996e-41 < y

   1. Initial program 81.7%

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

      1. expm1-log1p-u 44.1%

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

      2. expm1-udef 32.0%

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

   3. Applied egg-rr 32.0%

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

      1. expm1-def 44.1%

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

      2. expm1-log1p 81.7%

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

      3. associate-*r/ 97.6%

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

      4. associate-*l/ 97.6%

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

      5. *-commutative 97.6%

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

   5. Simplified 97.6%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{y \cdot \cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \end{array} \]

Alternative 2: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{\frac{z}{y}}{x}}{x} + z \cdot 0.5}{z \cdot \frac{z}{y \cdot x}}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-25} \lor \neg \left(x \leq 1.2 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.8e+134)
   (/ (+ (/ (* y (/ (/ z y) x)) x) (* z 0.5)) (* z (/ z (* y x))))
   (if (or (<= x -3e-25) (not (<= x 1.2e-101)))
     (* (/ (cosh x) z) (/ y x))
     (+ (* 0.5 (* y (/ x z))) (/ y (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.8e+134) {
		tmp = (((y * ((z / y) / x)) / x) + (z * 0.5)) / (z * (z / (y * x)));
	} else if ((x <= -3e-25) || !(x <= 1.2e-101)) {
		tmp = (cosh(x) / z) * (y / x);
	} 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 (x <= (-6.8d+134)) then
        tmp = (((y * ((z / y) / x)) / x) + (z * 0.5d0)) / (z * (z / (y * x)))
    else if ((x <= (-3d-25)) .or. (.not. (x <= 1.2d-101))) then
        tmp = (cosh(x) / z) * (y / x)
    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 (x <= -6.8e+134) {
		tmp = (((y * ((z / y) / x)) / x) + (z * 0.5)) / (z * (z / (y * x)));
	} else if ((x <= -3e-25) || !(x <= 1.2e-101)) {
		tmp = (Math.cosh(x) / z) * (y / x);
	} else {
		tmp = (0.5 * (y * (x / z))) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.8e+134:
		tmp = (((y * ((z / y) / x)) / x) + (z * 0.5)) / (z * (z / (y * x)))
	elif (x <= -3e-25) or not (x <= 1.2e-101):
		tmp = (math.cosh(x) / z) * (y / x)
	else:
		tmp = (0.5 * (y * (x / z))) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.8e+134)
		tmp = Float64(Float64(Float64(Float64(y * Float64(Float64(z / y) / x)) / x) + Float64(z * 0.5)) / Float64(z * Float64(z / Float64(y * x))));
	elseif ((x <= -3e-25) || !(x <= 1.2e-101))
		tmp = Float64(Float64(cosh(x) / z) * Float64(y / x));
	else
		tmp = Float64(Float64(0.5 * Float64(y * Float64(x / z))) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.8e+134)
		tmp = (((y * ((z / y) / x)) / x) + (z * 0.5)) / (z * (z / (y * x)));
	elseif ((x <= -3e-25) || ~((x <= 1.2e-101)))
		tmp = (cosh(x) / z) * (y / x);
	else
		tmp = (0.5 * (y * (x / z))) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.8e+134], N[(N[(N[(N[(y * N[(N[(z / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -3e-25], N[Not[LessEqual[x, 1.2e-101]], $MachinePrecision]], N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.80000000000000035e134

   1. Initial program 63.6%

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

      1. associate-*l/ 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in x around 0 56.8%

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

      1. clear-num 56.8%

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

      2. un-div-inv 56.8%

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

      3. associate-/r* 51.1%

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

   6. Applied egg-rr 51.1%

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

      1. +-commutative 51.1%

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

      2. associate-/r* 51.1%

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

      3. frac-add 56.7%

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

      4. associate-/l/ 56.7%

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

      5. *-commutative 56.7%

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

      6. associate-/l/ 70.7%

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

      7. *-commutative 70.7%

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

   8. Applied egg-rr 70.7%

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

      1. associate-*l/ 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-/r* 76.6%

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

   10. Applied egg-rr 76.6%

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

   if -6.80000000000000035e134 < x < -2.9999999999999998e-25 or 1.2e-101 < x

   1. Initial program 87.7%

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

      1. associate-*l/ 87.6%

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

   3. Simplified 87.6%

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

   if -2.9999999999999998e-25 < x < 1.2e-101

   1. Initial program 86.7%

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

      1. associate-*l/ 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in x around 0 93.3%

