Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.2% → 97.9%

Time: 7.4s

Alternatives: 13

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-104} \lor \neg \left(t_0 \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (cosh x) (/ y x)) z)))
   (if (or (<= t_0 -2e-104) (not (<= t_0 5e+302)))
     (* y (/ (/ (cosh x) z) x))
     t_0)))

C

double code(double x, double y, double z) {
	double t_0 = (cosh(x) * (y / x)) / z;
	double tmp;
	if ((t_0 <= -2e-104) || !(t_0 <= 5e+302)) {
		tmp = y * ((cosh(x) / z) / x);
	} 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 = (cosh(x) * (y / x)) / z
    if ((t_0 <= (-2d-104)) .or. (.not. (t_0 <= 5d+302))) then
        tmp = y * ((cosh(x) / z) / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (Math.cosh(x) * (y / x)) / z;
	double tmp;
	if ((t_0 <= -2e-104) || !(t_0 <= 5e+302)) {
		tmp = y * ((Math.cosh(x) / z) / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (math.cosh(x) * (y / x)) / z
	tmp = 0
	if (t_0 <= -2e-104) or not (t_0 <= 5e+302):
		tmp = y * ((math.cosh(x) / z) / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(cosh(x) * Float64(y / x)) / z)
	tmp = 0.0
	if ((t_0 <= -2e-104) || !(t_0 <= 5e+302))
		tmp = Float64(y * Float64(Float64(cosh(x) / z) / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (cosh(x) * (y / x)) / z;
	tmp = 0.0;
	if ((t_0 <= -2e-104) || ~((t_0 <= 5e+302)))
		tmp = y * ((cosh(x) / z) / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-104], N[Not[LessEqual[t$95$0, 5e+302]], $MachinePrecision]], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < -1.99999999999999985e-104 or 5e302 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 80.6%

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

      1. associate-*r/ 73.7%

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

      2. associate-/r* 80.7%

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

   3. Simplified 80.7%

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

      1. associate-/r* 73.7%

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

      2. associate-*r/ 80.6%

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

      3. associate-/l* 72.7%

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

      4. add-log-exp 61.8%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right)} \]

      5. *-un-lft-identity 61.8%

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

      6. log-prod 61.8%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right)} \]

      7. metadata-eval 61.8%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right) \]

      8. add-log-exp 72.7%

         \[\leadsto 0 + \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}} \]

      9. associate-/l* 80.6%

         \[\leadsto 0 + \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      10. associate-*r/ 73.7%

          \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

   5. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{0 + \cosh x \cdot \frac{\frac{y}{x}}{z}} \]
   6. Step-by-step derivation

      1. +-lft-identity 73.7%

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

      2. associate-*r/ 80.6%

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

      3. associate-*l/ 80.6%

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

      4. associate-*r/ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r/ 99.9%

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

   7. Simplified 99.9%

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

   if -1.99999999999999985e-104 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 5e302

   1. Initial program 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq -2 \cdot 10^{-104} \lor \neg \left(\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 5 \cdot 10^{+302}\right):\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array} \]

Alternative 2: 96.1% 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 84.5%

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

   1. associate-*r/ 79.1%

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

   2. associate-/r* 81.6%

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

3. Simplified 81.6%

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

   1. associate-/r* 79.1%

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

   2. associate-*r/ 84.5%

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

   3. associate-/l* 77.6%

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

   4. add-log-exp 51.7%

      \[\leadsto \color{blue}{\log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right)} \]

   5. *-un-lft-identity 51.7%

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

   6. log-prod 51.7%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right)} \]

   7. metadata-eval 51.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right) \]

   8. add-log-exp 77.6%

      \[\leadsto 0 + \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}} \]

   9. associate-/l* 84.5%

      \[\leadsto 0 + \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

   10. associate-*r/ 79.1%

       \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

5. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{0 + \cosh x \cdot \frac{\frac{y}{x}}{z}} \]
6. Step-by-step derivation

