Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.6% → 96.0%

Time: 9.9s

Alternatives: 16

Speedup: 1.0×

• 64.4% 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: 84.6% 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 82.5%

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

   1. associate-*r/ 95.8%

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

   2. associate-/l/ 83.3%

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

   3. associate-*l/ 82.2%

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

   4. *-commutative 82.2%

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

   5. *-commutative 82.2%

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

3. Simplified 82.2%

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

   1. *-commutative 82.2%

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

   2. associate-/r* 94.3%

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

   3. associate-*l/ 95.8%

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

5. Applied egg-rr 95.8%

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

6. Final simplification 95.8%

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

Alternative 2: 92.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

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

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 = y * (cosh(x) / (x * z));
	}
	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 = y * (Math.cosh(x) / (x * z));
	}
	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 = y * (math.cosh(x) / (x * z))
	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(y * Float64(cosh(x) / Float64(x * z)));
	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 = y * (cosh(x) / (x * z));
	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[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.2%

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

   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-*r/ 100.0%

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

      2. associate-/l/ 64.7%

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

      3. associate-*l/ 64.7%

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

      4. *-commutative 64.7%

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

      5. *-commutative 64.7%

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

   3. Simplified 64.7%

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

4. Final simplification 91.1%

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

Alternative 3: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+174}:\\ \;\;\;\;\frac{1}{x \cdot \frac{z}{x}} \cdot \frac{x \cdot \left(y \cdot \left(z \cdot 0.5\right)\right)}{z}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-40} \lor \neg \left(z \leq 9.5 \cdot 10^{-17}\right):\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.6e+174)
   (* (/ 1.0 (* x (/ z x))) (/ (* x (* y (* z 0.5))) z))
   (if (or (<= z -1.2e-40) (not (<= z 9.5e-17)))
     (* y (/ (cosh x) (* x z)))
     (* (cosh x) (/ (/ y z) x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+174) {
		tmp = (1.0 / (x * (z / x))) * ((x * (y * (z * 0.5))) / z);
	} else if ((z <= -1.2e-40) || !(z <= 9.5e-17)) {
		tmp = y * (cosh(x) / (x * z));
	} else {
		tmp = cosh(x) * ((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 <= (-1.6d+174)) then
        tmp = (1.0d0 / (x * (z / x))) * ((x * (y * (z * 0.5d0))) / z)
    else if ((z <= (-1.2d-40)) .or. (.not. (z <= 9.5d-17))) then
        tmp = y * (cosh(x) / (x * z))
    else
        tmp = cosh(x) * ((y / z) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.6e+174) {
		tmp = (1.0 / (x * (z / x))) * ((x * (y * (z * 0.5))) / z);
	} else if ((z <= -1.2e-40) || !(z <= 9.5e-17)) {
		tmp = y * (Math.cosh(x) / (x * z));
	} else {
		tmp = Math.cosh(x) * ((y / z) / x);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.6e+174:
		tmp = (1.0 / (x * (z / x))) * ((x * (y * (z * 0.5))) / z)
	elif (z <= -1.2e-40) or not (z <= 9.5e-17):
		tmp = y * (math.cosh(x) / (x * z))
	else:
		tmp = math.cosh(x) * ((y / z) / x)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.6e+174)
		tmp = Float64(Float64(1.0 / Float64(x * Float64(z / x))) * Float64(Float64(x * Float64(y * Float64(z * 0.5))) / z));
	elseif ((z <= -1.2e-40) || !(z <= 9.5e-17))
		tmp = Float64(y * Float64(cosh(x) / Float64(x * z)));
	else
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.6e+174)
		tmp = (1.0 / (x * (z / x))) * ((x * (y * (z * 0.5))) / z);
	elseif ((z <= -1.2e-40) || ~((z <= 9.5e-17)))
		tmp = y * (cosh(x) / (x * z));
	else
		tmp = cosh(x) * ((y / z) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.6e+174], N[(N[(1.0 / N[(x * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(y * N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[z, -1.2e-40], N[Not[LessEqual[z, 9.5e-17]], $MachinePrecision]], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e174