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

      1. associate-/l* 93.3%

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

      2. associate-/r/ 93.3%

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

   6. Applied egg-rr 93.3%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{\frac{z}{y}}{x}}{x} + z \cdot 0.5}{z \cdot \frac{z}{y \cdot x}}\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-25} \lor \neg \left(x \leq 1.2 \cdot 10^{-101}\right):\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right) + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 3: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+136} \lor \neg \left(z \leq 9.6 \cdot 10^{+238}\right):\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.4e+136) (not (<= z 9.6e+238)))
   (* (/ (cosh x) z) (/ y x))
   (/ (cosh x) (* x (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.4e+136) || !(z <= 9.6e+238)) {
		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 ((z <= (-4.4d+136)) .or. (.not. (z <= 9.6d+238))) 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 ((z <= -4.4e+136) || !(z <= 9.6e+238)) {
		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 (z <= -4.4e+136) or not (z <= 9.6e+238):
		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 ((z <= -4.4e+136) || !(z <= 9.6e+238))
		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 ((z <= -4.4e+136) || ~((z <= 9.6e+238)))
		tmp = (cosh(x) / z) * (y / x);
	else
		tmp = cosh(x) / (x * (z / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.4e+136], N[Not[LessEqual[z, 9.6e+238]], $MachinePrecision]], 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 z < -4.3999999999999999e136 or 9.6e238 < z

   1. Initial program 86.5%

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

      1. associate-*l/ 86.5%

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

   3. Simplified 86.5%

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

   if -4.3999999999999999e136 < z < 9.6e238

   1. Initial program 83.6%

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

      1. associate-/l* 82.6%

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

      2. associate-/r/ 91.6%

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

      3. associate-*l/ 86.4%

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

      4. *-commutative 86.4%

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

   3. Simplified 86.4%

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

      1. *-commutative 86.4%

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

      2. associate-/l* 82.6%

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

      3. associate-/r/ 91.6%

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

   5. Applied egg-rr 91.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{+136} \lor \neg \left(z \leq 9.6 \cdot 10^{+238}\right):\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 4: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+30}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.06e+30) (/ (cosh x) (* x (/ z y))) (/ (* y (/ (cosh x) x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.06e+30) {
		tmp = cosh(x) / (x * (z / y));
	} else {
		tmp = (y * (cosh(x) / x)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.06d+30)) then
        tmp = cosh(x) / (x * (z / y))
    else
        tmp = (y * (cosh(x) / x)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.06e+30) {
		tmp = Math.cosh(x) / (x * (z / y));
	} else {
		tmp = (y * (Math.cosh(x) / x)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.06e+30:
		tmp = math.cosh(x) / (x * (z / y))
	else:
		tmp = (y * (math.cosh(x) / x)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.06e+30)
		tmp = Float64(cosh(x) / Float64(x * Float64(z / y)));
	else
		tmp = Float64(Float64(y * Float64(cosh(x) / x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.06e+30)
		tmp = cosh(x) / (x * (z / y));
	else
		tmp = (y * (cosh(x) / x)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.06e+30], N[(N[Cosh[x], $MachinePrecision] / N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.06e30

   1. Initial program 86.6%

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

      1. associate-/l* 85.2%

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

      2. associate-/r/ 98.4%

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

      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* 85.2%

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

      3. associate-/r/ 98.4%

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

   5. Applied egg-rr 98.4%

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

   if -1.06e30 < y

   1. Initial program 83.3%

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

      1. expm1-log1p-u 43.4%

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

      2. expm1-udef 32.3%

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

   3. Applied egg-rr 32.3%

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

      1. expm1-def 43.4%

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

      2. expm1-log1p 83.3%

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

      3. associate-*r/ 97.8%

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

      4. associate-*l/ 97.8%

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

      5. *-commutative 97.8%

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

   5. Simplified 97.8%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{+30}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \end{array} \]