   1. +-lft-identity 79.1%

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

   2. associate-*r/ 84.5%

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

   3. associate-*l/ 84.5%

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

   4. associate-*r/ 93.8%

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

   5. *-commutative 93.8%

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

   6. associate-*r/ 96.6%

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

7. Simplified 96.6%

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

8. Final simplification 96.6%

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

Alternative 3: 67.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -390:\\ \;\;\;\;\frac{z \cdot \left(y + 0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{x \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right) + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -390.0)
   (/ (* z (+ y (* 0.5 (* y (* x x))))) (* x (* z z)))
   (+ (* y (* x (/ 0.5 z))) (/ y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -390.0) {
		tmp = (z * (y + (0.5 * (y * (x * x))))) / (x * (z * z));
	} else {
		tmp = (y * (x * (0.5 / z))) + (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 <= (-390.0d0)) then
        tmp = (z * (y + (0.5d0 * (y * (x * x))))) / (x * (z * z))
    else
        tmp = (y * (x * (0.5d0 / z))) + (y / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -390.0) {
		tmp = (z * (y + (0.5 * (y * (x * x))))) / (x * (z * z));
	} else {
		tmp = (y * (x * (0.5 / z))) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -390.0:
		tmp = (z * (y + (0.5 * (y * (x * x))))) / (x * (z * z))
	else:
		tmp = (y * (x * (0.5 / z))) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -390.0)
		tmp = Float64(Float64(z * Float64(y + Float64(0.5 * Float64(y * Float64(x * x))))) / Float64(x * Float64(z * z)));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.5 / z))) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -390.0)
		tmp = (z * (y + (0.5 * (y * (x * x))))) / (x * (z * z));
	else
		tmp = (y * (x * (0.5 / z))) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -390.0], N[(N[(z * N[(y + N[(0.5 * N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -390

   1. Initial program 74.6%

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

      1. associate-*r/ 64.8%

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

      2. associate-/r* 69.0%

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

   3. Simplified 69.0%

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

   4. Taylor expanded in x around 0 41.8%

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

      1. associate-*r/ 41.8%

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

      2. frac-add 46.1%

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

      3. *-commutative 46.1%

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

      4. *-commutative 46.1%

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

   6. Applied egg-rr 46.1%

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

      1. +-commutative 46.1%

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

      2. *-commutative 46.1%

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

      3. associate-*r* 47.4%

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

      4. distribute-rgt-out 47.4%

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

      5. *-commutative 47.4%

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

      6. associate-*l* 47.4%

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

      7. *-commutative 47.4%

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

      8. associate-*r* 47.4%

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

      9. *-commutative 47.4%

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

      10. associate-*r* 54.1%

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

      11. *-commutative 54.1%

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

      12. associate-*l* 54.1%

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

   8. Simplified 54.1%

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

   if -390 < x

   1. Initial program 88.3%

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

      1. associate-*r/ 84.5%

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

      2. associate-/r* 86.5%

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

   3. Simplified 86.5%

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

   4. Taylor expanded in x around 0 80.0%

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

      1. div-inv 80.0%

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

      2. *-commutative 80.0%

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

      3. associate-*l* 82.0%

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

   6. Applied egg-rr 82.0%

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

   7. Taylor expanded in y around 0 80.0%

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

      1. associate-*l/ 82.0%

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

      2. associate-*r* 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r/ 82.0%

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

      5. associate-*l/ 82.0%

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

      6. *-commutative 82.0%

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

   9. Simplified 82.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -390:\\ \;\;\;\;\frac{z \cdot \left(y + 0.5 \cdot \left(y \cdot \left(x \cdot x\right)\right)\right)}{x \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right) + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 4: 65.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.95e-134)
   (/ (+ (/ y x) (* 0.5 (* x y))) z)
   (+ (/ y (* x z)) (* 0.5 (/ y (/ z x))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.95e-134) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = (y / (x * z)) + (0.5 * (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 <= (-1.95d-134)) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else
        tmp = (y / (x * z)) + (0.5d0 * (y / (z / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.95e-134) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = (y / (x * z)) + (0.5 * (y / (z / x)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.95e-134:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	else:
		tmp = (y / (x * z)) + (0.5 * (y / (z / x)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.95e-134)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	else
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(0.5 * Float64(y / Float64(z / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.95e-134)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	else
		tmp = (y / (x * z)) + (0.5 * (y / (z / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.95e-134], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.95e-134

   1. Initial program 88.3%

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

      1. associate-*r/ 80.9%

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

      2. associate-/r* 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in x around 0 67.1%