   1. Initial program 80.3%

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

   2. Taylor expanded in x around 0 53.1%

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

      1. +-commutative 53.1%

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

      2. associate-/l* 37.9%

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

      3. *-commutative 37.9%

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

   4. Simplified 37.9%

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

      1. +-commutative 37.9%

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

      2. *-commutative 37.9%

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

      3. associate-*r/ 34.8%

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

      4. frac-add 4.4%

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

      5. *-commutative 4.4%

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

      6. *-commutative 4.4%

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

      7. *-commutative 4.4%

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

   6. Applied egg-rr 4.4%

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

      1. +-commutative 4.4%

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

      2. *-commutative 4.4%

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

      3. associate-*l* 4.4%

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

      4. distribute-lft-out 4.4%

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

      5. *-commutative 4.4%

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

      6. *-commutative 4.4%

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

      7. associate-*l* 4.4%

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

   8. Simplified 4.4%

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

   9. Taylor expanded in x around inf 5.2%

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

       1. associate-*r* 5.2%

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

       2. *-commutative 5.2%

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

       3. *-commutative 5.2%

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

       4. *-commutative 5.2%

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

       5. associate-*l* 5.1%

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

       6. *-commutative 5.1%

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

       7. associate-*l* 5.1%

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

       8. *-commutative 5.1%

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

   11. Simplified 5.1%

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

       1. *-un-lft-identity 5.1%

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

       2. *-commutative 5.1%

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

       3. times-frac 61.4%

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

   13. Applied egg-rr 61.4%

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

   if -1.6e174 < z < -1.19999999999999996e-40 or 9.50000000000000029e-17 < z

   1. Initial program 76.4%

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

      1. associate-*r/ 91.4%

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

      2. associate-/l/ 83.9%

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

      3. associate-*l/ 83.7%

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

      4. *-commutative 83.7%

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

      5. *-commutative 83.7%

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

   3. Simplified 83.7%

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

   if -1.19999999999999996e-40 < z < 9.50000000000000029e-17

   1. Initial program 87.7%

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

      1. associate-*r/ 87.7%

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

      2. associate-/l/ 89.7%

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

      3. associate-/r* 98.3%

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

   3. Simplified 98.3%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+174}:\\ \;\;\;\;\frac{1}{x \cdot \frac{z}{x}} \cdot \frac{x \cdot \left(y \cdot \left(z \cdot 0.5\right)\right)}{z}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-40} \lor \neg \left(z \leq 9.5 \cdot 10^{-17}\right):\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 4: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1e+28)
   (* (/ y x) (/ (cosh x) z))
   (if (<= z 1.1e-14) (* (cosh x) (/ (/ y z) x)) (* y (/ (cosh x) (* x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+28) {
		tmp = (y / x) * (cosh(x) / z);
	} else if (z <= 1.1e-14) {
		tmp = cosh(x) * ((y / z) / x);
	} else {
		tmp = y * (cosh(x) / (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 <= (-1d+28)) then
        tmp = (y / x) * (cosh(x) / z)
    else if (z <= 1.1d-14) then
        tmp = cosh(x) * ((y / z) / x)
    else
        tmp = y * (cosh(x) / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1e+28) {
		tmp = (y / x) * (Math.cosh(x) / z);
	} else if (z <= 1.1e-14) {
		tmp = Math.cosh(x) * ((y / z) / x);
	} else {
		tmp = y * (Math.cosh(x) / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1e+28:
		tmp = (y / x) * (math.cosh(x) / z)
	elif z <= 1.1e-14:
		tmp = math.cosh(x) * ((y / z) / x)
	else:
		tmp = y * (math.cosh(x) / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1e+28)
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	elseif (z <= 1.1e-14)
		tmp = Float64(cosh(x) * Float64(Float64(y / z) / x));
	else
		tmp = Float64(y * Float64(cosh(x) / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1e+28)
		tmp = (y / x) * (cosh(x) / z);
	elseif (z <= 1.1e-14)
		tmp = cosh(x) * ((y / z) / x);
	else
		tmp = y * (cosh(x) / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1e+28], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.1e-14], N[(N[Cosh[x], $MachinePrecision] * N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.99999999999999958e27