Alternative 5: 68.6% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{z}{y}}{x}\\ t_1 := \frac{\frac{y \cdot t_0}{x} + z \cdot 0.5}{z \cdot \frac{z}{y \cdot x}}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-242}:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{0.5}{t_0}\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+222}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot \left(x \cdot \left(y \cdot 0.5\right)\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ z y) x))
        (t_1 (/ (+ (/ (* y t_0) x) (* z 0.5)) (* z (/ z (* y x))))))
   (if (<= x -1.25e+88)
     t_1
     (if (<= x 1.3e-242)
       (+ (/ y (* x z)) (/ 0.5 t_0))
       (if (<= x 490.0)
         (/ (/ y z) x)
         (if (<= x 5.6e+222)
           (/ (+ (* (* x z) (* x (* y 0.5))) (* y z)) (* z (* x z)))
           t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = (z / y) / x;
	double t_1 = (((y * t_0) / x) + (z * 0.5)) / (z * (z / (y * x)));
	double tmp;
	if (x <= -1.25e+88) {
		tmp = t_1;
	} else if (x <= 1.3e-242) {
		tmp = (y / (x * z)) + (0.5 / t_0);
	} else if (x <= 490.0) {
		tmp = (y / z) / x;
	} else if (x <= 5.6e+222) {
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (z / y) / x
    t_1 = (((y * t_0) / x) + (z * 0.5d0)) / (z * (z / (y * x)))
    if (x <= (-1.25d+88)) then
        tmp = t_1
    else if (x <= 1.3d-242) then
        tmp = (y / (x * z)) + (0.5d0 / t_0)
    else if (x <= 490.0d0) then
        tmp = (y / z) / x
    else if (x <= 5.6d+222) then
        tmp = (((x * z) * (x * (y * 0.5d0))) + (y * z)) / (z * (x * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z / y) / x;
	double t_1 = (((y * t_0) / x) + (z * 0.5)) / (z * (z / (y * x)));
	double tmp;
	if (x <= -1.25e+88) {
		tmp = t_1;
	} else if (x <= 1.3e-242) {
		tmp = (y / (x * z)) + (0.5 / t_0);
	} else if (x <= 490.0) {
		tmp = (y / z) / x;
	} else if (x <= 5.6e+222) {
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / y) / x
	t_1 = (((y * t_0) / x) + (z * 0.5)) / (z * (z / (y * x)))
	tmp = 0
	if x <= -1.25e+88:
		tmp = t_1
	elif x <= 1.3e-242:
		tmp = (y / (x * z)) + (0.5 / t_0)
	elif x <= 490.0:
		tmp = (y / z) / x
	elif x <= 5.6e+222:
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / y) / x)
	t_1 = Float64(Float64(Float64(Float64(y * t_0) / x) + Float64(z * 0.5)) / Float64(z * Float64(z / Float64(y * x))))
	tmp = 0.0
	if (x <= -1.25e+88)
		tmp = t_1;
	elseif (x <= 1.3e-242)
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(0.5 / t_0));
	elseif (x <= 490.0)
		tmp = Float64(Float64(y / z) / x);
	elseif (x <= 5.6e+222)
		tmp = Float64(Float64(Float64(Float64(x * z) * Float64(x * Float64(y * 0.5))) + Float64(y * z)) / Float64(z * Float64(x * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z / y) / x;
	t_1 = (((y * t_0) / x) + (z * 0.5)) / (z * (z / (y * x)));
	tmp = 0.0;
	if (x <= -1.25e+88)
		tmp = t_1;
	elseif (x <= 1.3e-242)
		tmp = (y / (x * z)) + (0.5 / t_0);
	elseif (x <= 490.0)
		tmp = (y / z) / x;
	elseif (x <= 5.6e+222)
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / y), $MachinePrecision] / x), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(y * t$95$0), $MachinePrecision] / x), $MachinePrecision] + N[(z * 0.5), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e+88], t$95$1, If[LessEqual[x, 1.3e-242], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 490.0], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.6e+222], N[(N[(N[(N[(x * z), $MachinePrecision] * N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.24999999999999999e88 or 5.6000000000000003e222 < x

   1. Initial program 66.7%

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

      1. associate-*l/ 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in x around 0 62.1%

