Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.4% → 97.2%

Time: 8.5s

Alternatives: 14

Speedup: 1.0×

• 64.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.4% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+133}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.7e+133) (/ (* y (/ (cosh x) x)) z) (/ (cosh x) (* x (/ z y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.7e+133) {
		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 <= 3.7d+133) 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 <= 3.7e+133) {
		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 <= 3.7e+133:
		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 <= 3.7e+133)
		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 <= 3.7e+133)
		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, 3.7e+133], 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 < 3.70000000000000023e133

   1. Initial program 84.5%

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

      1. associate-*r/ 97.6%

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

      2. associate-/l/ 79.4%

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

      3. associate-*l/ 79.3%

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

      4. *-commutative 79.3%

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

      5. *-commutative 79.3%

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

   3. Simplified 79.3%

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

      1. *-commutative 79.3%

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

      2. associate-/r* 97.6%

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

      3. associate-*l/ 97.6%

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

   5. Applied egg-rr 97.6%

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

   if 3.70000000000000023e133 < y

   1. Initial program 89.1%

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

      1. associate-/l* 89.1%

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

      2. associate-/r/ 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{+133}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x \cdot \frac{z}{y}}\\ \end{array} \]

Alternative 2: 93.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < +inf.0

   1. Initial program 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-*r/ 95.8%

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

   3. Simplified 95.8%

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

   if +inf.0 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 0.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 50.0%

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

      3. associate-*l/ 50.0%

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

      4. *-commutative 50.0%

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

      5. *-commutative 50.0%

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

   3. Simplified 50.0%

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

      1. *-commutative 50.0%

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

      2. associate-/r* 100.0%

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

      3. associate-*l/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 72.2%

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

      1. unpow2 29.2%

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

   8. Simplified 72.2%

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

4. Final simplification 93.2%

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

Alternative 3: 92.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154} \lor \neg \left(x \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.35e+154) (not (<= x 1.35e+154)))
   (/ (* y (/ (+ 1.0 (* 0.5 (* x x))) x)) z)
   (* y (/ (cosh x) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.35e+154) || !(x <= 1.35e+154)) {
		tmp = (y * ((1.0 + (0.5 * (x * x))) / x)) / z;
	} 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 ((x <= (-1.35d+154)) .or. (.not. (x <= 1.35d+154))) then
        tmp = (y * ((1.0d0 + (0.5d0 * (x * x))) / x)) / z
    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 ((x <= -1.35e+154) || !(x <= 1.35e+154)) {
		tmp = (y * ((1.0 + (0.5 * (x * x))) / x)) / z;
	} else {
		tmp = y * (Math.cosh(x) / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.35e+154) or not (x <= 1.35e+154):
		tmp = (y * ((1.0 + (0.5 * (x * x))) / x)) / z
	else:
		tmp = y * (math.cosh(x) / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.35e+154) || !(x <= 1.35e+154))
		tmp = Float64(Float64(y * Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / x)) / z);
	else
		tmp = Float64(y * Float64(cosh(x) / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.35e+154) || ~((x <= 1.35e+154)))
		tmp = (y * ((1.0 + (0.5 * (x * x))) / x)) / z;
	else
		tmp = y * (cosh(x) / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.35e+154], N[Not[LessEqual[x, 1.35e+154]], $MachinePrecision]], N[(N[(y * N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e154 or 1.35000000000000003e154 < x

   1. Initial program 69.2%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 49.2%

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

      3. associate-*l/ 49.2%

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

      4. *-commutative 49.2%

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

      5. *-commutative 49.2%

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

   3. Simplified 49.2%

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

      1. *-commutative 49.2%

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

      2. associate-/r* 100.0%

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

      3. associate-*l/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. unpow2 49.2%

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

   8. Simplified 100.0%

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

   if -1.35000000000000003e154 < x < 1.35000000000000003e154

   1. Initial program 90.7%

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

      1. associate-*r/ 94.9%

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

      2. associate-/l/ 91.6%

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

      3. associate-*l/ 91.6%

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

      4. *-commutative 91.6%

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

      5. *-commutative 91.6%

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

   3. Simplified 91.6%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+154} \lor \neg \left(x \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \]

