Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 96.4%

Time: 12.2s

Alternatives: 12

Speedup: 1.0×

• 65.4% 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: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. *-commutative 84.8%

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

   2. associate-*l/ 96.9%

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

   3. associate-/l* 96.9%

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

   4. associate-/l* 98.4%

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

   5. associate-/r* 83.2%

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

3. Simplified 83.2%

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

   1. clear-num 83.2%

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

   2. un-div-inv 83.2%

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

   3. *-commutative 83.2%

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

   4. associate-/l* 98.5%

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

6. Applied egg-rr 98.5%

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

7. Final simplification 98.5%

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

Alternative 2: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 1.42e+30)
   (* y (/ (cosh x) (* z x)))
   (+ (* 0.5 (/ (* y x) z)) (/ y (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.42e+30) {
		tmp = y * (cosh(x) / (z * x));
	} else {
		tmp = (0.5 * ((y * x) / z)) + (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 (y <= 1.42d+30) then
        tmp = y * (cosh(x) / (z * x))
    else
        tmp = (0.5d0 * ((y * x) / z)) + (y / (z * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 1.42e+30) {
		tmp = y * (Math.cosh(x) / (z * x));
	} else {
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 1.42e+30:
		tmp = y * (math.cosh(x) / (z * x))
	else:
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 1.42e+30)
		tmp = Float64(y * Float64(cosh(x) / Float64(z * x)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / z)) + Float64(y / Float64(z * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 1.42e+30)
		tmp = y * (cosh(x) / (z * x));
	else
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 1.42e+30], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.41999999999999991e30

   1. Initial program 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-*l/ 97.5%

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

      3. associate-/l* 97.5%

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

      4. associate-/l* 98.0%

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

      5. associate-/r* 82.9%

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

   3. Simplified 82.9%

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

   if 1.41999999999999991e30 < y

   1. Initial program 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-*l/ 94.9%

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

      3. associate-/l* 94.9%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 84.0%

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

   3. Simplified 84.0%

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

   5. Taylor expanded in x around 0 91.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.42 \cdot 10^{+30}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-18}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 5e-18) (* y (/ (cosh x) (* z x))) (* (cosh x) (/ (/ y x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 5e-18) {
		tmp = y * (cosh(x) / (z * x));
	} else {
		tmp = cosh(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 (y <= 5d-18) then
        tmp = y * (cosh(x) / (z * x))
    else
        tmp = cosh(x) * ((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-18) {
		tmp = y * (Math.cosh(x) / (z * x));
	} else {
		tmp = Math.cosh(x) * ((y / x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 5e-18:
		tmp = y * (math.cosh(x) / (z * x))
	else:
		tmp = math.cosh(x) * ((y / x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 5e-18)
		tmp = Float64(y * Float64(cosh(x) / Float64(z * x)));
	else
		tmp = Float64(cosh(x) * Float64(Float64(y / x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 5e-18)
		tmp = y * (cosh(x) / (z * x));
	else
		tmp = cosh(x) * ((y / x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 5e-18], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.00000000000000036e-18

   1. Initial program 81.0%

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

      1. *-commutative 81.0%

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

      2. associate-*l/ 97.4%

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

      3. associate-/l* 97.4%

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

      4. associate-/l* 97.9%

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

      5. associate-/r* 84.1%

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

   3. Simplified 84.1%

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

   if 5.00000000000000036e-18 < y

   1. Initial program 95.7%

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

      1. associate-/l* 88.2%

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

   3. Simplified 88.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-18}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.65e+67) (* y (/ (cosh x) (* z x))) (* (/ (cosh x) z) (/ y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.65e+67) {
		tmp = y * (cosh(x) / (z * x));
	} else {
		tmp = (cosh(x) / 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) :: tmp
    if (z <= 1.65d+67) then
        tmp = y * (cosh(x) / (z * x))
    else
        tmp = (cosh(x) / z) * (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.65e+67) {
		tmp = y * (Math.cosh(x) / (z * x));
	} else {
		tmp = (Math.cosh(x) / z) * (y / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.65e+67:
		tmp = y * (math.cosh(x) / (z * x))
	else:
		tmp = (math.cosh(x) / z) * (y / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.65e+67)
		tmp = Float64(y * Float64(cosh(x) / Float64(z * x)));
	else
		tmp = Float64(Float64(cosh(x) / z) * Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.65e+67)
		tmp = y * (cosh(x) / (z * x));
	else
		tmp = (cosh(x) / z) * (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.65e+67], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.6500000000000001e67