   1. Initial program 82.3%

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

      1. *-commutative 82.3%

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

      2. associate-*r/ 82.2%

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

   3. Simplified 82.2%

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

   if -9.99999999999999958e27 < z < 1.1e-14

   1. Initial program 87.8%

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

      1. associate-*r/ 87.8%

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

      2. associate-/l/ 89.7%

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

      3. associate-/r* 97.7%

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

   3. Simplified 97.7%

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

   if 1.1e-14 < z

   1. Initial program 69.0%

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

      1. associate-*r/ 89.4%

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

      2. associate-/l/ 75.8%

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

      3. associate-*l/ 75.6%

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

      4. *-commutative 75.6%

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

      5. *-commutative 75.6%

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

   3. Simplified 75.6%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+28}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;\cosh x \cdot \frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \]

Alternative 5: 83.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

   1. associate-*r/ 95.8%

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

   2. associate-/l/ 83.3%

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

   3. associate-*l/ 82.2%

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

   4. *-commutative 82.2%

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

   5. *-commutative 82.2%

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

3. Simplified 82.2%

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

4. Final simplification 82.2%

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

Alternative 6: 70.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x \cdot \frac{z}{x}} \cdot \frac{x \cdot \left(y \cdot \left(z \cdot 0.5\right)\right)}{z}\\ \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+193}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z}{x}} \cdot \frac{z \cdot 0.5}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 (* x (/ z x))) (/ (* x (* y (* z 0.5))) z))))
   (if (<= x -1300000.0)
     t_0
     (if (<= x 1.4)
       (/ (/ y z) x)
       (if (<= x 2.95e+193)
         (* (/ (* x y) (/ z x)) (/ (* z 0.5) (* x z)))
         t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (1.0 / (x * (z / x))) * ((x * (y * (z * 0.5))) / z);
	double tmp;
	if (x <= -1300000.0) {
		tmp = t_0;
	} else if (x <= 1.4) {
		tmp = (y / z) / x;
	} else if (x <= 2.95e+193) {
		tmp = ((x * y) / (z / x)) * ((z * 0.5) / (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 = (1.0d0 / (x * (z / x))) * ((x * (y * (z * 0.5d0))) / z)
    if (x <= (-1300000.0d0)) then
        tmp = t_0
    else if (x <= 1.4d0) then
        tmp = (y / z) / x
    else if (x <= 2.95d+193) then
        tmp = ((x * y) / (z / x)) * ((z * 0.5d0) / (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 = (1.0 / (x * (z / x))) * ((x * (y * (z * 0.5))) / z);
	double tmp;
	if (x <= -1300000.0) {
		tmp = t_0;
	} else if (x <= 1.4) {
		tmp = (y / z) / x;
	} else if (x <= 2.95e+193) {
		tmp = ((x * y) / (z / x)) * ((z * 0.5) / (x * z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (1.0 / (x * (z / x))) * ((x * (y * (z * 0.5))) / z)
	tmp = 0
	if x <= -1300000.0:
		tmp = t_0
	elif x <= 1.4:
		tmp = (y / z) / x
	elif x <= 2.95e+193:
		tmp = ((x * y) / (z / x)) * ((z * 0.5) / (x * z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(1.0 / Float64(x * Float64(z / x))) * Float64(Float64(x * Float64(y * Float64(z * 0.5))) / z))
	tmp = 0.0
	if (x <= -1300000.0)
		tmp = t_0;
	elseif (x <= 1.4)
		tmp = Float64(Float64(y / z) / x);
	elseif (x <= 2.95e+193)
		tmp = Float64(Float64(Float64(x * y) / Float64(z / x)) * Float64(Float64(z * 0.5) / Float64(x * z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (1.0 / (x * (z / x))) * ((x * (y * (z * 0.5))) / z);
	tmp = 0.0;
	if (x <= -1300000.0)
		tmp = t_0;
	elseif (x <= 1.4)
		tmp = (y / z) / x;
	elseif (x <= 2.95e+193)
		tmp = ((x * y) / (z / x)) * ((z * 0.5) / (x * z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(1.0 / N[(x * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x * N[(y * N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1300000.0], t$95$0, If[LessEqual[x, 1.4], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.95e+193], N[(N[(N[(x * y), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision] * N[(N[(z * 0.5), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3e6 or 2.9499999999999999e193 < x

