Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 83.8% → 99.7%

Time: 9.7s

Alternatives: 14

Speedup: 1.0×

• 63.8% 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: 83.8% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+54} \lor \neg \left(z \leq 20000\right):\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4e+54) (not (<= z 20000.0)))
   (* y (/ (/ (cosh x) z) x))
   (* (/ (cosh x) x) (/ y z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4e+54) || !(z <= 20000.0)) {
		tmp = y * ((cosh(x) / z) / 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 <= (-4d+54)) .or. (.not. (z <= 20000.0d0))) then
        tmp = y * ((cosh(x) / z) / 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 <= -4e+54) || !(z <= 20000.0)) {
		tmp = y * ((Math.cosh(x) / z) / x);
	} else {
		tmp = (Math.cosh(x) / x) * (y / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4e+54) or not (z <= 20000.0):
		tmp = y * ((math.cosh(x) / z) / x)
	else:
		tmp = (math.cosh(x) / x) * (y / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4e+54) || !(z <= 20000.0))
		tmp = Float64(y * Float64(Float64(cosh(x) / z) / 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 <= -4e+54) || ~((z <= 20000.0)))
		tmp = y * ((cosh(x) / z) / x);
	else
		tmp = (cosh(x) / x) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4e+54], N[Not[LessEqual[z, 20000.0]], $MachinePrecision]], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] / 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 < -4.0000000000000003e54 or 2e4 < z

   1. Initial program 77.2%

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

      1. associate-/l* 62.4%

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

      2. associate-/r/ 63.3%

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

      3. associate-*l/ 66.7%

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

      4. *-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-/l* 62.4%

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

      3. associate-/l* 77.2%

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

      4. expm1-log1p-u 55.9%

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

      5. expm1-udef 37.5%

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

      6. associate-*r/ 40.8%

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

      7. associate-/l/ 33.5%

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

      8. *-commutative 33.5%

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

      9. associate-*l/ 33.5%

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

      10. *-commutative 33.5%

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

   5. Applied egg-rr 33.5%

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

      1. expm1-def 51.9%

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

      2. expm1-log1p 74.9%

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

      3. *-commutative 74.9%

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

      4. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

   if -4.0000000000000003e54 < z < 2e4

   1. Initial program 88.8%

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

      1. associate-/l* 88.0%

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

      2. associate-/r/ 93.2%

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

      3. associate-*l/ 86.9%

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

      4. *-commutative 86.9%

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

   3. Simplified 86.9%

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

      1. associate-/l* 93.2%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+54} \lor \neg \left(z \leq 20000\right):\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \end{array} \]

Alternative 2: 96.0% accurate, 1.0× speedup

Math

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

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(y * Float64(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[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

