Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 96.0%

Time: 10.3s

Alternatives: 12

Speedup: 1.0×

• 64.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. associate-*r/ 80.1%

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

   2. associate-/r* 76.7%

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

3. Simplified 76.7%

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

   1. associate-*r/ 81.0%

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

   2. *-commutative 81.0%

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

   3. frac-times 87.5%

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

   4. associate-*l/ 98.7%

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

5. Applied egg-rr 98.7%

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

6. Final simplification 98.7%

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

Alternative 2: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-157} \lor \neg \left(z \leq 2.4 \cdot 10^{-246}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.15e-157) (not (<= z 2.4e-246)))
   (* (cosh x) (/ y (* x z)))
   (+ (* 0.5 (/ y (/ z x))) (/ (/ y x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.15e-157) || !(z <= 2.4e-246)) {
		tmp = cosh(x) * (y / (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 ((z <= (-1.15d-157)) .or. (.not. (z <= 2.4d-246))) then
        tmp = cosh(x) * (y / (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 ((z <= -1.15e-157) || !(z <= 2.4e-246)) {
		tmp = Math.cosh(x) * (y / (x * z));
	} else {
		tmp = (0.5 * (y / (z / x))) + ((y / x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.15e-157) or not (z <= 2.4e-246):
		tmp = math.cosh(x) * (y / (x * z))
	else:
		tmp = (0.5 * (y / (z / x))) + ((y / x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.15e-157) || !(z <= 2.4e-246))
		tmp = Float64(cosh(x) * Float64(y / Float64(x * z)));
	else
		tmp = Float64(Float64(0.5 * Float64(y / Float64(z / x))) + Float64(Float64(y / x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.15e-157) || ~((z <= 2.4e-246)))
		tmp = cosh(x) * (y / (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[z, -1.15e-157], N[Not[LessEqual[z, 2.4e-246]], $MachinePrecision]], N[(N[Cosh[x], $MachinePrecision] * N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999994e-157 or 2.3999999999999998e-246 < z

   1. Initial program 88.5%

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

      1. associate-*r/ 79.4%

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

      2. associate-/r* 76.4%

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

   3. Simplified 76.4%

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

   if -1.14999999999999994e-157 < z < 2.3999999999999998e-246

   1. Initial program 83.1%

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

      1. associate-*r/ 83.1%

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

      2. associate-/r* 78.2%

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

   3. Simplified 78.2%

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

   4. Taylor expanded in x around 0 60.8%

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

      1. associate-/l/ 80.3%

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

      2. +-commutative 80.3%

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

      3. associate-/l* 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-157} \lor \neg \left(z \leq 2.4 \cdot 10^{-246}\right):\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 3: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e+106) (* (/ (cosh x) z) (/ y x)) (* (/ (cosh x) x) (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e+106) {
		tmp = (cosh(x) / z) * (y / x);
	} else {
		tmp = (cosh(x) / x) * (y / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.3d+106)) then
        tmp = (cosh(x) / z) * (y / x)
    else
        tmp = (cosh(x) / x) * (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e+106) {
		tmp = (Math.cosh(x) / z) * (y / x);
	} else {
		tmp = (Math.cosh(x) / x) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.3e+106:
		tmp = (math.cosh(x) / z) * (y / x)
	else:
		tmp = (math.cosh(x) / x) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e+106)
		tmp = Float64(Float64(cosh(x) / z) * Float64(y / x));
	else
		tmp = Float64(Float64(cosh(x) / x) * Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.3e+106)
		tmp = (cosh(x) / z) * (y / x);
	else
		tmp = (cosh(x) / x) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e+106], N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3000000000000001e106

   1. Initial program 84.3%

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

      1. associate-*l/ 84.4%

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

   3. Simplified 84.4%

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

   if -1.3000000000000001e106 < z

   1. Initial program 88.1%

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

      1. associate-/l* 82.5%

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

   3. Simplified 82.5%

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

      1. associate-/l* 88.1%

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

      2. associate-*r/ 98.2%

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

      3. associate-/r* 85.0%

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

      4. times-frac 95.8%

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

   5. Applied egg-rr 95.8%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 4: 90.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. associate-/l* 79.6%