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

      2. associate-*l/ 97.1%

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

      3. associate-/l* 97.1%

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

      4. associate-/l* 98.1%

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

      5. associate-/r* 88.9%

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

   3. Simplified 88.9%

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

   if 1.6500000000000001e67 < z

   1. Initial program 89.9%

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

      1. associate-*r/ 96.0%

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

      2. associate-/l/ 59.0%

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

      3. times-frac 89.8%

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

   4. Applied egg-rr 89.8%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.65 \cdot 10^{+67}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. *-commutative 84.8%

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

   2. associate-*l/ 96.9%

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

   3. associate-/l* 96.9%

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

   4. associate-/l* 98.4%

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

   5. associate-/r* 83.2%

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

3. Simplified 83.2%

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

   1. clear-num 83.2%

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

   2. un-div-inv 83.2%

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

   3. *-commutative 83.2%

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

   4. associate-/l* 98.5%

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

6. Applied egg-rr 98.5%

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

   1. clear-num 98.2%

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

   2. associate-/r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-/r* 98.4%

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

   5. clear-num 98.4%

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

8. Applied egg-rr 98.4%

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

9. Final simplification 98.4%

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

Alternative 6: 67.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ \mathbf{if}\;z \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;t\_0 + y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(z \cdot x\right) + y \cdot z}{z \cdot \left(z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (* z x))))
   (if (<= z 1.9e-36)
     (+ t_0 (* y (* x (/ 0.5 z))))
     (if (<= z 2.8e+68)
       (/ (+ (* (* y (* x 0.5)) (* z x)) (* y z)) (* z (* z x)))
       (+ (* 0.5 (/ (* y x) z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = y / (z * x);
	double tmp;
	if (z <= 1.9e-36) {
		tmp = t_0 + (y * (x * (0.5 / z)));
	} else if (z <= 2.8e+68) {
		tmp = (((y * (x * 0.5)) * (z * x)) + (y * z)) / (z * (z * x));
	} else {
		tmp = (0.5 * ((y * x) / z)) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (z * x)
    if (z <= 1.9d-36) then
        tmp = t_0 + (y * (x * (0.5d0 / z)))
    else if (z <= 2.8d+68) then
        tmp = (((y * (x * 0.5d0)) * (z * x)) + (y * z)) / (z * (z * x))
    else
        tmp = (0.5d0 * ((y * x) / z)) + 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);
	double tmp;
	if (z <= 1.9e-36) {
		tmp = t_0 + (y * (x * (0.5 / z)));
	} else if (z <= 2.8e+68) {
		tmp = (((y * (x * 0.5)) * (z * x)) + (y * z)) / (z * (z * x));
	} else {
		tmp = (0.5 * ((y * x) / z)) + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (z * x)
	tmp = 0
	if z <= 1.9e-36:
		tmp = t_0 + (y * (x * (0.5 / z)))
	elif z <= 2.8e+68:
		tmp = (((y * (x * 0.5)) * (z * x)) + (y * z)) / (z * (z * x))
	else:
		tmp = (0.5 * ((y * x) / z)) + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(z * x))
	tmp = 0.0
	if (z <= 1.9e-36)
		tmp = Float64(t_0 + Float64(y * Float64(x * Float64(0.5 / z))));
	elseif (z <= 2.8e+68)
		tmp = Float64(Float64(Float64(Float64(y * Float64(x * 0.5)) * Float64(z * x)) + Float64(y * z)) / Float64(z * Float64(z * x)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / z)) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (z * x);
	tmp = 0.0;
	if (z <= 1.9e-36)
		tmp = t_0 + (y * (x * (0.5 / z)));
	elseif (z <= 2.8e+68)
		tmp = (((y * (x * 0.5)) * (z * x)) + (y * z)) / (z * (z * x));
	else
		tmp = (0.5 * ((y * x) / z)) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, 1.9e-36], N[(t$95$0 + N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.8e+68], N[(N[(N[(N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] * N[(z * x), $MachinePrecision]), $MachinePrecision] + N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < 1.89999999999999985e-36