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

   1. associate-/l* 75.9%

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

   2. associate-/r/ 79.0%

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

   3. associate-*l/ 77.3%

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

   4. *-commutative 77.3%

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

3. Simplified 77.3%

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

   1. *-commutative 77.3%

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

   2. associate-/l* 75.9%

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

   3. associate-/l* 83.3%

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

   4. expm1-log1p-u 49.3%

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

   5. expm1-udef 38.4%

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

   6. associate-*r/ 43.5%

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

   7. associate-/l/ 38.4%

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

   8. *-commutative 38.4%

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

   9. associate-*l/ 38.4%

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

   10. *-commutative 38.4%

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

5. Applied egg-rr 38.4%

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

   1. expm1-def 49.3%

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

   2. expm1-log1p 84.3%

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

   3. *-commutative 84.3%

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

   4. associate-/r* 96.5%

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

7. Simplified 96.5%

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

8. Final simplification 96.5%

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

Alternative 3: 67.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-210}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{z \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1e-72)
   (+ (* 0.5 (/ (* y x) z)) (/ y (* z x)))
   (if (<= y 1.9e-210)
     (/ (+ (* 0.5 (* y x)) (/ y x)) z)
     (* y (+ (* 0.5 (/ x z)) (/ 1.0 (* z x)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-72) {
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x));
	} else if (y <= 1.9e-210) {
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (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 <= (-1d-72)) then
        tmp = (0.5d0 * ((y * x) / z)) + (y / (z * x))
    else if (y <= 1.9d-210) then
        tmp = ((0.5d0 * (y * x)) + (y / x)) / z
    else
        tmp = y * ((0.5d0 * (x / z)) + (1.0d0 / (z * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1e-72) {
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x));
	} else if (y <= 1.9e-210) {
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (z * x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1e-72:
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x))
	elif y <= 1.9e-210:
		tmp = ((0.5 * (y * x)) + (y / x)) / z
	else:
		tmp = y * ((0.5 * (x / z)) + (1.0 / (z * x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1e-72)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / z)) + Float64(y / Float64(z * x)));
	elseif (y <= 1.9e-210)
		tmp = Float64(Float64(Float64(0.5 * Float64(y * x)) + Float64(y / x)) / z);
	else
		tmp = Float64(y * Float64(Float64(0.5 * Float64(x / z)) + Float64(1.0 / Float64(z * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1e-72)
		tmp = (0.5 * ((y * x) / z)) + (y / (z * x));
	elseif (y <= 1.9e-210)
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	else
		tmp = y * ((0.5 * (x / z)) + (1.0 / (z * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1e-72], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.9e-210], N[(N[(N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.9999999999999997e-73

   1. Initial program 90.4%

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

      1. associate-/l* 84.7%

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

      2. associate-/r/ 92.8%

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

      3. associate-*l/ 87.0%

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

      4. *-commutative 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in x around 0 76.6%

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

   if -9.9999999999999997e-73 < y < 1.90000000000000002e-210

   1. Initial program 74.5%

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

   2. Taylor expanded in x around 0 58.0%

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

   if 1.90000000000000002e-210 < y

   1. Initial program 85.2%

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

      1. associate-/l* 78.6%

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

      2. associate-/r/ 85.0%

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

      3. associate-*l/ 84.8%

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

      4. *-commutative 84.8%

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

   3. Simplified 84.8%

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

      1. *-commutative 84.8%

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

      2. associate-/l* 78.6%

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

      3. associate-/l* 85.2%

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

      4. expm1-log1p-u 49.1%

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

      5. expm1-udef 40.4%

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

      6. associate-*r/ 45.0%

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

      7. associate-/l/ 40.9%

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

      8. *-commutative 40.9%

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

      9. associate-*l/ 40.9%

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

      10. *-commutative 40.9%

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

   5. Applied egg-rr 40.9%

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

      1. expm1-def 49.7%

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

      2. expm1-log1p 88.5%

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

      3. *-commutative 88.5%

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

      4. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around 0 61.5%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{-72}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-210}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{z \cdot x}\right)\\ \end{array} \]

Alternative 4: 67.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{z \cdot x}\\ \mathbf{if}\;y \leq -5 \cdot 10^{-75}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + t_0\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-211}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0 + y \cdot \frac{x \cdot 0.5}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (* z x))))
   (if (<= y -5e-75)
     (+ (* 0.5 (/ (* y x) z)) t_0)
     (if (<= y 5.9e-211)
       (/ (+ (* 0.5 (* y x)) (/ y x)) z)
       (+ t_0 (* y (/ (* x 0.5) z)))))))

C

double code(double x, double y, double z) {
	double t_0 = y / (z * x);
	double tmp;
	if (y <= -5e-75) {
		tmp = (0.5 * ((y * x) / z)) + t_0;
	} else if (y <= 5.9e-211) {
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	} else {
		tmp = t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = y / (z * x)
    if (y <= (-5d-75)) then
        tmp = (0.5d0 * ((y * x) / z)) + t_0
    else if (y <= 5.9d-211) then
        tmp = ((0.5d0 * (y * x)) + (y / x)) / z
    else
        tmp = t_0 + (y * ((x * 0.5d0) / z))
    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 (y <= -5e-75) {
		tmp = (0.5 * ((y * x) / z)) + t_0;
	} else if (y <= 5.9e-211) {
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	} else {
		tmp = t_0 + (y * ((x * 0.5) / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (z * x)
	tmp = 0
	if y <= -5e-75:
		tmp = (0.5 * ((y * x) / z)) + t_0
	elif y <= 5.9e-211:
		tmp = ((0.5 * (y * x)) + (y / x)) / z
	else:
		tmp = t_0 + (y * ((x * 0.5) / z))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(z * x))
	tmp = 0.0
	if (y <= -5e-75)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * x) / z)) + t_0);
	elseif (y <= 5.9e-211)
		tmp = Float64(Float64(Float64(0.5 * Float64(y * x)) + Float64(y / x)) / z);
	else
		tmp = Float64(t_0 + Float64(y * Float64(Float64(x * 0.5) / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (z * x);
	tmp = 0.0;
	if (y <= -5e-75)
		tmp = (0.5 * ((y * x) / z)) + t_0;
	elseif (y <= 5.9e-211)
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	else
		tmp = t_0 + (y * ((x * 0.5) / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5e-75], N[(N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[y, 5.9e-211], N[(N[(N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(t$95$0 + N[(y * N[(N[(x * 0.5), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.99999999999999979e-75