Alternative 4: 68.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+117} \lor \neg \left(y \leq 2.6 \cdot 10^{+218}\right):\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -4.4e+117) (not (<= y 2.6e+218)))
   (/ (* y (+ (* x 0.5) (/ 1.0 x))) z)
   (* y (/ (+ 1.0 (* 0.5 (* x x))) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.4e+117) || !(y <= 2.6e+218)) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	} else {
		tmp = y * ((1.0 + (0.5 * (x * 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 <= (-4.4d+117)) .or. (.not. (y <= 2.6d+218))) then
        tmp = (y * ((x * 0.5d0) + (1.0d0 / x))) / z
    else
        tmp = y * ((1.0d0 + (0.5d0 * (x * x))) / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -4.4e+117) || !(y <= 2.6e+218)) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	} else {
		tmp = y * ((1.0 + (0.5 * (x * x))) / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -4.4e+117) or not (y <= 2.6e+218):
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z
	else:
		tmp = y * ((1.0 + (0.5 * (x * x))) / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -4.4e+117) || !(y <= 2.6e+218))
		tmp = Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z);
	else
		tmp = Float64(y * Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -4.4e+117) || ~((y <= 2.6e+218)))
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	else
		tmp = y * ((1.0 + (0.5 * (x * x))) / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -4.4e+117], N[Not[LessEqual[y, 2.6e+218]], $MachinePrecision]], N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.40000000000000028e117 or 2.60000000000000002e218 < y

   1. Initial program 94.8%

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

      1. associate-*r/ 94.8%

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

      2. associate-/l/ 72.7%

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

      3. associate-*l/ 72.7%

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

      4. *-commutative 72.7%

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

      5. *-commutative 72.7%

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

   3. Simplified 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-/r* 100.0%

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

      3. associate-*l/ 94.8%

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

   5. Applied egg-rr 94.8%

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

   6. Taylor expanded in x around 0 87.9%

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

   if -4.40000000000000028e117 < y < 2.60000000000000002e218

   1. Initial program 82.7%

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

      1. associate-*r/ 96.6%

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

      2. associate-/l/ 83.1%

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

      3. associate-*l/ 83.0%

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

      4. *-commutative 83.0%

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

      5. *-commutative 83.0%

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

   3. Simplified 83.0%

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

   4. Taylor expanded in x around 0 65.0%

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

      1. unpow2 65.0%

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

   6. Simplified 65.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+117} \lor \neg \left(y \leq 2.6 \cdot 10^{+218}\right):\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{x \cdot z}\\ \end{array} \]

Alternative 5: 69.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-266} \lor \neg \left(x \leq 2.1 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x 1.85e-266) (not (<= x 2.1e-150)))
   (* (/ y x) (/ (+ 1.0 (* 0.5 (* x x))) z))
   (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= 1.85e-266) || !(x <= 2.1e-150)) {
		tmp = (y / x) * ((1.0 + (0.5 * (x * 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 ((x <= 1.85d-266) .or. (.not. (x <= 2.1d-150))) then
        tmp = (y / x) * ((1.0d0 + (0.5d0 * (x * 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 ((x <= 1.85e-266) || !(x <= 2.1e-150)) {
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= 1.85e-266) or not (x <= 2.1e-150):
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z)
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= 1.85e-266) || !(x <= 2.1e-150))
		tmp = Float64(Float64(y / x) * Float64(Float64(1.0 + Float64(0.5 * Float64(x * 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 ((x <= 1.85e-266) || ~((x <= 2.1e-150)))
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, 1.85e-266], N[Not[LessEqual[x, 2.1e-150]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001e-266 or 2.1000000000000001e-150 < x

   1. Initial program 85.9%

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

      1. *-commutative 85.9%

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

      2. associate-*r/ 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in x around 0 66.2%