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l/ 97.0%

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

      3. associate-/l* 97.0%

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

      4. associate-/l* 98.0%

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

      5. associate-/r* 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in x around 0 67.4%

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

      1. associate-/l* 66.0%

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

      2. *-commutative 66.0%

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

   7. Applied egg-rr 66.0%

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

      1. associate-*l/ 67.4%

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

      2. clear-num 67.4%

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

      3. associate-/l/ 71.6%

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

      4. un-div-inv 71.6%

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

      5. associate-/l/ 67.4%

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

      6. *-commutative 67.4%

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

   9. Applied egg-rr 67.4%

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

       1. associate-/r/ 67.4%

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

       2. associate-*r* 72.0%

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

   11. Simplified 72.0%

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

   if 1.89999999999999985e-36 < z < 2.8e68

   1. Initial program 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*l/ 100.0%

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

      3. associate-/l* 99.8%

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

      4. associate-/l* 99.7%

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

      5. associate-/r* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 46.6%

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

      1. associate-*r/ 46.6%

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

      2. frac-add 67.8%

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

      3. *-commutative 67.8%

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

      4. *-commutative 67.8%

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

      5. associate-*l* 67.8%

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

      6. *-commutative 67.8%

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

      7. *-commutative 67.8%

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

   7. Applied egg-rr 67.8%

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

   if 2.8e68 < z

   1. Initial program 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 95.9%

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

      3. associate-/l* 95.9%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 58.1%

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

   3. Simplified 58.1%

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

   5. Taylor expanded in x around 0 63.3%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{z \cdot x} + y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(z \cdot x\right) + y \cdot z}{z \cdot \left(z \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.2e+46)
   (* y (/ (+ (* x 0.5) (/ 1.0 x)) z))
   (+ (* 0.5 (/ (* y x) z)) (/ y (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.2e+46) {
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	} else {
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.2d+46) then
        tmp = y * (((x * 0.5d0) + (1.0d0 / x)) / z)
    else
        tmp = (0.5d0 * ((y * x) / z)) + (y / (z * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.2e+46) {
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	} else {
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.2e+46:
		tmp = y * (((x * 0.5) + (1.0 / x)) / z)
	else:
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.2e+46)
		tmp = Float64(y * Float64(Float64(Float64(x * 0.5) + Float64(1.0 / x)) / z));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / z)) + Float64(y / Float64(z * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.2e+46)
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	else
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.2e+46], N[(y * N[(N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.2e46

   1. Initial program 83.5%

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

      1. *-commutative 83.5%

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

      2. associate-*l/ 97.1%

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

      3. associate-/l* 97.1%

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

      4. associate-/l* 98.1%

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

      5. associate-/r* 88.9%

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

   3. Simplified 88.9%

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

      1. clear-num 88.8%

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

      2. un-div-inv 88.9%

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

      3. *-commutative 88.9%

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

      4. associate-/l* 98.1%

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

   6. Applied egg-rr 98.1%

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

      1. clear-num 98.1%

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

      2. associate-/r/ 98.1%

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

      3. *-commutative 98.1%

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

      4. associate-/r* 98.1%

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

      5. clear-num 98.1%

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

   8. Applied egg-rr 98.1%

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

   9. Taylor expanded in x around 0 71.1%

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

   if 2.2e46 < z

   1. Initial program 90.1%

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

      1. *-commutative 90.1%

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

      2. associate-*l/ 96.1%

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

      3. associate-/l* 96.0%

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

      4. associate-/l* 99.8%

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

      5. associate-/r* 59.8%

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

   3. Simplified 59.8%

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

   5. Taylor expanded in x around 0 62.9%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.9% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = y / (z * x)
    else
        tmp = 0.5d0 * ((y * x) / z)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-*l/ 95.8%