   1. Initial program 90.4%

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

      1. associate-/l* 84.7%

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

      2. associate-/r/ 92.8%

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

      3. associate-*l/ 87.0%

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

      4. *-commutative 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in x around 0 76.6%

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

   if -4.99999999999999979e-75 < y < 5.9000000000000002e-211

   1. Initial program 74.5%

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

   2. Taylor expanded in x around 0 58.0%

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

   if 5.9000000000000002e-211 < y

   1. Initial program 85.2%

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

      1. associate-/l* 78.6%

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

      2. associate-/r/ 85.0%

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

      3. associate-*l/ 84.8%

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

      4. *-commutative 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in x around 0 60.7%

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

      1. clear-num 60.7%

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

      2. un-div-inv 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/r* 59.8%

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

   6. Applied egg-rr 59.8%

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

      1. clear-num 59.8%

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

      2. associate-/l/ 60.7%

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

      3. *-commutative 60.7%

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

      4. associate-/r* 60.7%

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

      5. clear-num 60.7%

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

      6. associate-*l* 60.7%

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

      7. *-un-lft-identity 60.7%

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

      8. times-frac 61.6%

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

      9. /-rgt-identity 61.6%

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

   8. Applied egg-rr 61.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{-75}:\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z} + \frac{y}{z \cdot x}\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-211}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x} + y \cdot \frac{x \cdot 0.5}{z}\\ \end{array} \]

Alternative 5: 64.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{1}{\frac{z \cdot x}{y}}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -3.4e+126)
   (/ 1.0 (/ (* z x) y))
   (if (<= z 1.2e+51)
     (* (/ y z) (+ (* x 0.5) (/ 1.0 x)))
     (* y (/ (/ 1.0 z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+126) {
		tmp = 1.0 / ((z * x) / y);
	} else if (z <= 1.2e+51) {
		tmp = (y / z) * ((x * 0.5) + (1.0 / x));
	} else {
		tmp = y * ((1.0 / 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 <= (-3.4d+126)) then
        tmp = 1.0d0 / ((z * x) / y)
    else if (z <= 1.2d+51) then
        tmp = (y / z) * ((x * 0.5d0) + (1.0d0 / x))
    else
        tmp = y * ((1.0d0 / z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -3.4e+126) {
		tmp = 1.0 / ((z * x) / y);
	} else if (z <= 1.2e+51) {
		tmp = (y / z) * ((x * 0.5) + (1.0 / x));
	} else {
		tmp = y * ((1.0 / z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -3.4e+126:
		tmp = 1.0 / ((z * x) / y)
	elif z <= 1.2e+51:
		tmp = (y / z) * ((x * 0.5) + (1.0 / x))
	else:
		tmp = y * ((1.0 / z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -3.4e+126)
		tmp = Float64(1.0 / Float64(Float64(z * x) / y));
	elseif (z <= 1.2e+51)
		tmp = Float64(Float64(y / z) * Float64(Float64(x * 0.5) + Float64(1.0 / x)));
	else
		tmp = Float64(y * Float64(Float64(1.0 / z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -3.4e+126)
		tmp = 1.0 / ((z * x) / y);
	elseif (z <= 1.2e+51)
		tmp = (y / z) * ((x * 0.5) + (1.0 / x));
	else
		tmp = y * ((1.0 / z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -3.4e+126], N[(1.0 / N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.2e+51], N[(N[(y / z), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.39999999999999989e126