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

3. Simplified 79.6%

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

   1. associate-/l* 87.5%

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

   2. associate-*r/ 98.1%

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

   3. associate-/r* 81.0%

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

   4. times-frac 92.1%

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

5. Applied egg-rr 92.1%

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

6. Final simplification 92.1%

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

Alternative 5: 67.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-17)
   (+ (* 0.5 (/ y (/ z x))) (/ (/ y x) z))
   (/ (+ (/ y x) (* 0.5 (* y x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-17) {
		tmp = (0.5 * (y / (z / x))) + ((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 (y <= 5d-17) then
        tmp = (0.5d0 * (y / (z / x))) + ((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 (y <= 5e-17) {
		tmp = (0.5 * (y / (z / x))) + ((y / x) / z);
	} else {
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e-17:
		tmp = (0.5 * (y / (z / x))) + ((y / x) / z)
	else:
		tmp = ((y / x) + (0.5 * (y * x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-17)
		tmp = Float64(Float64(0.5 * Float64(y / Float64(z / x))) + 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 (y <= 5e-17)
		tmp = (0.5 * (y / (z / x))) + ((y / x) / z);
	else
		tmp = ((y / x) + (0.5 * (y * x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e-17], N[(N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y / x), $MachinePrecision] / z), $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 y < 4.9999999999999999e-17

   1. Initial program 85.0%

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

      1. associate-*r/ 76.3%

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

      2. associate-/r* 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in x around 0 57.4%

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

      1. associate-/l/ 62.3%

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

      2. +-commutative 62.3%

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

      3. associate-/l* 67.1%

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

   6. Simplified 67.1%

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

   if 4.9999999999999999e-17 < y

   1. Initial program 95.4%

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

   2. Taylor expanded in x around 0 71.8%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(y \cdot x\right)}{z}\\ \end{array} \]

Alternative 6: 67.1% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{x}}{z}\\ \mathbf{if}\;y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{0.5}{\frac{z}{y \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (/ y x) z)))
   (if (<= y 2e-17)
     (+ (* 0.5 (/ y (/ z x))) t_0)
     (+ t_0 (/ 0.5 (/ z (* y x)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y / x) / z;
	double tmp;
	if (y <= 2e-17) {
		tmp = (0.5 * (y / (z / x))) + t_0;
	} else {
		tmp = t_0 + (0.5 / (z / (y * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / x) / z
    if (y <= 2d-17) then
        tmp = (0.5d0 * (y / (z / x))) + t_0
    else
        tmp = t_0 + (0.5d0 / (z / (y * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / x) / z;
	double tmp;
	if (y <= 2e-17) {
		tmp = (0.5 * (y / (z / x))) + t_0;
	} else {
		tmp = t_0 + (0.5 / (z / (y * x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / x) / z
	tmp = 0
	if y <= 2e-17:
		tmp = (0.5 * (y / (z / x))) + t_0
	else:
		tmp = t_0 + (0.5 / (z / (y * x)))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / x) / z)
	tmp = 0.0
	if (y <= 2e-17)
		tmp = Float64(Float64(0.5 * Float64(y / Float64(z / x))) + t_0);
	else
		tmp = Float64(t_0 + Float64(0.5 / Float64(z / Float64(y * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / x) / z;
	tmp = 0.0;
	if (y <= 2e-17)
		tmp = (0.5 * (y / (z / x))) + t_0;
	else
		tmp = t_0 + (0.5 / (z / (y * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[y, 2e-17], N[(N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(t$95$0 + N[(0.5 / N[(z / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000014e-17

   1. Initial program 85.0%

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

      1. associate-*r/ 76.2%

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

      2. associate-/r* 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in x around 0 57.2%

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

      1. associate-/l/ 62.1%

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

      2. +-commutative 62.1%

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

      3. associate-/l* 66.9%

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

   6. Simplified 66.9%

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

   if 2.00000000000000014e-17 < y

   1. Initial program 95.5%

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

      1. associate-*r/ 92.3%

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

      2. associate-/r* 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in x around 0 76.7%