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

      1. clear-num 62.1%

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

      2. un-div-inv 62.1%

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

      3. associate-/r* 54.6%

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

   6. Applied egg-rr 54.6%

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

      1. +-commutative 54.6%

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

      2. associate-/r* 54.6%

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

      3. frac-add 57.5%

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

      4. associate-/l/ 57.5%

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

      5. *-commutative 57.5%

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

      6. associate-/l/ 70.8%

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

      7. *-commutative 70.8%

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

   8. Applied egg-rr 70.8%

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

      1. associate-*l/ 70.8%

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

      2. *-commutative 70.8%

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

      3. associate-/r* 75.4%

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

   10. Applied egg-rr 75.4%

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

   if -1.24999999999999999e88 < x < 1.30000000000000009e-242

   1. Initial program 91.1%

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

      1. associate-*l/ 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in x around 0 84.0%

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

      1. clear-num 84.0%

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

      2. un-div-inv 84.0%

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

      3. associate-/r* 84.0%

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

   6. Applied egg-rr 84.0%

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

   7. Taylor expanded in z around 0 84.0%

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

      1. *-commutative 84.0%

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

   9. Simplified 84.0%

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

   10. Taylor expanded in z around 0 84.0%

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

       1. associate-/l/ 84.0%

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

   12. Simplified 84.0%

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

   if 1.30000000000000009e-242 < x < 490

   1. Initial program 90.7%

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

      1. associate-*l/ 90.5%

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

   3. Simplified 90.5%

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

      1. associate-*r/ 97.9%

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

      2. associate-*l/ 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 96.9%

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

   if 490 < x < 5.6000000000000003e222

   1. Initial program 86.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 32.4%

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

      1. associate-*r/ 32.4%

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

      2. frac-add 53.4%

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

      3. *-commutative 53.4%

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

      4. associate-*l* 53.4%

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

   6. Applied egg-rr 53.4%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(y \cdot 0.5\right)\right) \cdot \left(x \cdot z\right) + z \cdot y}{z \cdot \left(x \cdot z\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{\frac{z}{y}}{x}}{x} + z \cdot 0.5}{z \cdot \frac{z}{y \cdot x}}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-242}:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{0.5}{\frac{\frac{z}{y}}{x}}\\ \mathbf{elif}\;x \leq 490:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{+222}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot \left(x \cdot \left(y \cdot 0.5\right)\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \frac{\frac{z}{y}}{x}}{x} + z \cdot 0.5}{z \cdot \frac{z}{y \cdot x}}\\ \end{array} \]

Alternative 6: 66.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{y \cdot x}\\ \mathbf{if}\;x \leq -3 \cdot 10^{+90}:\\ \;\;\;\;\frac{z \cdot 0.5 + \frac{y}{x} \cdot t_0}{z \cdot t_0}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{0.5}{\frac{\frac{z}{y}}{x}}\\ \mathbf{elif}\;x \leq 17:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot \left(x \cdot \left(y \cdot 0.5\right)\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ z (* y x))))
   (if (<= x -3e+90)
     (/ (+ (* z 0.5) (* (/ y x) t_0)) (* z t_0))
     (if (<= x 3.3e-251)
       (+ (/ y (* x z)) (/ 0.5 (/ (/ z y) x)))
       (if (<= x 17.0)
         (/ (/ y z) x)
         (/ (+ (* (* x z) (* x (* y 0.5))) (* y z)) (* z (* x z))))))))