   1. Initial program 65.8%

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

   2. Taylor expanded in x around 0 42.1%

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

      1. +-commutative 42.1%

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

      2. associate-/l* 40.7%

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

      3. *-commutative 40.7%

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

   4. Simplified 40.7%

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

      1. +-commutative 40.7%

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

      2. *-commutative 40.7%

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

      3. associate-*r/ 39.6%

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

      4. frac-add 40.6%

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

      5. *-commutative 40.6%

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

      6. *-commutative 40.6%

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

      7. *-commutative 40.6%

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

   6. Applied egg-rr 40.6%

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

      1. +-commutative 40.6%

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

      2. *-commutative 40.6%

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

      3. associate-*l* 40.6%

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

      4. distribute-lft-out 40.6%

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

      5. *-commutative 40.6%

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

      6. *-commutative 40.6%

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

      7. associate-*l* 47.4%

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

   8. Simplified 47.4%

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

   9. Taylor expanded in x around inf 47.4%

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

       1. associate-*r* 47.4%

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

       2. *-commutative 47.4%

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

       3. *-commutative 47.4%

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

       4. *-commutative 47.4%

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

       5. associate-*l* 36.6%

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

       6. *-commutative 36.6%

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

       7. associate-*l* 36.6%

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

       8. *-commutative 36.6%

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

   11. Simplified 36.6%

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

       1. *-un-lft-identity 36.6%

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

       2. *-commutative 36.6%

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

       3. times-frac 58.8%

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

   13. Applied egg-rr 58.8%

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

   if -1.3e6 < x < 1.3999999999999999

   1. Initial program 92.3%

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

      1. associate-*r/ 92.3%

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

      2. associate-/l/ 91.6%

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

      3. associate-*l/ 89.5%

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

      4. *-commutative 89.5%

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

      5. *-commutative 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around 0 89.4%

