Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.3% → 96.0%

Time: 9.6s

Alternatives: 13

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: 84.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. associate-*l/ 88.6%

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

3. Simplified 88.6%

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

   1. associate-/r/ 82.3%

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

   2. associate-/l* 77.8%

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

   3. *-commutative 77.8%

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

   4. expm1-log1p-u 42.2%

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

   5. expm1-udef 29.0%

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

   6. associate-/l* 31.7%

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

   7. times-frac 38.6%

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

5. Applied egg-rr 38.6%

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

   1. expm1-def 47.1%

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

   2. expm1-log1p 87.7%

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

   3. associate-*r/ 99.1%

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

7. Simplified 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{z}{x \cdot 0.5} + x \cdot z\right) \cdot \left(-y\right)}{x \cdot \left(z \cdot \left(-2 \cdot \frac{z}{x}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 INFINITY)
     (/ t_0 z)
     (/ (* (+ (/ z (* x 0.5)) (* x z)) (- y)) (* x (* z (* -2.0 (/ z x))))))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = (((z / (x * 0.5)) + (x * z)) * -y) / (x * (z * (-2.0 * (z / x))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / z;
	} else {
		tmp = (((z / (x * 0.5)) + (x * z)) * -y) / (x * (z * (-2.0 * (z / x))));
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 / z
	else:
		tmp = (((z / (x * 0.5)) + (x * z)) * -y) / (x * (z * (-2.0 * (z / x))))
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(Float64(Float64(z / Float64(x * 0.5)) + Float64(x * z)) * Float64(-y)) / Float64(x * Float64(z * Float64(-2.0 * Float64(z / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 / z;
	else
		tmp = (((z / (x * 0.5)) + (x * z)) * -y) / (x * (z * (-2.0 * (z / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 / z), $MachinePrecision], N[(N[(N[(N[(z / N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision] / N[(x * N[(z * N[(-2.0 * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.0%

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

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

   1. Initial program 0.0%

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

      1. associate-*l/ 0.0%

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

   3. Simplified 0.0%

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

   4. Taylor expanded in x around 0 10.4%

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

      1. div-inv 10.4%

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

      2. associate-*l* 10.2%

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

      3. *-commutative 10.2%

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

      4. associate-*l/ 10.2%

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

      5. *-un-lft-identity 10.2%

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

   6. Applied egg-rr 10.2%

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

      1. *-commutative 10.2%

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

      2. associate-*r/ 10.4%

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

      3. *-commutative 10.4%

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

      4. associate-/r/ 10.4%

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

      5. associate-/l* 31.3%

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

      6. frac-2neg 31.3%

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

      7. frac-add 22.6%

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

      8. distribute-rgt-neg-in 22.6%

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

      9. associate-/l/ 22.6%

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

      10. *-commutative 22.6%

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

      11. associate-/l/ 22.6%

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

      12. *-commutative 22.6%

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

      13. distribute-rgt-neg-in 22.6%

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

   8. Applied egg-rr 22.6%

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

      1. *-commutative 22.6%

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

      2. associate-*l* 44.8%

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

   10. Simplified 44.8%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq \infty:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{z}{x \cdot 0.5} + x \cdot z\right) \cdot \left(-y\right)}{x \cdot \left(z \cdot \left(-2 \cdot \frac{z}{x}\right)\right)}\\ \end{array} \]