C

double code(double x, double y, double z) {
	double t_0 = z / (y * x);
	double tmp;
	if (x <= -3e+90) {
		tmp = ((z * 0.5) + ((y / x) * t_0)) / (z * t_0);
	} else if (x <= 3.3e-251) {
		tmp = (y / (x * z)) + (0.5 / ((z / y) / x));
	} else if (x <= 17.0) {
		tmp = (y / z) / x;
	} else {
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z / (y * x)
    if (x <= (-3d+90)) then
        tmp = ((z * 0.5d0) + ((y / x) * t_0)) / (z * t_0)
    else if (x <= 3.3d-251) then
        tmp = (y / (x * z)) + (0.5d0 / ((z / y) / x))
    else if (x <= 17.0d0) then
        tmp = (y / z) / x
    else
        tmp = (((x * z) * (x * (y * 0.5d0))) + (y * z)) / (z * (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 <= -3e+90) {
		tmp = ((z * 0.5) + ((y / x) * t_0)) / (z * t_0);
	} else if (x <= 3.3e-251) {
		tmp = (y / (x * z)) + (0.5 / ((z / y) / x));
	} else if (x <= 17.0) {
		tmp = (y / z) / x;
	} else {
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z / (y * x)
	tmp = 0
	if x <= -3e+90:
		tmp = ((z * 0.5) + ((y / x) * t_0)) / (z * t_0)
	elif x <= 3.3e-251:
		tmp = (y / (x * z)) + (0.5 / ((z / y) / x))
	elif x <= 17.0:
		tmp = (y / z) / x
	else:
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z / Float64(y * x))
	tmp = 0.0
	if (x <= -3e+90)
		tmp = Float64(Float64(Float64(z * 0.5) + Float64(Float64(y / x) * t_0)) / Float64(z * t_0));
	elseif (x <= 3.3e-251)
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(0.5 / Float64(Float64(z / y) / x)));
	elseif (x <= 17.0)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(Float64(Float64(x * z) * Float64(x * Float64(y * 0.5))) + Float64(y * z)) / Float64(z * 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 <= -3e+90)
		tmp = ((z * 0.5) + ((y / x) * t_0)) / (z * t_0);
	elseif (x <= 3.3e-251)
		tmp = (y / (x * z)) + (0.5 / ((z / y) / x));
	elseif (x <= 17.0)
		tmp = (y / z) / x;
	else
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (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, -3e+90], N[(N[(N[(z * 0.5), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(z * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.3e-251], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 / N[(N[(z / y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 17.0], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(x * z), $MachinePrecision] * N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.99999999999999979e90

   1. Initial program 69.2%

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

      1. associate-*l/ 69.2%

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

   3. Simplified 69.2%

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

   4. Taylor expanded in x around 0 60.9%

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

      1. clear-num 60.9%

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

      2. un-div-inv 60.9%

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

      3. associate-/r* 51.3%

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

   6. Applied egg-rr 51.3%

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

      1. +-commutative 51.3%

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

      2. associate-/r* 51.3%

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

      3. frac-add 56.0%

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

      4. associate-/l/ 56.0%

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

      5. *-commutative 56.0%

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

      6. associate-/l/ 72.7%

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

      7. *-commutative 72.7%

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

   8. Applied egg-rr 72.7%

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

   if -2.99999999999999979e90 < x < 3.3e-251

   1. Initial program 91.1%

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

      1. associate-*l/ 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in x around 0 84.0%

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

      1. clear-num 84.0%

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

      2. un-div-inv 84.0%

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

      3. associate-/r* 84.0%

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

   6. Applied egg-rr 84.0%

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

   7. Taylor expanded in z around 0 84.0%

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

      1. *-commutative 84.0%

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

   9. Simplified 84.0%

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

   10. Taylor expanded in z around 0 84.0%

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

       1. associate-/l/ 84.0%

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

   12. Simplified 84.0%

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

   if 3.3e-251 < x < 17

   1. Initial program 90.7%

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

      1. associate-*l/ 90.5%

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

   3. Simplified 90.5%

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

      1. associate-*r/ 97.9%

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

      2. associate-*l/ 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 96.9%

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

   if 17 < x

   1. Initial program 77.9%

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

      1. associate-*l/ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in x around 0 43.5%