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

      1. div-inv 91.6%

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

      2. associate-/r* 93.5%

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

   6. Applied egg-rr 93.5%

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

   if 1.3999999999999999 < x < 2.9499999999999999e193

   1. Initial program 79.5%

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

   2. Taylor expanded in x around 0 17.4%

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

      1. +-commutative 17.4%

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

      2. associate-/l* 28.0%

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

      3. *-commutative 28.0%

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

   4. Simplified 28.0%

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

      1. +-commutative 28.0%

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

      2. *-commutative 28.0%

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

      3. associate-*r/ 28.0%

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

      4. frac-add 27.4%

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

      5. *-commutative 27.4%

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

      6. *-commutative 27.4%

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

      7. *-commutative 27.4%

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

   6. Applied egg-rr 27.4%

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

      1. +-commutative 27.4%

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

      2. *-commutative 27.4%

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

      3. associate-*l* 27.4%

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

      4. distribute-lft-out 27.4%

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

      5. *-commutative 27.4%

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

      6. *-commutative 27.4%

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

      7. associate-*l* 29.7%

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

   8. Simplified 29.7%

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

   9. Taylor expanded in x around inf 29.7%

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

       1. associate-*r* 29.7%

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

       2. *-commutative 29.7%

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

       3. *-commutative 29.7%

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

       4. *-commutative 29.7%

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

       5. associate-*l* 25.3%

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

       6. *-commutative 25.3%

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

       7. associate-*l* 25.3%

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

       8. *-commutative 25.3%

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

   11. Simplified 25.3%

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

       1. associate-*r* 27.6%

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

       2. *-commutative 27.6%

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

       3. *-commutative 27.6%

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

       4. associate-*r* 27.5%

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

       5. times-frac 40.4%

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

       6. *-commutative 40.4%

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

   13. Applied egg-rr 40.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1300000:\\ \;\;\;\;\frac{1}{x \cdot \frac{z}{x}} \cdot \frac{x \cdot \left(y \cdot \left(z \cdot 0.5\right)\right)}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{+193}:\\ \;\;\;\;\frac{x \cdot y}{\frac{z}{x}} \cdot \frac{z \cdot 0.5}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x \cdot \frac{z}{x}} \cdot \frac{x \cdot \left(y \cdot \left(z \cdot 0.5\right)\right)}{z}\\ \end{array} \]

Alternative 7: 70.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot z}\\ t_1 := t_0 + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-261}:\\ \;\;\;\;t_0 + y \cdot \frac{0.5}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ y (* x z))) (t_1 (+ t_0 (* 0.5 (/ (* x y) z)))))
   (if (<= y -2e+37)
     t_1
     (if (<= y 1.25e-261)
       (+ t_0 (* y (/ 0.5 (/ z x))))
       (if (<= y 5e-11) (/ (+ (/ y x) (* x (* y 0.5))) z) t_1)))))

C

double code(double x, double y, double z) {
	double t_0 = y / (x * z);
	double t_1 = t_0 + (0.5 * ((x * y) / z));
	double tmp;
	if (y <= -2e+37) {
		tmp = t_1;
	} else if (y <= 1.25e-261) {
		tmp = t_0 + (y * (0.5 / (z / x)));
	} else if (y <= 5e-11) {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y / (x * z)
    t_1 = t_0 + (0.5d0 * ((x * y) / z))
    if (y <= (-2d+37)) then
        tmp = t_1
    else if (y <= 1.25d-261) then
        tmp = t_0 + (y * (0.5d0 / (z / x)))
    else if (y <= 5d-11) then
        tmp = ((y / x) + (x * (y * 0.5d0))) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y / (x * z);
	double t_1 = t_0 + (0.5 * ((x * y) / z));
	double tmp;
	if (y <= -2e+37) {
		tmp = t_1;
	} else if (y <= 1.25e-261) {
		tmp = t_0 + (y * (0.5 / (z / x)));
	} else if (y <= 5e-11) {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y / (x * z)
	t_1 = t_0 + (0.5 * ((x * y) / z))
	tmp = 0
	if y <= -2e+37:
		tmp = t_1
	elif y <= 1.25e-261:
		tmp = t_0 + (y * (0.5 / (z / x)))
	elif y <= 5e-11:
		tmp = ((y / x) + (x * (y * 0.5))) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y / Float64(x * z))
	t_1 = Float64(t_0 + Float64(0.5 * Float64(Float64(x * y) / z)))
	tmp = 0.0
	if (y <= -2e+37)
		tmp = t_1;
	elseif (y <= 1.25e-261)
		tmp = Float64(t_0 + Float64(y * Float64(0.5 / Float64(z / x))));
	elseif (y <= 5e-11)
		tmp = Float64(Float64(Float64(y / x) + Float64(x * Float64(y * 0.5))) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y / (x * z);
	t_1 = t_0 + (0.5 * ((x * y) / z));
	tmp = 0.0;
	if (y <= -2e+37)
		tmp = t_1;
	elseif (y <= 1.25e-261)
		tmp = t_0 + (y * (0.5 / (z / x)));
	elseif (y <= 5e-11)
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+37], t$95$1, If[LessEqual[y, 1.25e-261], N[(t$95$0 + N[(y * N[(0.5 / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-11], N[(N[(N[(y / x), $MachinePrecision] + N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.99999999999999991e37 or 5.00000000000000018e-11 < y