Alternative 3: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot 2 + x \cdot \left(x \cdot \frac{y}{z}\right)}{x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4e+187)
   (/ (+ (* (/ y z) 2.0) (* x (* x (/ y z)))) (* x 2.0))
   (* (/ y x) (/ (cosh x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e+187) {
		tmp = (((y / z) * 2.0) + (x * (x * (y / z)))) / (x * 2.0);
	} else {
		tmp = (y / x) * (cosh(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 <= (-1.4d+187)) then
        tmp = (((y / z) * 2.0d0) + (x * (x * (y / z)))) / (x * 2.0d0)
    else
        tmp = (y / x) * (cosh(x) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4e+187) {
		tmp = (((y / z) * 2.0) + (x * (x * (y / z)))) / (x * 2.0);
	} else {
		tmp = (y / x) * (Math.cosh(x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4e+187:
		tmp = (((y / z) * 2.0) + (x * (x * (y / z)))) / (x * 2.0)
	else:
		tmp = (y / x) * (math.cosh(x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4e+187)
		tmp = Float64(Float64(Float64(Float64(y / z) * 2.0) + Float64(x * Float64(x * Float64(y / z)))) / Float64(x * 2.0));
	else
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4e+187)
		tmp = (((y / z) * 2.0) + (x * (x * (y / z)))) / (x * 2.0);
	else
		tmp = (y / x) * (cosh(x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4e+187], N[(N[(N[(N[(y / z), $MachinePrecision] * 2.0), $MachinePrecision] + N[(x * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999995e187

   1. Initial program 48.0%

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

      1. associate-*l/ 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in x around 0 46.3%

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

      1. *-commutative 46.3%

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

      2. associate-/l* 38.7%

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

      3. associate-*l/ 38.7%

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

   6. Applied egg-rr 38.7%

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

      1. +-commutative 38.7%

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

      2. associate-/l/ 38.7%

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

      3. div-inv 38.7%

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

      4. clear-num 38.7%

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

      5. *-commutative 38.7%

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

      6. metadata-eval 38.7%

         \[\leadsto \frac{\frac{y}{z}}{x} + \frac{y}{z} \cdot \left(x \cdot \color{blue}{\frac{1}{2}}\right) \]

      7. div-inv 38.7%

         \[\leadsto \frac{\frac{y}{z}}{x} + \frac{y}{z} \cdot \color{blue}{\frac{x}{2}} \]

      8. associate-*r/ 38.7%

         \[\leadsto \frac{\frac{y}{z}}{x} + \color{blue}{\frac{\frac{y}{z} \cdot x}{2}} \]

      9. frac-add 72.6%

         \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot 2 + x \cdot \left(\frac{y}{z} \cdot x\right)}{x \cdot 2}} \]

   8. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot 2 + x \cdot \left(\frac{y}{z} \cdot x\right)}{x \cdot 2}} \]

   if -1.39999999999999995e187 < x

   1. Initial program 93.0%

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

      1. associate-*l/ 93.0%

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

   3. Simplified 93.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot 2 + x \cdot \left(x \cdot \frac{y}{z}\right)}{x \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \]

Alternative 4: 69.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+18} \lor \neg \left(x \leq 3.4 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{\left(\frac{z}{x \cdot 0.5} + x \cdot z\right) \cdot \left(-y\right)}{x \cdot \left(z \cdot \left(-2 \cdot \frac{z}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.6e+18) (not (<= x 3.4e+48)))
   (/ (* (+ (/ z (* x 0.5)) (* x z)) (- y)) (* x (* z (* -2.0 (/ z x)))))
   (/ (+ (/ y x) (* 0.5 (* x y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.6e+18) || !(x <= 3.4e+48)) {
		tmp = (((z / (x * 0.5)) + (x * z)) * -y) / (x * (z * (-2.0 * (z / x))));
	} else {
		tmp = ((y / x) + (0.5 * (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 ((x <= (-3.6d+18)) .or. (.not. (x <= 3.4d+48))) then
        tmp = (((z / (x * 0.5d0)) + (x * z)) * -y) / (x * (z * ((-2.0d0) * (z / x))))
    else
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.6e+18) || !(x <= 3.4e+48)) {
		tmp = (((z / (x * 0.5)) + (x * z)) * -y) / (x * (z * (-2.0 * (z / x))));
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.6e+18) or not (x <= 3.4e+48):
		tmp = (((z / (x * 0.5)) + (x * z)) * -y) / (x * (z * (-2.0 * (z / x))))
	else:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.6e+18) || !(x <= 3.4e+48))
		tmp = Float64(Float64(Float64(Float64(z / Float64(x * 0.5)) + Float64(x * z)) * Float64(-y)) / Float64(x * Float64(z * Float64(-2.0 * Float64(z / x)))));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.6e+18) || ~((x <= 3.4e+48)))
		tmp = (((z / (x * 0.5)) + (x * z)) * -y) / (x * (z * (-2.0 * (z / x))));
	else
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.6e+18], N[Not[LessEqual[x, 3.4e+48]], $MachinePrecision]], N[(N[(N[(N[(z / N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(x * z), $MachinePrecision]), $MachinePrecision] * (-y)), $MachinePrecision] / N[(x * N[(z * N[(-2.0 * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e18 or 3.4000000000000003e48 < x