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

      1. associate-*r/ 43.5%

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

      2. frac-add 53.9%

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

      3. *-commutative 53.9%

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

      4. associate-*l* 53.9%

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

   6. Applied egg-rr 53.9%

      \[\leadsto \color{blue}{\frac{\left(x \cdot \left(y \cdot 0.5\right)\right) \cdot \left(x \cdot z\right) + z \cdot y}{z \cdot \left(x \cdot z\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+90}:\\ \;\;\;\;\frac{z \cdot 0.5 + \frac{y}{x} \cdot \frac{z}{y \cdot x}}{z \cdot \frac{z}{y \cdot x}}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-251}:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{0.5}{\frac{\frac{z}{y}}{x}}\\ \mathbf{elif}\;x \leq 17:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot \left(x \cdot \left(y \cdot 0.5\right)\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 7: 64.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-245}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot \left(x \cdot \left(y \cdot 0.5\right)\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 6.2e-245)
   (+ (/ y (* x z)) (* 0.5 (/ (* y x) z)))
   (if (<= x 370.0)
     (/ (/ y z) x)
     (/ (+ (* (* x z) (* x (* y 0.5))) (* y z)) (* z (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.2e-245) {
		tmp = (y / (x * z)) + (0.5 * ((y * x) / z));
	} else if (x <= 370.0) {
		tmp = (y / z) / x;
	} else {
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 6.2d-245) then
        tmp = (y / (x * z)) + (0.5d0 * ((y * x) / z))
    else if (x <= 370.0d0) then
        tmp = (y / z) / x
    else
        tmp = (((x * z) * (x * (y * 0.5d0))) + (y * z)) / (z * (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 6.2e-245) {
		tmp = (y / (x * z)) + (0.5 * ((y * x) / z));
	} else if (x <= 370.0) {
		tmp = (y / z) / x;
	} else {
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 6.2e-245:
		tmp = (y / (x * z)) + (0.5 * ((y * x) / z))
	elif x <= 370.0:
		tmp = (y / z) / x
	else:
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 6.2e-245)
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(0.5 * Float64(Float64(y * x) / z)));
	elseif (x <= 370.0)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(Float64(Float64(x * z) * Float64(x * Float64(y * 0.5))) + Float64(y * z)) / Float64(z * Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 6.2e-245)
		tmp = (y / (x * z)) + (0.5 * ((y * x) / z));
	elseif (x <= 370.0)
		tmp = (y / z) / x;
	else
		tmp = (((x * z) * (x * (y * 0.5))) + (y * z)) / (z * (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 6.2e-245], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 370.0], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(N[(x * z), $MachinePrecision] * N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.20000000000000006e-245

   1. Initial program 84.9%

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

      1. associate-*l/ 84.9%

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

   3. Simplified 84.9%

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

   4. Taylor expanded in x around 0 77.4%

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

   if 6.20000000000000006e-245 < x < 370

   1. Initial program 90.7%

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

      1. associate-*l/ 90.5%

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

   3. Simplified 90.5%

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

      1. associate-*r/ 97.9%

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

      2. associate-*l/ 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 96.9%

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

   if 370 < x

   1. Initial program 77.9%

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

      1. associate-*l/ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in x around 0 43.5%

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

      1. associate-*r/ 43.5%

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

      2. frac-add 53.9%

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

      3. *-commutative 53.9%

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

      4. associate-*l* 53.9%

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

   6. Applied egg-rr 53.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-245}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{elif}\;x \leq 370:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot \left(x \cdot \left(y \cdot 0.5\right)\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \end{array} \]

Alternative 8: 69.3% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+82} \lor \neg \left(z \leq 3 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.7e+82) (not (<= z 3e-45)))
   (+ (/ y (* x z)) (* 0.5 (/ (* y x) z)))
   (/ (+ (/ y x) (* 0.5 (* y x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.7e+82) || !(z <= 3e-45)) {
		tmp = (y / (x * z)) + (0.5 * ((y * x) / z));
	} else {
		tmp = ((y / x) + (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 <= (-4.7d+82)) .or. (.not. (z <= 3d-45))) then
        tmp = (y / (x * z)) + (0.5d0 * ((y * x) / z))
    else
        tmp = ((y / x) + (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 <= -4.7e+82) || !(z <= 3e-45)) {
		tmp = (y / (x * z)) + (0.5 * ((y * x) / z));
	} else {
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.7e+82) or not (z <= 3e-45):
		tmp = (y / (x * z)) + (0.5 * ((y * x) / z))
	else:
		tmp = ((y / x) + (0.5 * (y * x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.7e+82) || !(z <= 3e-45))
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(0.5 * Float64(Float64(y * x) / z)));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(y * x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.7e+82) || ~((z <= 3e-45)))
		tmp = (y / (x * z)) + (0.5 * ((y * x) / z));
	else
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.7e+82], N[Not[LessEqual[z, 3e-45]], $MachinePrecision]], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.7e82 or 3.00000000000000011e-45 < z

   1. Initial program 75.6%

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

      1. associate-*l/ 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in x around 0 66.1%

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

   if -4.7e82 < z < 3.00000000000000011e-45

   1. Initial program 90.9%

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

   2. Taylor expanded in x around 0 80.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+82} \lor \neg \left(z \leq 3 \cdot 10^{-45}\right):\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]