   1. Initial program 90.8%

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

   2. Taylor expanded in x around 0 82.4%

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

   if -1.99999999999999991e37 < y < 1.24999999999999995e-261

   1. Initial program 78.2%

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

   2. Taylor expanded in x around 0 55.5%

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

      1. +-commutative 55.5%

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

      2. associate-/l* 64.1%

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

      3. *-commutative 64.1%

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

   4. Simplified 64.1%

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

      1. clear-num 64.1%

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

      2. un-div-inv 64.1%

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

   6. Applied egg-rr 64.1%

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

      1. associate-/r/ 65.2%

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

   8. Applied egg-rr 65.2%

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

   if 1.24999999999999995e-261 < y < 5.00000000000000018e-11

   1. Initial program 72.5%

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

   2. Taylor expanded in x around 0 61.2%

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

      1. associate-*r* 61.2%

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

   4. Simplified 61.2%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-261}:\\ \;\;\;\;\frac{y}{x \cdot z} + y \cdot \frac{0.5}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]

Alternative 8: 68.5% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.3999999999999999 < x

   1. Initial program 70.9%

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

   2. Taylor expanded in x around 0 32.8%

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

      1. +-commutative 32.8%

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

      2. associate-/l* 35.9%

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

      3. *-commutative 35.9%

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

   4. Simplified 35.9%

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

      1. +-commutative 35.9%

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

      2. *-commutative 35.9%

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

      3. associate-*r/ 35.2%

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

      4. frac-add 35.6%

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

      5. *-commutative 35.6%

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

      6. *-commutative 35.6%

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

      7. *-commutative 35.6%

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

   6. Applied egg-rr 35.6%

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

      1. +-commutative 35.6%

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

      2. *-commutative 35.6%

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

      3. associate-*l* 35.6%

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

      4. distribute-lft-out 35.6%

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

      5. *-commutative 35.6%

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

      6. *-commutative 35.6%

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

      7. associate-*l* 40.8%

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

   8. Simplified 40.8%

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

   9. Taylor expanded in x around inf 40.8%

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

       1. associate-*r* 40.8%

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

       2. *-commutative 40.8%

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

       3. *-commutative 40.8%

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

       4. *-commutative 40.8%

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

       5. associate-*l* 32.4%

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

       6. *-commutative 32.4%

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

       7. associate-*l* 32.4%

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

       8. *-commutative 32.4%

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

   11. Simplified 32.4%

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

       1. associate-*r* 38.4%

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

       2. *-commutative 38.4%

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

       3. *-commutative 38.4%

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

       4. associate-*r* 35.8%

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

       5. times-frac 44.0%

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

       6. *-commutative 44.0%

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

   13. Applied egg-rr 44.0%

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

   if -1.3999999999999999 < x < 1.3999999999999999

   1. Initial program 92.3%

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

      1. associate-*r/ 92.3%

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

      2. associate-/l/ 91.6%

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

      3. associate-*l/ 89.5%

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

      4. *-commutative 89.5%

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

      5. *-commutative 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around 0 89.4%