   1. Initial program 78.5%

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

      1. associate-*l/ 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in x around 0 41.3%

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

      1. div-inv 41.3%

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

      2. associate-*l* 34.2%

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

      3. *-commutative 34.2%

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

      4. associate-*l/ 34.2%

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

      5. *-un-lft-identity 34.2%

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

   6. Applied egg-rr 34.2%

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

      1. *-commutative 34.2%

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

      2. associate-*r/ 41.3%

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

      3. *-commutative 41.3%

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

      4. associate-/r/ 41.3%

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

      5. associate-/l* 42.0%

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

      6. frac-2neg 42.0%

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

      7. frac-add 44.0%

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

      8. distribute-rgt-neg-in 44.0%

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

      9. associate-/l/ 44.0%

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

      10. *-commutative 44.0%

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

      11. associate-/l/ 44.0%

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

      12. *-commutative 44.0%

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

      13. distribute-rgt-neg-in 44.0%

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

   8. Applied egg-rr 44.0%

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

      1. *-commutative 44.0%

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

      2. associate-*l* 58.5%

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

   10. Simplified 58.5%

       \[\leadsto \color{blue}{\frac{-y \cdot \left(\frac{z}{x \cdot 0.5} + x \cdot z\right)}{x \cdot \left(z \cdot \left(-2 \cdot \frac{z}{x}\right)\right)}} \]

   if -3.6e18 < x < 3.4000000000000003e48

   1. Initial program 97.6%

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

   2. Taylor expanded in x around 0 87.6%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+18} \lor \neg \left(x \leq 3.4 \cdot 10^{+48}\right):\\ \;\;\;\;\frac{\left(\frac{z}{x \cdot 0.5} + x \cdot z\right) \cdot \left(-y\right)}{x \cdot \left(z \cdot \left(-2 \cdot \frac{z}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 5: 72.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+132} \lor \neg \left(z \leq 1.5 \cdot 10^{-92}\right):\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot 2 + x \cdot \left(x \cdot \frac{y}{z}\right)}{x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4.8e+132) (not (<= z 1.5e-92)))
   (+ (* 0.5 (/ (* x y) z)) (/ y (* x z)))
   (/ (+ (* (/ y z) 2.0) (* x (* x (/ y z)))) (* x 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.8e+132) || !(z <= 1.5e-92)) {
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	} else {
		tmp = (((y / z) * 2.0) + (x * (x * (y / z)))) / (x * 2.0);
	}
	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) :: tmp
    if ((z <= (-4.8d+132)) .or. (.not. (z <= 1.5d-92))) then
        tmp = (0.5d0 * ((x * y) / z)) + (y / (x * z))
    else
        tmp = (((y / z) * 2.0d0) + (x * (x * (y / z)))) / (x * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4.8e+132) || !(z <= 1.5e-92)) {
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	} else {
		tmp = (((y / z) * 2.0) + (x * (x * (y / z)))) / (x * 2.0);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4.8e+132) or not (z <= 1.5e-92):
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z))
	else:
		tmp = (((y / z) * 2.0) + (x * (x * (y / z)))) / (x * 2.0)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4.8e+132) || !(z <= 1.5e-92))
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z)) + Float64(y / Float64(x * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) * 2.0) + Float64(x * Float64(x * Float64(y / z)))) / Float64(x * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4.8e+132) || ~((z <= 1.5e-92)))
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	else
		tmp = (((y / z) * 2.0) + (x * (x * (y / z)))) / (x * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4.8e+132], N[Not[LessEqual[z, 1.5e-92]], $MachinePrecision]], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] * 2.0), $MachinePrecision] + N[(x * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.8000000000000002e132 or 1.50000000000000007e-92 < z