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

      3. associate-/l* 95.8%

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

      4. associate-/l* 97.9%

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

      5. associate-/r* 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around 0 64.9%

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

   if 1.3999999999999999 < x

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0 38.5%

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

   4. Taylor expanded in x around inf 38.5%

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

4. Final simplification 57.8%

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

Alternative 9: 56.6% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = y / (z * x)
    else
        tmp = y * ((x * 0.5d0) / z)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.1%

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

      1. *-commutative 85.1%

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

      2. associate-*l/ 95.8%

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

      3. associate-/l* 95.8%

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

      4. associate-/l* 97.9%

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

      5. associate-/r* 90.9%

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

   3. Simplified 90.9%

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

   5. Taylor expanded in x around 0 64.9%

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

   if 1.3999999999999999 < x

   1. Initial program 84.0%

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

   3. Taylor expanded in x around 0 38.5%

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

   4. Taylor expanded in x around inf 38.5%

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

      1. associate-*r/ 38.5%

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

      2. associate-*r* 38.5%

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

      3. *-commutative 38.5%

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

      4. *-commutative 38.5%

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

      5. associate-*r/ 32.9%

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

      6. *-commutative 32.9%

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

   6. Simplified 32.9%

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

4. Final simplification 56.2%

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

Alternative 10: 65.0% accurate, 9.7× speedup

Math

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

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(y * Float64(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[(y * N[(N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

   1. *-commutative 84.8%

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

   2. associate-*l/ 96.9%

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

   3. associate-/l* 96.9%

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

   4. associate-/l* 98.4%

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

   5. associate-/r* 83.2%

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

3. Simplified 83.2%

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

   1. clear-num 83.2%

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

   2. un-div-inv 83.2%

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

   3. *-commutative 83.2%

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

   4. associate-/l* 98.5%

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

6. Applied egg-rr 98.5%

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

   1. clear-num 98.2%

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

   2. associate-/r/ 98.4%

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

   3. *-commutative 98.4%

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

   4. associate-/r* 98.4%

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

   5. clear-num 98.4%

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

8. Applied egg-rr 98.4%

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

9. Taylor expanded in x around 0 66.9%

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

10. Final simplification 66.9%

    \[\leadsto y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z} \]
11. Add Preprocessing

Alternative 11: 51.8% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7e-50) (/ (/ y x) z) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-50) {
		tmp = (y / x) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 7d-50) then
        tmp = (y / x) / z
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7e-50) {
		tmp = (y / x) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7e-50:
		tmp = (y / x) / z
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7e-50)
		tmp = Float64(Float64(y / 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 (y <= 7e-50)
		tmp = (y / x) / z;
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7e-50], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.99999999999999993e-50

   1. Initial program 79.9%

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

   3. Taylor expanded in x around 0 46.0%

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

   if 6.99999999999999993e-50 < y

   1. Initial program 96.2%

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

      1. *-commutative 96.2%

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

      2. associate-*l/ 96.2%

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

      3. associate-/l* 96.2%

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

      4. associate-/l* 99.9%

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

      5. associate-/r* 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in x around 0 57.4%

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

      1. div-inv 57.5%

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

      2. *-commutative 57.5%

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

      3. associate-/r* 63.6%

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

   7. Applied egg-rr 63.6%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.4% accurate, 21.4× speedup

Math

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

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(y / Float64(z * x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. *-commutative 84.8%

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

   2. associate-*l/ 96.9%

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

   3. associate-/l* 96.9%

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

   4. associate-/l* 98.4%

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

   5. associate-/r* 83.2%

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

3. Simplified 83.2%

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

5. Taylor expanded in x around 0 49.9%

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

6. Final simplification 49.9%

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

Developer target: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024040 
(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))