Alternative 9: 65.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{\frac{z}{0.5}}\\ \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x (/ z 0.5)))))
   (if (<= x -1.45)
     t_0
     (if (<= x 5.2e-248) (/ y (* x z)) (if (<= x 1.45) (/ (/ y z) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / (z / 0.5));
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 5.2e-248) {
		tmp = y / (x * z);
	} else if (x <= 1.45) {
		tmp = (y / z) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x / (z / 0.5d0))
    if (x <= (-1.45d0)) then
        tmp = t_0
    else if (x <= 5.2d-248) then
        tmp = y / (x * z)
    else if (x <= 1.45d0) then
        tmp = (y / z) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x / (z / 0.5));
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 5.2e-248) {
		tmp = y / (x * z);
	} else if (x <= 1.45) {
		tmp = (y / z) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / (z / 0.5))
	tmp = 0
	if x <= -1.45:
		tmp = t_0
	elif x <= 5.2e-248:
		tmp = y / (x * z)
	elif x <= 1.45:
		tmp = (y / z) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / Float64(z / 0.5)))
	tmp = 0.0
	if (x <= -1.45)
		tmp = t_0;
	elseif (x <= 5.2e-248)
		tmp = Float64(y / Float64(x * z));
	elseif (x <= 1.45)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / (z / 0.5));
	tmp = 0.0;
	if (x <= -1.45)
		tmp = t_0;
	elseif (x <= 5.2e-248)
		tmp = y / (x * z);
	elseif (x <= 1.45)
		tmp = (y / z) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / N[(z / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.45], t$95$0, If[LessEqual[x, 5.2e-248], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996 or 1.44999999999999996 < x

   1. Initial program 78.9%

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

   2. Taylor expanded in x around 0 48.4%

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

   3. Taylor expanded in x around inf 48.4%

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

      1. associate-*l/ 46.1%

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

      2. *-commutative 46.1%

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

      3. associate-*r* 46.1%

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

      4. *-commutative 46.1%

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

      5. associate-*r* 46.1%

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

      6. associate-*r/ 46.1%

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

      7. *-commutative 46.1%

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

      8. associate-/l* 46.1%

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

   5. Simplified 46.1%

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

   if -1.44999999999999996 < x < 5.20000000000000013e-248

   1. Initial program 88.7%

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

      1. associate-*l/ 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in x around 0 95.3%

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

   if 5.20000000000000013e-248 < x < 1.44999999999999996

   1. Initial program 90.7%

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

      1. associate-*l/ 90.5%

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

   3. Simplified 90.5%

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

      1. associate-*r/ 97.9%

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

      2. associate-*l/ 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 96.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;y \cdot \frac{x}{\frac{z}{0.5}}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-248}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z}{0.5}}\\ \end{array} \]

Alternative 10: 65.7% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 10^{-252}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y (* x 0.5)) z)))
   (if (<= x -1.45)
     t_0
     (if (<= x 1e-252) (/ y (* x z)) (if (<= x 1.45) (/ (/ y z) x) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (y * (x * 0.5)) / z;
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 1e-252) {
		tmp = y / (x * z);
	} else if (x <= 1.45) {
		tmp = (y / z) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * (x * 0.5d0)) / z
    if (x <= (-1.45d0)) then
        tmp = t_0
    else if (x <= 1d-252) then
        tmp = y / (x * z)
    else if (x <= 1.45d0) then
        tmp = (y / z) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * (x * 0.5)) / z;
	double tmp;
	if (x <= -1.45) {
		tmp = t_0;
	} else if (x <= 1e-252) {
		tmp = y / (x * z);
	} else if (x <= 1.45) {
		tmp = (y / z) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * (x * 0.5)) / z
	tmp = 0
	if x <= -1.45:
		tmp = t_0
	elif x <= 1e-252:
		tmp = y / (x * z)
	elif x <= 1.45:
		tmp = (y / z) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * Float64(x * 0.5)) / z)
	tmp = 0.0
	if (x <= -1.45)
		tmp = t_0;
	elseif (x <= 1e-252)
		tmp = Float64(y / Float64(x * z));
	elseif (x <= 1.45)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * (x * 0.5)) / z;
	tmp = 0.0;
	if (x <= -1.45)
		tmp = t_0;
	elseif (x <= 1e-252)
		tmp = y / (x * z);
	elseif (x <= 1.45)
		tmp = (y / z) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -1.45], t$95$0, If[LessEqual[x, 1e-252], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.45], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996 or 1.44999999999999996 < x