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

   5. Taylor expanded in z around 0 67.1%

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

   if -1.95e-134 < z

   1. Initial program 82.3%

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

      1. associate-*r/ 78.0%

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

      2. associate-/r* 83.9%

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

   3. Simplified 83.9%

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

   4. Taylor expanded in x around 0 70.7%

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

      1. div-inv 70.7%

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

      2. *-commutative 70.7%

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

      3. associate-*l* 77.7%

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

   6. Applied egg-rr 77.7%

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

      1. associate-*r* 70.7%

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

      2. div-inv 70.7%

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

      3. associate-/l* 77.7%

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

   8. Applied egg-rr 77.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 5: 68.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e-63)
   (+ (/ y (* x z)) (* 0.5 (/ (* x y) z)))
   (* y (+ (* 0.5 (/ x z)) (/ 1.0 (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-63) {
		tmp = (y / (x * z)) + (0.5 * ((x * y) / z));
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2d-63)) then
        tmp = (y / (x * z)) + (0.5d0 * ((x * y) / z))
    else
        tmp = y * ((0.5d0 * (x / z)) + (1.0d0 / (x * z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e-63) {
		tmp = (y / (x * z)) + (0.5 * ((x * y) / z));
	} else {
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e-63:
		tmp = (y / (x * z)) + (0.5 * ((x * y) / z))
	else:
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e-63)
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(0.5 * Float64(Float64(x * y) / z)));
	else
		tmp = Float64(y * Float64(Float64(0.5 * Float64(x / z)) + Float64(1.0 / Float64(x * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2e-63)
		tmp = (y / (x * z)) + (0.5 * ((x * y) / z));
	else
		tmp = y * ((0.5 * (x / z)) + (1.0 / (x * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e-63], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000013e-63

   1. Initial program 88.9%

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

      1. associate-*r/ 80.3%

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

      2. associate-/r* 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in x around 0 67.6%

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

   if -2.00000000000000013e-63 < z

   1. Initial program 82.5%

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

      1. associate-*r/ 78.5%

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

      2. associate-/r* 83.5%

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

   3. Simplified 83.5%

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

      1. associate-/r* 78.5%

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

      2. associate-*r/ 82.5%

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

      3. associate-/l* 77.5%

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

      4. add-log-exp 53.3%

         \[\leadsto \color{blue}{\log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right)} \]

      5. *-un-lft-identity 53.3%

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

      6. log-prod 53.3%

         \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right)} \]

      7. metadata-eval 53.3%

         \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right) \]

      8. add-log-exp 77.5%

         \[\leadsto 0 + \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}} \]

      9. associate-/l* 82.5%

         \[\leadsto 0 + \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

      10. associate-*r/ 78.5%

          \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

   5. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{0 + \cosh x \cdot \frac{\frac{y}{x}}{z}} \]
   6. Step-by-step derivation

      1. +-lft-identity 78.5%

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

      2. associate-*r/ 82.5%

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

      3. associate-*l/ 82.5%

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

      4. associate-*r/ 93.2%

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

      5. *-commutative 93.2%

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

      6. associate-*r/ 95.1%

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

   7. Simplified 95.1%

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

   8. Taylor expanded in x around 0 76.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z} + \frac{1}{x \cdot z}\right)\\ \end{array} \]

Alternative 6: 68.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+78}:\\ \;\;\;\;t_0 + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right) + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (* x z))))
   (if (<= z -2e+78)
     (+ t_0 (* 0.5 (/ (* x y) z)))
     (+ (* y (* x (/ 0.5 z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = y / (x * z);
	double tmp;
	if (z <= -2e+78) {
		tmp = t_0 + (0.5 * ((x * y) / z));
	} else {
		tmp = (y * (x * (0.5 / z))) + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * z)
    if (z <= (-2d+78)) then
        tmp = t_0 + (0.5d0 * ((x * y) / z))
    else
        tmp = (y * (x * (0.5d0 / z))) + t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (x * z);
	double tmp;
	if (z <= -2e+78) {
		tmp = t_0 + (0.5 * ((x * y) / z));
	} else {
		tmp = (y * (x * (0.5 / z))) + t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (x * z)
	tmp = 0
	if z <= -2e+78:
		tmp = t_0 + (0.5 * ((x * y) / z))
	else:
		tmp = (y * (x * (0.5 / z))) + t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(x * z))
	tmp = 0.0
	if (z <= -2e+78)
		tmp = Float64(t_0 + Float64(0.5 * Float64(Float64(x * y) / z)));
	else
		tmp = Float64(Float64(y * Float64(x * Float64(0.5 / z))) + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (x * z);
	tmp = 0.0;
	if (z <= -2e+78)
		tmp = t_0 + (0.5 * ((x * y) / z));
	else
		tmp = (y * (x * (0.5 / z))) + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e+78], N[(t$95$0 + N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000002e78