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

      1. unpow2 59.6%

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

   6. Simplified 66.2%

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

   if 1.8500000000000001e-266 < x < 2.1000000000000001e-150

   1. Initial program 80.2%

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

      1. associate-*r/ 80.2%

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

      2. associate-/l/ 99.8%

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

      3. associate-*l/ 99.4%

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

      4. *-commutative 99.4%

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

      5. *-commutative 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 99.8%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-266} \lor \neg \left(x \leq 2.1 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 6: 69.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-269} \lor \neg \left(x \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x 9e-269) (not (<= x 1.35e+154)))
   (* (/ y x) (/ (+ 1.0 (* 0.5 (* x x))) z))
   (+ (* 0.5 (/ y (/ z x))) (/ y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= 9e-269) || !(x <= 1.35e+154)) {
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	} else {
		tmp = (0.5 * (y / (z / x))) + (y / (x * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= 9d-269) .or. (.not. (x <= 1.35d+154))) then
        tmp = (y / x) * ((1.0d0 + (0.5d0 * (x * x))) / z)
    else
        tmp = (0.5d0 * (y / (z / x))) + (y / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= 9e-269) || !(x <= 1.35e+154)) {
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	} else {
		tmp = (0.5 * (y / (z / x))) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= 9e-269) or not (x <= 1.35e+154):
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z)
	else:
		tmp = (0.5 * (y / (z / x))) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= 9e-269) || !(x <= 1.35e+154))
		tmp = Float64(Float64(y / x) * Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / z));
	else
		tmp = Float64(Float64(0.5 * Float64(y / Float64(z / x))) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= 9e-269) || ~((x <= 1.35e+154)))
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	else
		tmp = (0.5 * (y / (z / x))) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, 9e-269], N[Not[LessEqual[x, 1.35e+154]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.0000000000000003e-269 or 1.35000000000000003e154 < x

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*r/ 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in x around 0 69.3%

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

      1. unpow2 60.4%

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

   6. Simplified 69.3%

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

   if 9.0000000000000003e-269 < x < 1.35000000000000003e154

   1. Initial program 88.7%

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

      1. associate-*r/ 93.3%

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

      2. associate-/l/ 93.0%

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

      3. associate-*l/ 92.9%

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

      4. *-commutative 92.9%

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

      5. *-commutative 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in x around 0 72.7%

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

      1. associate-/r* 68.0%

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

      2. +-commutative 68.0%

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

      3. associate-/l* 66.9%

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

      4. associate-/r* 71.6%

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

   6. Simplified 71.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-269} \lor \neg \left(x \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 7: 69.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;z \leq -5.4 \cdot 10^{+17} \lor \neg \left(z \leq 4 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{\frac{y}{x} \cdot t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t_0}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 0.5 (* x x)))))
   (if (or (<= z -5.4e+17) (not (<= z 4e-188)))
     (/ (* (/ y x) t_0) z)
     (* y (/ t_0 (* x z))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if ((z <= -5.4e+17) || !(z <= 4e-188)) {
		tmp = ((y / x) * t_0) / z;
	} else {
		tmp = y * (t_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 = 1.0d0 + (0.5d0 * (x * x))
    if ((z <= (-5.4d+17)) .or. (.not. (z <= 4d-188))) then
        tmp = ((y / x) * t_0) / z
    else
        tmp = y * (t_0 / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (0.5 * (x * x));
	double tmp;
	if ((z <= -5.4e+17) || !(z <= 4e-188)) {
		tmp = ((y / x) * t_0) / z;
	} else {
		tmp = y * (t_0 / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (0.5 * (x * x))
	tmp = 0
	if (z <= -5.4e+17) or not (z <= 4e-188):
		tmp = ((y / x) * t_0) / z
	else:
		tmp = y * (t_0 / (x * z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(0.5 * Float64(x * x)))
	tmp = 0.0
	if ((z <= -5.4e+17) || !(z <= 4e-188))
		tmp = Float64(Float64(Float64(y / x) * t_0) / z);
	else
		tmp = Float64(y * Float64(t_0 / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (0.5 * (x * x));
	tmp = 0.0;
	if ((z <= -5.4e+17) || ~((z <= 4e-188)))
		tmp = ((y / x) * t_0) / z;
	else
		tmp = y * (t_0 / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[z, -5.4e+17], N[Not[LessEqual[z, 4e-188]], $MachinePrecision]], N[(N[(N[(y / x), $MachinePrecision] * t$95$0), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(t$95$0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4e17 or 3.9999999999999998e-188 < z