   1. Initial program 77.5%

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

      1. associate-/l* 54.8%

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

      2. associate-/r/ 52.9%

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

      3. associate-*l/ 58.9%

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

      4. *-commutative 58.9%

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

   3. Simplified 58.9%

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

   4. Taylor expanded in x around 0 48.5%

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

   if -3.39999999999999989e126 < z < 1.1999999999999999e51

   1. Initial program 90.1%

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

      1. associate-/l* 89.5%

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

      2. associate-/r/ 93.4%

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

      3. associate-*l/ 88.4%

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

      4. *-commutative 88.4%

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

   3. Simplified 88.4%

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

   4. Taylor expanded in x around 0 64.9%

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

      1. clear-num 64.9%

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

      2. un-div-inv 64.9%

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

      3. *-commutative 64.9%

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

      4. associate-/r* 63.7%

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

   6. Applied egg-rr 63.7%

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

   7. Taylor expanded in y around 0 66.1%

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

      1. *-commutative 66.1%

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

      2. +-commutative 66.1%

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

      3. associate-*r/ 66.1%

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

      4. *-commutative 66.1%

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

      5. distribute-rgt-in 66.1%

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

      6. associate-/r/ 66.1%

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

      7. *-commutative 66.1%

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

      8. associate-*l/ 71.6%

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

      9. associate-/r/ 71.0%

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

      10. associate-/r/ 71.0%

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

      11. *-commutative 71.0%

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

      12. associate-*l/ 71.0%

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

      13. associate-*r* 69.8%

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

      14. associate-*l/ 69.8%

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

      15. associate-*r/ 69.8%

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

      16. associate-*r/ 68.6%

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

      17. associate-*r* 68.6%

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

      18. *-commutative 68.6%

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

      19. distribute-rgt-out 68.6%

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

   9. Simplified 68.6%

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

   if 1.1999999999999999e51 < z

   1. Initial program 69.7%

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

      1. associate-/l* 55.8%

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

      2. associate-/r/ 60.8%

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

      3. associate-*l/ 61.7%

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

      4. *-commutative 61.7%

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

   3. Simplified 61.7%

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

      1. *-commutative 61.7%

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

      2. associate-/l* 55.8%

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

      3. associate-/l* 69.7%

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

      4. expm1-log1p-u 50.0%

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

      5. expm1-udef 35.4%

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

      6. associate-*r/ 40.5%

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

      7. associate-/l/ 31.2%

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

      8. *-commutative 31.2%

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

      9. associate-*l/ 31.2%

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

      10. *-commutative 31.2%

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

   5. Applied egg-rr 31.2%

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

      1. expm1-def 45.7%

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

      2. expm1-log1p 75.5%

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

      3. *-commutative 75.5%

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

      4. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 52.2%

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

      1. *-commutative 52.2%

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

      2. associate-/r* 52.4%

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

   10. Simplified 52.4%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+126}:\\ \;\;\;\;\frac{1}{\frac{z \cdot x}{y}}\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+51}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]

Alternative 6: 65.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{z \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.4e+66)
   (/ (+ (* 0.5 (* y x)) (/ y x)) z)
   (* y (+ (* 0.5 (/ x z)) (/ 1.0 (* z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e+66) {
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (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 <= 1.4d+66) then
        tmp = ((0.5d0 * (y * x)) + (y / x)) / z
    else
        tmp = y * ((0.5d0 * (x / z)) + (1.0d0 / (z * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.4e+66) {
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (z * x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.4e+66:
		tmp = ((0.5 * (y * x)) + (y / x)) / z
	else:
		tmp = y * ((0.5 * (x / z)) + (1.0 / (z * x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.4e+66)
		tmp = Float64(Float64(Float64(0.5 * Float64(y * x)) + Float64(y / x)) / z);
	else
		tmp = Float64(y * Float64(Float64(0.5 * Float64(x / z)) + Float64(1.0 / Float64(z * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.4e+66)
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	else
		tmp = y * ((0.5 * (x / z)) + (1.0 / (z * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.4e+66], N[(N[(N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e66

   1. Initial program 87.5%

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

   2. Taylor expanded in x around 0 65.2%

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

   if 1.4e66 < z

   1. Initial program 68.0%

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

      1. associate-/l* 53.4%

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

      2. associate-/r/ 60.4%

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

      3. associate-*l/ 59.7%

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

      4. *-commutative 59.7%

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

   3. Simplified 59.7%

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

      1. *-commutative 59.7%

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

      2. associate-/l* 53.4%

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

      3. associate-/l* 68.0%

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

      4. expm1-log1p-u 50.9%

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

      5. expm1-udef 37.2%

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

      6. associate-*r/ 42.6%

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

      7. associate-/l/ 32.8%

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

      8. *-commutative 32.8%

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

      9. associate-*l/ 32.8%

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

      10. *-commutative 32.8%

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

   5. Applied egg-rr 32.8%

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

      1. expm1-def 46.4%

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

      2. expm1-log1p 74.2%

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

      3. *-commutative 74.2%

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

      4. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 50.2%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+66}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{z \cdot x}\right)\\ \end{array} \]