   1. Initial program 87.1%

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

      1. associate-*l/ 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around 0 58.9%

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

   if -4.8000000000000002e132 < z < 1.50000000000000007e-92

   1. Initial program 89.8%

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

      1. associate-*l/ 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in x around 0 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-/l* 62.6%

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

      3. associate-*l/ 62.6%

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

   6. Applied egg-rr 62.6%

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

      1. +-commutative 62.6%

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

      2. associate-/l/ 71.2%

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

      3. div-inv 71.2%

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

      4. clear-num 71.2%

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

      5. *-commutative 71.2%

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

      6. metadata-eval 71.2%

         \[\leadsto \frac{\frac{y}{z}}{x} + \frac{y}{z} \cdot \left(x \cdot \color{blue}{\frac{1}{2}}\right) \]

      7. div-inv 71.2%

         \[\leadsto \frac{\frac{y}{z}}{x} + \frac{y}{z} \cdot \color{blue}{\frac{x}{2}} \]

      8. associate-*r/ 71.2%

         \[\leadsto \frac{\frac{y}{z}}{x} + \color{blue}{\frac{\frac{y}{z} \cdot x}{2}} \]

      9. frac-add 83.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{z} \cdot 2 + x \cdot \left(\frac{y}{z} \cdot x\right)}{x \cdot 2}} \]

   8. Applied egg-rr 83.2%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.8 \cdot 10^{+132} \lor \neg \left(z \leq 1.5 \cdot 10^{-92}\right):\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot 2 + x \cdot \left(x \cdot \frac{y}{z}\right)}{x \cdot 2}\\ \end{array} \]

Alternative 6: 60.3% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.3999999999999999 < x

   1. Initial program 80.7%

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

      1. associate-*l/ 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in x around 0 39.0%

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

   5. Taylor expanded in x around inf 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-/l* 39.0%

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

      4. *-commutative 39.0%

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

   7. Simplified 39.0%

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

      1. div-inv 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-*l* 32.3%

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

      4. clear-num 32.3%

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

   9. Applied egg-rr 32.3%

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

   if -1.3999999999999999 < x < 1.3999999999999999

   1. Initial program 98.1%

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

   2. Taylor expanded in x around 0 97.4%

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

4. Final simplification 61.8%

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

Alternative 7: 64.7% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.3999999999999999 < x

   1. Initial program 80.7%

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

      1. associate-*l/ 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in x around 0 39.0%

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

   5. Taylor expanded in x around inf 39.0%

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

      1. associate-*r/ 39.0%

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

      2. *-commutative 39.0%

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

      3. associate-/l* 39.0%

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

      4. *-commutative 39.0%

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

   7. Simplified 39.0%

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

      1. div-inv 39.0%

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

      2. associate-*l* 39.6%

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

      3. clear-num 39.6%

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

   9. Applied egg-rr 39.6%

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

   if -1.3999999999999999 < x < 1.3999999999999999

   1. Initial program 98.1%

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

   2. Taylor expanded in x around 0 97.4%

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

4. Final simplification 65.8%

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

Alternative 8: 64.6% accurate, 9.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4) {
		tmp = y * (x * (0.5 / z));
	} else if (x <= 1.4) {
		tmp = (y / x) / z;
	} else {
		tmp = (x * y) * (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 * (x * (0.5d0 / z))
    else if (x <= 1.4d0) then
        tmp = (y / x) / z
    else
        tmp = (x * y) * (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 * (x * (0.5 / z));
	} else if (x <= 1.4) {
		tmp = (y / x) / z;
	} else {
		tmp = (x * y) * (0.5 / z);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 79.5%