   1. Initial program 78.9%

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

   2. Taylor expanded in x around 0 48.4%

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

   3. Taylor expanded in x around inf 48.4%

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

      1. associate-*r* 48.4%

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

      2. *-commutative 48.4%

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

   5. Simplified 48.4%

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

   if -1.44999999999999996 < x < 9.99999999999999943e-253

   1. Initial program 88.7%

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

      1. associate-*l/ 88.7%

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

   3. Simplified 88.7%

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

   4. Taylor expanded in x around 0 95.3%

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

   if 9.99999999999999943e-253 < x < 1.44999999999999996

   1. Initial program 90.7%

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

      1. associate-*l/ 90.5%

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

   3. Simplified 90.5%

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

      1. associate-*r/ 97.9%

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

      2. associate-*l/ 97.9%

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

   5. Applied egg-rr 97.9%

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

   6. Taylor expanded in x around 0 96.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{elif}\;x \leq 10^{-252}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.45:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5\right)}{z}\\ \end{array} \]

Alternative 11: 65.9% accurate, 8.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (0.5 * (y * (x / z))) + (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 = (0.5d0 * (y * (x / z))) + (y / (x * z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

   1. associate-*l/ 84.2%

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

3. Simplified 84.2%

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

4. Taylor expanded in x around 0 70.6%

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

   1. associate-/l* 66.5%

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

   2. associate-/r/ 69.5%

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

6. Applied egg-rr 69.5%

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

7. Final simplification 69.5%

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

Alternative 12: 65.6% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in x around 0 68.8%

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

3. Taylor expanded in y around 0 68.7%

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

4. Final simplification 68.7%

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

Alternative 13: 65.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return ((y / x) + (0.5 * (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) + (0.5d0 * (y * x))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.2%

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

2. Taylor expanded in x around 0 68.8%

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

3. Final simplification 68.8%

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

Alternative 14: 57.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+26} \lor \neg \left(y \leq 1.9 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -8e+26) (not (<= y 1.9e-64))) (/ (/ y z) x) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -8e+26) || !(y <= 1.9e-64)) {
		tmp = (y / z) / x;
	} 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 <= (-8d+26)) .or. (.not. (y <= 1.9d-64))) then
        tmp = (y / z) / x
    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 <= -8e+26) || !(y <= 1.9e-64)) {
		tmp = (y / z) / x;
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -8e+26) or not (y <= 1.9e-64):
		tmp = (y / z) / x
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -8e+26) || !(y <= 1.9e-64))
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -8e+26) || ~((y <= 1.9e-64)))
		tmp = (y / z) / x;
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -8e+26], N[Not[LessEqual[y, 1.9e-64]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.00000000000000038e26 or 1.9000000000000001e-64 < y

   1. Initial program 90.6%

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

      1. associate-*l/ 90.6%

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

   3. Simplified 90.6%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 65.3%

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

   if -8.00000000000000038e26 < y < 1.9000000000000001e-64

   1. Initial program 76.4%

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

   2. Taylor expanded in x around 0 51.7%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+26} \lor \neg \left(y \leq 1.9 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 15: 51.0% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-60}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.9e-60) (/ y (* x z)) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.9e-60) {
		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 <= (-2.9d-60)) 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 <= -2.9e-60) {
		tmp = y / (x * z);
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.9e-60:
		tmp = y / (x * z)
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.9e-60)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.9e-60)
		tmp = y / (x * z);
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.9e-60], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8999999999999999e-60

   1. Initial program 89.7%

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

      1. associate-*l/ 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in x around 0 51.8%

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

   if -2.8999999999999999e-60 < y

   1. Initial program 81.2%

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

   2. Taylor expanded in x around 0 52.9%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-60}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 16: 49.7% 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 84.2%

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

   1. associate-*l/ 84.2%

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

3. Simplified 84.2%

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

4. Taylor expanded in x around 0 50.7%

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

5. Final simplification 50.7%

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

Developer target: 96.9% 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 2023308 
(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))