Alternative 7: 65.1% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.25 \cdot 10^{+67}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2.25e+67)
   (/ (* y (+ (* x 0.5) (/ 1.0 x))) z)
   (* y (/ (/ 1.0 z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.25e+67) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	} else {
		tmp = y * ((1.0 / 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 <= 2.25d+67) then
        tmp = (y * ((x * 0.5d0) + (1.0d0 / x))) / z
    else
        tmp = y * ((1.0d0 / z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 2.25e+67) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	} else {
		tmp = y * ((1.0 / z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2.25e+67:
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z
	else:
		tmp = y * ((1.0 / z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2.25e+67)
		tmp = Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z);
	else
		tmp = Float64(y * Float64(Float64(1.0 / z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 2.25e+67)
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	else
		tmp = y * ((1.0 / z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2.25e+67], N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.2499999999999999e67

   1. Initial program 87.1%

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

   2. Taylor expanded in x around 0 64.9%

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

   3. Taylor expanded in y around 0 64.8%

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

   if 2.2499999999999999e67 < z

   1. Initial program 69.3%

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

      1. associate-/l* 54.4%

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

      2. associate-/r/ 61.6%

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

      3. associate-*l/ 60.8%

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

      4. *-commutative 60.8%

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

   3. Simplified 60.8%

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

      1. *-commutative 60.8%

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

      2. associate-/l* 54.4%

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

      3. associate-/l* 69.3%

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

      4. expm1-log1p-u 51.8%

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

      5. expm1-udef 37.9%

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

      6. associate-*r/ 43.4%

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

      7. associate-/l/ 33.4%

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

      8. *-commutative 33.4%

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

      9. associate-*l/ 33.4%

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

      10. *-commutative 33.4%

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

   5. Applied egg-rr 33.4%

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

      1. expm1-def 47.3%

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

      2. expm1-log1p 73.7%

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

      3. *-commutative 73.7%

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

      4. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 50.5%

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

      1. *-commutative 50.5%

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

      2. associate-/r* 50.7%

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

   10. Simplified 50.7%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.25 \cdot 10^{+67}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]

Alternative 8: 65.3% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.04 \cdot 10^{+89}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.04e+89) (/ (+ (* 0.5 (* y x)) (/ y x)) z) (* y (/ (/ 1.0 z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.04e+89) {
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	} else {
		tmp = y * ((1.0 / 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 <= 1.04d+89) then
        tmp = ((0.5d0 * (y * x)) + (y / x)) / z
    else
        tmp = y * ((1.0d0 / z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.04e+89) {
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	} else {
		tmp = y * ((1.0 / z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.04e+89:
		tmp = ((0.5 * (y * x)) + (y / x)) / z
	else:
		tmp = y * ((1.0 / z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.04e+89)
		tmp = Float64(Float64(Float64(0.5 * Float64(y * x)) + Float64(y / x)) / z);
	else
		tmp = Float64(y * Float64(Float64(1.0 / z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.04e+89)
		tmp = ((0.5 * (y * x)) + (y / x)) / z;
	else
		tmp = y * ((1.0 / z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.04e+89], N[(N[(N[(0.5 * N[(y * x), $MachinePrecision]), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.04e89

   1. Initial program 87.3%

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

   2. Taylor expanded in x around 0 65.1%

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

   if 1.04e89 < z

   1. Initial program 66.8%

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

      1. associate-/l* 52.8%

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

      2. associate-/r/ 62.4%

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

      3. associate-*l/ 59.7%

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

      4. *-commutative 59.7%

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

   3. Simplified 59.7%

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

      1. *-commutative 59.7%

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

      2. associate-/l* 52.8%

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

      3. associate-/l* 66.8%

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

      4. expm1-log1p-u 50.1%

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

      5. expm1-udef 37.0%

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

      6. associate-*r/ 43.0%

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

      7. associate-/l/ 32.1%

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

      8. *-commutative 32.1%

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

      9. associate-*l/ 32.1%

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

      10. *-commutative 32.1%

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

   5. Applied egg-rr 32.1%

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

      1. expm1-def 45.2%

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

      2. expm1-log1p 71.6%

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

      3. *-commutative 71.6%

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

      4. associate-/r* 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 48.5%

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

      1. *-commutative 48.5%

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

      2. associate-/r* 48.8%

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

   10. Simplified 48.8%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.04 \cdot 10^{+89}:\\ \;\;\;\;\frac{0.5 \cdot \left(y \cdot x\right) + \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]