   1. Initial program 79.7%

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

      1. associate-*r/ 63.8%

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

      2. associate-/r* 61.1%

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

   3. Simplified 61.1%

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

   4. Taylor expanded in x around 0 64.3%

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

   if -2.00000000000000002e78 < z

   1. Initial program 85.5%

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

      1. associate-*r/ 82.2%

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

      2. associate-/r* 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in x around 0 70.4%

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

      1. div-inv 70.4%

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

      2. *-commutative 70.4%

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

      3. associate-*l* 76.2%

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

   6. Applied egg-rr 76.2%

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

   7. Taylor expanded in y around 0 70.4%

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

      1. associate-*l/ 76.2%

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

      2. associate-*r* 76.2%

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

      3. *-commutative 76.2%

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

      4. associate-*r/ 76.2%

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

      5. associate-*l/ 76.2%

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

      6. *-commutative 76.2%

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

   9. Simplified 76.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+78}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right) + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 7: 65.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -8e+101) (* 0.5 (* y (/ x z))) (/ (* y (+ (* x 0.5) (/ 1.0 x))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e+101) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-8d+101)) then
        tmp = 0.5d0 * (y * (x / z))
    else
        tmp = (y * ((x * 0.5d0) + (1.0d0 / x))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -8e+101) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -8e+101:
		tmp = 0.5 * (y * (x / z))
	else:
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -8e+101)
		tmp = Float64(0.5 * Float64(y * Float64(x / z)));
	else
		tmp = Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -8e+101)
		tmp = 0.5 * (y * (x / z));
	else
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -8e+101], N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999998e101

   1. Initial program 66.7%

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

      1. associate-*r/ 55.6%

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

      2. associate-/r* 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in x around 0 53.1%

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

   5. Taylor expanded in x around inf 53.1%

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

      1. associate-*l/ 65.9%

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

      2. *-commutative 65.9%

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

   7. Simplified 65.9%

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

   if -7.9999999999999998e101 < x

   1. Initial program 88.3%

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

      1. associate-*r/ 84.1%

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

      2. associate-/r* 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in x around 0 72.8%

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

   5. Taylor expanded in z around 0 71.6%

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

   6. Taylor expanded in y around 0 71.6%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+101}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \end{array} \]

Alternative 8: 65.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -7e+102) (* 0.5 (* y (/ x z))) (/ (+ (/ y x) (* 0.5 (* x y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -7e+102) {
		tmp = 0.5 * (y * (x / z));
	} 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 <= (-7d+102)) then
        tmp = 0.5d0 * (y * (x / z))
    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 <= -7e+102) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -7e+102:
		tmp = 0.5 * (y * (x / z))
	else:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -7e+102)
		tmp = Float64(0.5 * Float64(y * Float64(x / z)));
	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 <= -7e+102)
		tmp = 0.5 * (y * (x / z));
	else
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -7e+102], N[(0.5 * N[(y * N[(x / z), $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 < -7.00000000000000021e102

   1. Initial program 66.7%

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

      1. associate-*r/ 55.6%

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

      2. associate-/r* 64.4%

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

   3. Simplified 64.4%

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

   4. Taylor expanded in x around 0 53.1%

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

   5. Taylor expanded in x around inf 53.1%

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

      1. associate-*l/ 65.9%

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

      2. *-commutative 65.9%

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

   7. Simplified 65.9%

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

   if -7.00000000000000021e102 < x

   1. Initial program 88.3%

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

      1. associate-*r/ 84.1%

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

      2. associate-/r* 85.3%

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

   3. Simplified 85.3%

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

   4. Taylor expanded in x around 0 72.8%

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

   5. Taylor expanded in z around 0 71.6%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+102}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 9: 66.1% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

   1. associate-*r/ 79.1%

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

   2. associate-/r* 81.6%

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

3. Simplified 81.6%

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

   1. associate-/r* 79.1%

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

   2. associate-*r/ 84.5%

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

   3. associate-/l* 77.6%

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

   4. add-log-exp 51.7%

      \[\leadsto \color{blue}{\log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right)} \]

   5. *-un-lft-identity 51.7%

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

   6. log-prod 51.7%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right)} \]