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

      1. associate-/l/ 72.3%

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

      2. +-commutative 72.3%

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

      3. associate-/l* 64.8%

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

   6. Simplified 64.8%

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

      1. clear-num 64.8%

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

      2. un-div-inv 64.8%

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

      3. associate-/l/ 72.3%

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

   8. Applied egg-rr 72.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}} + \frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z} + \frac{0.5}{\frac{z}{y \cdot x}}\\ \end{array} \]

Alternative 7: 65.6% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d0)) .or. (.not. (x <= 1.42d0))) then
        tmp = 0.5d0 * (y * (x / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.4199999999999999 < x

   1. Initial program 81.4%

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

   2. Taylor expanded in x around 0 40.8%

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

   3. Taylor expanded in x around inf 40.8%

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

      1. associate-*r/ 44.1%

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

   5. Simplified 44.1%

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

   if -1.3999999999999999 < x < 1.4199999999999999

   1. Initial program 95.5%

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

      1. associate-*r/ 95.6%

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

      2. associate-/r* 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r/ 89.5%

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

      2. *-commutative 89.5%

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

      3. frac-times 95.4%

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

      4. associate-*l/ 97.1%

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

   5. Applied egg-rr 97.1%

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

   6. Taylor expanded in x around 0 96.9%

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

4. Final simplification 67.0%

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

Alternative 8: 65.5% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.42\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.42)))
   (* 0.5 (* y (/ x z)))
   (* (/ y z) (/ 1.0 x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y / z) * (1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d0)) .or. (.not. (x <= 1.42d0))) then
        tmp = 0.5d0 * (y * (x / z))
    else
        tmp = (y / z) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.42)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y / z) * (1.0 / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.42):
		tmp = 0.5 * (y * (x / z))
	else:
		tmp = (y / z) * (1.0 / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.42))
		tmp = Float64(0.5 * Float64(y * Float64(x / z)));
	else
		tmp = Float64(Float64(y / z) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.42)))
		tmp = 0.5 * (y * (x / z));
	else
		tmp = (y / z) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4], N[Not[LessEqual[x, 1.42]], $MachinePrecision]], N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.4199999999999999 < x

   1. Initial program 81.4%

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

   2. Taylor expanded in x around 0 40.8%

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

   3. Taylor expanded in x around inf 40.8%

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

      1. associate-*r/ 44.1%

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

   5. Simplified 44.1%

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

   if -1.3999999999999999 < x < 1.4199999999999999

   1. Initial program 95.5%

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

      1. associate-*r/ 95.6%

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

      2. associate-/r* 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around 0 89.2%

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

      1. associate-/r* 96.9%

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

      2. div-inv 96.9%

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

   6. Applied egg-rr 96.9%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{1}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.42\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 9: 65.8% 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 87.5%

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

2. Taylor expanded in x around 0 64.5%

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

3. Final simplification 64.5%

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

Alternative 10: 48.9% 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 87.5%

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

   1. associate-*r/ 80.1%

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

   2. associate-/r* 76.7%

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

3. Simplified 76.7%

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

4. Taylor expanded in x around 0 43.2%

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

5. Final simplification 43.2%

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

Alternative 11: 48.5% accurate, 21.4× speedup

Math

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

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(Float64(y / x) / z)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. associate-*r/ 80.1%

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

   2. associate-/r* 76.7%

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

3. Simplified 76.7%

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

4. Taylor expanded in x around 0 43.2%

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

   1. associate-/l/ 45.9%

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

6. Simplified 45.9%

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

7. Final simplification 45.9%

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

Alternative 12: 53.0% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y, z):
	return (y / z) / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.5%

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

   1. associate-*r/ 80.1%

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

   2. associate-/r* 76.7%

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

3. Simplified 76.7%

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

   1. associate-*r/ 81.0%

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

   2. *-commutative 81.0%

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

   3. frac-times 87.5%

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

   4. associate-*l/ 98.7%

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

5. Applied egg-rr 98.7%

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

6. Taylor expanded in x around 0 52.1%

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

7. Final simplification 52.1%

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

Developer target: 96.6% 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 2023274 
(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))