Alternative 9: 62.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -90000 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -90000.0) (not (<= x 1.4)))
   (* 0.5 (* x (/ y z)))
   (/ y (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -90000.0) || !(x <= 1.4)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-90000.0d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = 0.5d0 * (x * (y / z))
    else
        tmp = y / (z * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -90000.0) || !(x <= 1.4)) {
		tmp = 0.5 * (x * (y / z));
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -90000.0) or not (x <= 1.4):
		tmp = 0.5 * (x * (y / z))
	else:
		tmp = y / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -90000.0) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(x * Float64(y / z)));
	else
		tmp = Float64(y / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -90000.0) || ~((x <= 1.4)))
		tmp = 0.5 * (x * (y / z));
	else
		tmp = y / (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -90000.0], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9e4 or 1.3999999999999999 < x

   1. Initial program 77.3%

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

      1. associate-/l* 63.6%

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

      2. associate-/r/ 68.9%

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

      3. associate-*l/ 63.6%

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

      4. *-commutative 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in x around 0 32.2%

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

   5. Taylor expanded in x around inf 32.2%

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

      1. associate-*r/ 27.9%

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

   7. Simplified 27.9%

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

   if -9e4 < x < 1.3999999999999999

   1. Initial program 89.7%

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

      1. associate-/l* 88.9%

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

      2. associate-/r/ 89.8%

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

      3. associate-*l/ 91.9%

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

      4. *-commutative 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in x around 0 91.1%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -90000 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 10: 66.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -90000 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -90000.0) (not (<= x 1.4)))
   (* 0.5 (/ (* y x) z))
   (/ y (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -90000.0) || !(x <= 1.4)) {
		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 <= (-90000.0d0)) .or. (.not. (x <= 1.4d0))) 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 <= -90000.0) || !(x <= 1.4)) {
		tmp = 0.5 * ((y * x) / z);
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -90000.0) or not (x <= 1.4):
		tmp = 0.5 * ((y * x) / z)
	else:
		tmp = y / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -90000.0) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(Float64(y * x) / z));
	else
		tmp = Float64(y / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -90000.0) || ~((x <= 1.4)))
		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, -90000.0], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9e4 or 1.3999999999999999 < x

   1. Initial program 77.3%

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

      1. associate-/l* 63.6%

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

      2. associate-/r/ 68.9%

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

      3. associate-*l/ 63.6%

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

      4. *-commutative 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in x around 0 32.2%

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

   5. Taylor expanded in x around inf 32.2%

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

   if -9e4 < x < 1.3999999999999999

   1. Initial program 89.7%

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

      1. associate-/l* 88.9%

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

      2. associate-/r/ 89.8%

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

      3. associate-*l/ 91.9%

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

      4. *-commutative 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in x around 0 91.1%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -90000 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 11: 65.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -90000 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -90000.0) (not (<= x 1.4)))
   (* 0.5 (/ (* y x) z))
   (* y (/ (/ 1.0 z) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -90000.0) || !(x <= 1.4)) {
		tmp = 0.5 * ((y * x) / z);
	} else {
		tmp = y * ((1.0 / 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 <= (-90000.0d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = 0.5d0 * ((y * x) / z)
    else
        tmp = y * ((1.0d0 / z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -90000.0) || !(x <= 1.4)) {
		tmp = 0.5 * ((y * x) / z);
	} else {
		tmp = y * ((1.0 / z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -90000.0) or not (x <= 1.4):
		tmp = 0.5 * ((y * x) / z)
	else:
		tmp = y * ((1.0 / z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -90000.0) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(Float64(y * x) / z));
	else
		tmp = Float64(y * Float64(Float64(1.0 / z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -90000.0) || ~((x <= 1.4)))
		tmp = 0.5 * ((y * x) / z);
	else
		tmp = y * ((1.0 / z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -90000.0], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9e4 or 1.3999999999999999 < x