   1. Initial program 83.3%

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

   2. Taylor expanded in x around 0 62.7%

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

      1. unpow2 51.6%

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

   4. Simplified 62.7%

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

   if -5.4e17 < z < 3.9999999999999998e-188

   1. Initial program 88.8%

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

      1. associate-*r/ 99.9%

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

      2. associate-/l/ 96.8%

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

      3. associate-*l/ 96.8%

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

      4. *-commutative 96.8%

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

      5. *-commutative 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in x around 0 87.3%

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

      1. unpow2 87.3%

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

   6. Simplified 87.3%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+17} \lor \neg \left(z \leq 4 \cdot 10^{-188}\right):\\ \;\;\;\;\frac{\frac{y}{x} \cdot \left(1 + 0.5 \cdot \left(x \cdot x\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{x \cdot z}\\ \end{array} \]

Alternative 8: 77.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return (y * ((1.0 + (0.5 * (x * x))) / 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 * ((1.0d0 + (0.5d0 * (x * x))) / x)) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. associate-*r/ 96.2%

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

   2. associate-/l/ 80.9%

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

   3. associate-*l/ 80.8%

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

   4. *-commutative 80.8%

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

   5. *-commutative 80.8%

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

3. Simplified 80.8%

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

   1. *-commutative 80.8%

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

   2. associate-/r* 98.0%

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

   3. associate-*l/ 96.2%

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

5. Applied egg-rr 96.2%

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

6. Taylor expanded in x around 0 76.0%

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

   1. unpow2 64.1%

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

8. Simplified 76.0%

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

9. Final simplification 76.0%

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

Alternative 9: 65.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -28 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -28.0) (not (<= x 1.4))) (* 0.5 (/ y (/ z x))) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -28.0) || !(x <= 1.4)) {
		tmp = 0.5 * (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 ((x <= (-28.0d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = 0.5d0 * (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 ((x <= -28.0) || !(x <= 1.4)) {
		tmp = 0.5 * (y / (z / x));
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -28.0) or not (x <= 1.4):
		tmp = 0.5 * (y / (z / x))
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -28.0) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(y / Float64(z / x)));
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -28.0) || ~((x <= 1.4)))
		tmp = 0.5 * (y / (z / x));
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -28.0], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -28 or 1.3999999999999999 < x

   1. Initial program 79.7%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 68.1%

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

      3. associate-*l/ 68.1%

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

      4. *-commutative 68.1%

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

      5. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around 0 37.7%

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

      1. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in x around inf 42.7%

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

      1. associate-/l* 37.9%

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

   9. Simplified 37.9%

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

   if -28 < x < 1.3999999999999999

   1. Initial program 91.8%

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

      1. associate-*r/ 91.8%

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

      2. associate-/l/ 95.8%

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

      3. associate-*l/ 95.6%

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

      4. *-commutative 95.6%

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

      5. *-commutative 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around 0 95.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -28 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 10: 65.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -28 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;\frac{0.5}{z} \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -28.0) (not (<= x 1.4))) (* (/ 0.5 z) (* y x)) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -28.0) || !(x <= 1.4)) {
		tmp = (0.5 / z) * (y * 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 ((x <= (-28.0d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = (0.5d0 / z) * (y * 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 ((x <= -28.0) || !(x <= 1.4)) {
		tmp = (0.5 / z) * (y * x);
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -28.0) or not (x <= 1.4):
		tmp = (0.5 / z) * (y * x)
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -28.0) || !(x <= 1.4))
		tmp = Float64(Float64(0.5 / z) * Float64(y * x));
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -28.0) || ~((x <= 1.4)))
		tmp = (0.5 / z) * (y * x);
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -28.0], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(N[(0.5 / z), $MachinePrecision] * N[(y * x), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -28 or 1.3999999999999999 < x