   7. metadata-eval 51.7%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}}\right) \]

   8. add-log-exp 77.6%

      \[\leadsto 0 + \color{blue}{\frac{\cosh x}{\frac{z}{\frac{y}{x}}}} \]

   9. associate-/l* 84.5%

      \[\leadsto 0 + \color{blue}{\frac{\cosh x \cdot \frac{y}{x}}{z}} \]

   10. associate-*r/ 79.1%

       \[\leadsto 0 + \color{blue}{\cosh x \cdot \frac{\frac{y}{x}}{z}} \]

5. Applied egg-rr 79.1%

   \[\leadsto \color{blue}{0 + \cosh x \cdot \frac{\frac{y}{x}}{z}} \]
6. Step-by-step derivation

   1. +-lft-identity 79.1%

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

   2. associate-*r/ 84.5%

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

   3. associate-*l/ 84.5%

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

   4. associate-*r/ 93.8%

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

   5. *-commutative 93.8%

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

   6. associate-*r/ 96.6%

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

7. Simplified 96.6%

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

8. Taylor expanded in x around 0 72.1%

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

9. Final simplification 72.1%

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

Alternative 10: 66.3% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.5) (not (<= x 1.4))) (* 0.5 (* y (/ x z))) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.5) || !(x <= 1.4)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2.5d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = 0.5d0 * (y * (x / z))
    else
        tmp = y / (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.5) || !(x <= 1.4)) {
		tmp = 0.5 * (y * (x / z));
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2.5) or not (x <= 1.4):
		tmp = 0.5 * (y * (x / z))
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.5) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(y * Float64(x / z)));
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2.5) || ~((x <= 1.4)))
		tmp = 0.5 * (y * (x / z));
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.5], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5 or 1.3999999999999999 < x

   1. Initial program 78.0%

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

      1. associate-*r/ 67.4%

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

      2. associate-/r* 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in x around 0 47.0%

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

   5. Taylor expanded in x around inf 47.0%

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

      1. associate-*l/ 52.8%

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

      2. *-commutative 52.8%

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

   7. Simplified 52.8%

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

   if -2.5 < x < 1.3999999999999999

   1. Initial program 91.4%

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

      1. associate-*r/ 91.4%

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

      2. associate-/r* 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 92.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 11: 51.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.5e-91) (/ (/ y x) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-91) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.5d-91) then
        tmp = (y / x) / z
    else
        tmp = y / (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.5e-91) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.5e-91:
		tmp = (y / x) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.5e-91)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.5e-91)
		tmp = (y / x) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.5e-91], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4999999999999999e-91

   1. Initial program 82.6%

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

      1. associate-*l/ 82.5%

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

   3. Simplified 82.5%

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

   4. Taylor expanded in x around 0 50.0%

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

      1. associate-*l/ 50.0%

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

      2. *-un-lft-identity 50.0%

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

   6. Applied egg-rr 50.0%

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

   if 3.4999999999999999e-91 < y

   1. Initial program 89.0%

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

      1. associate-*r/ 81.2%

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

      2. associate-/r* 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in x around 0 48.5%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 12: 55.8% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.35 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.35e+41) (/ (/ y z) x) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.35e+41) {
		tmp = (y / z) / x;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 3.35d+41) then
        tmp = (y / z) / x
    else
        tmp = y / (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.35e+41) {
		tmp = (y / z) / x;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.35e+41:
		tmp = (y / z) / x
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.35e+41)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.35e+41)
		tmp = (y / z) / x;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.35e+41], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.3499999999999998e41

   1. Initial program 86.3%

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

      1. associate-*r/ 82.8%

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

      2. associate-/r* 85.8%

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

   3. Simplified 85.8%

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

   4. Taylor expanded in x around 0 44.0%

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

      1. *-commutative 44.0%

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

      2. associate-/r* 52.7%

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

   6. Simplified 52.7%

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

   if 3.3499999999999998e41 < z

   1. Initial program 77.7%

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

      1. associate-*r/ 64.2%

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

      2. associate-/r* 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in x around 0 62.1%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.35 \cdot 10^{+41}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 13: 49.8% 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 84.5%

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

   1. associate-*r/ 79.1%

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

   2. associate-/r* 81.6%

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

3. Simplified 81.6%

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

4. Taylor expanded in x around 0 47.7%

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

5. Final simplification 47.7%

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

Developer target: 96.8% 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 2023290 
(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))