   1. Initial program 77.3%

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

      1. associate-/l* 63.6%

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

      2. associate-/r/ 68.9%

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

      3. associate-*l/ 63.6%

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

      4. *-commutative 63.6%

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

   3. Simplified 63.6%

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

   4. Taylor expanded in x around 0 32.2%

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

   5. Taylor expanded in x around inf 32.2%

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

   if -9e4 < x < 1.3999999999999999

   1. Initial program 89.7%

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

      1. associate-/l* 88.9%

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

      2. associate-/r/ 89.8%

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

      3. associate-*l/ 91.9%

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

      4. *-commutative 91.9%

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

   3. Simplified 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-/l* 88.9%

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

      3. associate-/l* 89.7%

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

      4. expm1-log1p-u 56.7%

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

      5. expm1-udef 34.1%

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

      6. associate-*r/ 34.1%

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

      7. associate-/l/ 34.9%

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

      8. *-commutative 34.9%

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

      9. associate-*l/ 34.9%

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

      10. *-commutative 34.9%

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

   5. Applied egg-rr 34.9%

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

      1. expm1-def 57.4%

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

      2. expm1-log1p 92.6%

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

      3. *-commutative 92.6%

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

      4. associate-/r* 92.8%

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

   7. Simplified 92.8%

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

   8. Taylor expanded in x around 0 91.0%

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

      1. *-commutative 91.0%

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

      2. associate-/r* 91.1%

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

   10. Simplified 91.1%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -90000 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \frac{y \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \end{array} \]

Alternative 12: 56.3% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+54} \lor \neg \left(z \leq 4.5 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4e+54) (not (<= z 4.5e+21))) (/ y (* z x)) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4e+54) || !(z <= 4.5e+21)) {
		tmp = y / (z * x);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-4d+54)) .or. (.not. (z <= 4.5d+21))) then
        tmp = y / (z * x)
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4e+54) || !(z <= 4.5e+21)) {
		tmp = y / (z * x);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4e+54) or not (z <= 4.5e+21):
		tmp = y / (z * x)
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4e+54) || !(z <= 4.5e+21))
		tmp = Float64(y / Float64(z * x));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4e+54) || ~((z <= 4.5e+21)))
		tmp = y / (z * x);
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4e+54], N[Not[LessEqual[z, 4.5e+21]], $MachinePrecision]], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.0000000000000003e54 or 4.5e21 < z

   1. Initial program 76.7%

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

      1. associate-/l* 61.4%

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

      2. associate-/r/ 62.4%

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

      3. associate-*l/ 65.9%

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

      4. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in x around 0 53.2%

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

   if -4.0000000000000003e54 < z < 4.5e21

   1. Initial program 89.0%

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

      1. associate-/l* 88.3%

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

      2. associate-/r/ 93.3%

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

      3. associate-*l/ 87.1%

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

      4. *-commutative 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around 0 41.8%

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

      1. *-un-lft-identity 41.8%

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

      2. times-frac 53.5%

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

   6. Applied egg-rr 53.5%

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

      1. associate-*l/ 53.5%

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

      2. *-un-lft-identity 53.5%

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

   8. Applied egg-rr 53.5%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+54} \lor \neg \left(z \leq 4.5 \cdot 10^{+21}\right):\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 13: 50.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.7e+87) (/ (/ y x) z) (/ y (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.7e+87) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.7d+87) then
        tmp = (y / x) / z
    else
        tmp = y / (z * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.7e+87) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (z * x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.7e+87:
		tmp = (y / x) / z
	else:
		tmp = y / (z * x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.7e+87)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(y / Float64(z * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.7e+87)
		tmp = (y / x) / z;
	else
		tmp = y / (z * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.7e+87], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(z * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.7000000000000001e87

   1. Initial program 87.2%

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

   2. Taylor expanded in x around 0 48.9%

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

   if 1.7000000000000001e87 < z

   1. Initial program 68.1%

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

      1. associate-/l* 52.7%

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

      2. associate-/r/ 61.9%

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

      3. associate-*l/ 59.3%

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

      4. *-commutative 59.3%

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

   3. Simplified 59.3%

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

   4. Taylor expanded in x around 0 48.7%

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

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]

Alternative 14: 49.4% 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 83.3%

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

   1. associate-/l* 75.9%

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

   2. associate-/r/ 79.0%

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

   3. associate-*l/ 77.3%

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

   4. *-commutative 77.3%

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

3. Simplified 77.3%

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

4. Taylor expanded in x around 0 47.1%

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

5. Final simplification 47.1%

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

Developer target: 97.7% 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 2023321 
(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))