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

      1. associate-*l/ 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in x around 0 37.3%

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

   5. Taylor expanded in x around inf 37.3%

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

      1. associate-*r/ 37.3%

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

      2. *-commutative 37.3%

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

      3. associate-/l* 37.3%

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

      4. *-commutative 37.3%

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

   7. Simplified 37.3%

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

      1. div-inv 37.3%

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

      2. associate-*l* 39.6%

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

      3. clear-num 39.6%

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

   9. Applied egg-rr 39.6%

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

   if -1.3999999999999999 < x < 1.3999999999999999

   1. Initial program 98.1%

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

   2. Taylor expanded in x around 0 97.4%

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

   if 1.3999999999999999 < x

   1. Initial program 82.2%

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

      1. associate-*l/ 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in x around 0 41.2%

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

   5. Taylor expanded in x around inf 41.2%

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

      1. associate-*r/ 41.2%

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

      2. *-commutative 41.2%

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

      3. associate-/l* 41.2%

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

      4. *-commutative 41.2%

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

   7. Simplified 41.2%

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

      1. clear-num 41.2%

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

      2. associate-/r/ 41.2%

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

      3. clear-num 41.2%

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

      4. *-commutative 41.2%

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

   9. Applied egg-rr 41.2%

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

4. Final simplification 66.2%

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

Alternative 9: 64.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. associate-*l/ 88.6%

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

3. Simplified 88.6%

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

   1. associate-/r/ 82.3%

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

   2. associate-/l* 77.8%

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

   3. *-commutative 77.8%

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

   4. expm1-log1p-u 42.2%

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

   5. expm1-udef 29.0%

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

   6. associate-/l* 31.7%

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

   7. times-frac 38.6%

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

5. Applied egg-rr 38.6%

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

   1. expm1-def 47.1%

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

   2. expm1-log1p 87.7%

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

   3. associate-*r/ 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in x around 0 65.3%

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

9. Final simplification 65.3%

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

Alternative 10: 64.5% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

2. Taylor expanded in x around 0 65.7%

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

3. Final simplification 65.7%

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

Alternative 11: 55.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+38} \lor \neg \left(y \leq 4 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1e+38) (not (<= y 4e+86))) (/ (/ y z) x) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+38) || !(y <= 4e+86)) {
		tmp = (y / z) / x;
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1d+38)) .or. (.not. (y <= 4d+86))) then
        tmp = (y / z) / x
    else
        tmp = (y / x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1e+38) || !(y <= 4e+86)) {
		tmp = (y / z) / x;
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1e+38) or not (y <= 4e+86):
		tmp = (y / z) / x
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1e+38) || !(y <= 4e+86))
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1e+38) || ~((y <= 4e+86)))
		tmp = (y / z) / x;
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1e+38], N[Not[LessEqual[y, 4e+86]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.99999999999999977e37 or 4.0000000000000001e86 < y

   1. Initial program 98.0%

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

      1. associate-*l/ 98.0%

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

   3. Simplified 98.0%

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

      1. associate-*r/ 99.8%

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

      2. associate-*l/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 58.0%

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

   if -9.99999999999999977e37 < y < 4.0000000000000001e86

   1. Initial program 82.8%

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

   2. Taylor expanded in x around 0 50.8%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+38} \lor \neg \left(y \leq 4 \cdot 10^{+86}\right):\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 12: 47.9% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

   1. associate-*l/ 88.6%

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

3. Simplified 88.6%

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

4. Taylor expanded in x around 0 43.1%

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

5. Final simplification 43.1%

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

Alternative 13: 47.9% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.6%

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

2. Taylor expanded in x around 0 47.3%

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

3. Final simplification 47.3%

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

Developer target: 97.0% 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 2023320 
(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))