   1. Initial program 79.7%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 68.1%

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

      3. associate-*l/ 68.1%

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

      4. *-commutative 68.1%

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

      5. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in x around 0 37.7%

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

      1. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in x around inf 42.7%

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

      1. associate-*r/ 42.7%

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

      2. *-commutative 42.7%

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

      3. associate-*l* 42.7%

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

   9. Simplified 42.7%

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

       1. associate-/l* 37.9%

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

       2. div-inv 37.9%

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

       3. clear-num 37.9%

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

       4. *-commutative 37.9%

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

       5. associate-*r/ 37.9%

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

       6. *-commutative 37.9%

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

       7. associate-*r* 37.9%

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

       8. clear-num 37.9%

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

       9. div-inv 37.9%

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

       10. *-commutative 37.9%

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

       11. associate-*r/ 37.9%

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

       12. div-inv 37.9%

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

       13. times-frac 42.7%

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

   11. Applied egg-rr 42.7%

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

   12. Taylor expanded in y around 0 42.7%

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

   if -28 < x < 1.3999999999999999

   1. Initial program 91.8%

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

      1. associate-*r/ 91.8%

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

      2. associate-/l/ 95.8%

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

      3. associate-*l/ 95.6%

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

      4. *-commutative 95.6%

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

      5. *-commutative 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in x around 0 95.2%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -28 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;\frac{0.5}{z} \cdot \left(y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 11: 65.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. associate-*r/ 96.2%

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

   2. associate-/l/ 80.9%

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

   3. associate-*l/ 80.8%

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

   4. *-commutative 80.8%

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

   5. *-commutative 80.8%

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

3. Simplified 80.8%

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

   1. *-commutative 80.8%

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

   2. associate-/r* 98.0%

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

   3. associate-*l/ 96.2%

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

5. Applied egg-rr 96.2%

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

6. Taylor expanded in x around 0 65.0%

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

7. Final simplification 65.0%

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

Alternative 12: 65.7% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in x around 0 65.1%

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

   1. associate-*r* 65.1%

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

4. Simplified 65.1%

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

5. Final simplification 65.1%

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

Alternative 13: 55.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.4e-69) (/ (/ y z) x) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e-69) {
		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 (z <= 1.4d-69) 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 (z <= 1.4e-69) {
		tmp = (y / z) / x;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.4e-69:
		tmp = (y / z) / x
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.4e-69)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.4e-69)
		tmp = (y / z) / x;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.4e-69], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.3999999999999999e-69

   1. Initial program 87.1%

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

      1. associate-*r/ 97.1%

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

      2. associate-/l/ 84.8%

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

      3. associate-*l/ 84.7%

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

      4. *-commutative 84.7%

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

      5. *-commutative 84.7%

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

   3. Simplified 84.7%

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

   4. Taylor expanded in x around 0 48.0%

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

      1. div-inv 48.1%

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

      2. associate-/r* 56.2%

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

   6. Applied egg-rr 56.2%

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

   if 1.3999999999999999e-69 < z

   1. Initial program 81.6%

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

      1. associate-*r/ 94.4%

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

      2. associate-/l/ 73.1%

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

      3. associate-*l/ 73.0%

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

      4. *-commutative 73.0%

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

      5. *-commutative 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in x around 0 45.4%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 14: 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 85.3%

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

   1. associate-*r/ 96.2%

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

   2. associate-/l/ 80.9%

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

   3. associate-*l/ 80.8%

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

   4. *-commutative 80.8%

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

   5. *-commutative 80.8%

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

3. Simplified 80.8%

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

4. Taylor expanded in x around 0 47.2%

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

5. Final simplification 47.2%

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

Developer target: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023199 
(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))