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

      1. div-inv 91.6%

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

      2. associate-/r* 93.5%

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

   6. Applied egg-rr 93.5%

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

4. Final simplification 70.9%

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

Alternative 9: 68.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{x \cdot y}{\frac{z}{x}} \cdot \frac{z \cdot 0.5}{x \cdot z}\\ \mathbf{elif}\;x \leq 225:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{\frac{z}{x} + 0.5 \cdot \left(x \cdot z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.4)
   (* (/ (* x y) (/ z x)) (/ (* z 0.5) (* x z)))
   (if (<= x 225.0)
     (/ (/ y z) x)
     (/ y (/ (* z z) (+ (/ z x) (* 0.5 (* x z))))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4) {
		tmp = ((x * y) / (z / x)) * ((z * 0.5) / (x * z));
	} else if (x <= 225.0) {
		tmp = (y / z) / x;
	} else {
		tmp = y / ((z * z) / ((z / x) + (0.5 * (x * z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-1.4d0)) then
        tmp = ((x * y) / (z / x)) * ((z * 0.5d0) / (x * z))
    else if (x <= 225.0d0) then
        tmp = (y / z) / x
    else
        tmp = y / ((z * z) / ((z / x) + (0.5d0 * (x * z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.4) {
		tmp = ((x * y) / (z / x)) * ((z * 0.5) / (x * z));
	} else if (x <= 225.0) {
		tmp = (y / z) / x;
	} else {
		tmp = y / ((z * z) / ((z / x) + (0.5 * (x * z))));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.4:
		tmp = ((x * y) / (z / x)) * ((z * 0.5) / (x * z))
	elif x <= 225.0:
		tmp = (y / z) / x
	else:
		tmp = y / ((z * z) / ((z / x) + (0.5 * (x * z))))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(Float64(x * y) / Float64(z / x)) * Float64(Float64(z * 0.5) / Float64(x * z)));
	elseif (x <= 225.0)
		tmp = Float64(Float64(y / z) / x);
	else
		tmp = Float64(y / Float64(Float64(z * z) / Float64(Float64(z / x) + Float64(0.5 * Float64(x * z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.4)
		tmp = ((x * y) / (z / x)) * ((z * 0.5) / (x * z));
	elseif (x <= 225.0)
		tmp = (y / z) / x;
	else
		tmp = y / ((z * z) / ((z / x) + (0.5 * (x * z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.4], N[(N[(N[(x * y), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision] * N[(N[(z * 0.5), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 225.0], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(N[(z * z), $MachinePrecision] / N[(N[(z / x), $MachinePrecision] + N[(0.5 * N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 69.0%

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

   2. Taylor expanded in x around 0 37.1%

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

      1. +-commutative 37.1%

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

      2. associate-/l* 38.6%

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

      3. *-commutative 38.6%

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

   4. Simplified 38.6%

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

      1. +-commutative 38.6%

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

      2. *-commutative 38.6%

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

      3. associate-*r/ 38.6%

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

      4. frac-add 40.7%

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

      5. *-commutative 40.7%

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

      6. *-commutative 40.7%

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

      7. *-commutative 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. +-commutative 40.7%

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

      2. *-commutative 40.7%

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

      3. associate-*l* 40.7%

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

      4. distribute-lft-out 40.7%

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

      5. *-commutative 40.7%

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

      6. *-commutative 40.7%

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

      7. associate-*l* 44.1%

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

   8. Simplified 44.1%

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

   9. Taylor expanded in x around inf 44.1%

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

       1. associate-*r* 44.1%

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

       2. *-commutative 44.1%

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

       3. *-commutative 44.1%

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

       4. *-commutative 44.1%

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

       5. associate-*l* 33.9%

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

       6. *-commutative 33.9%

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

       7. associate-*l* 33.9%

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

       8. *-commutative 33.9%

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

   11. Simplified 33.9%

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

       1. associate-*r* 42.6%

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

       2. *-commutative 42.6%

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

       3. *-commutative 42.6%

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

       4. associate-*r* 40.8%

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

       5. times-frac 47.6%

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

       6. *-commutative 47.6%

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

   13. Applied egg-rr 47.6%

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

   if -1.3999999999999999 < x < 225

   1. Initial program 92.3%

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

      1. associate-*r/ 92.3%

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

      2. associate-/l/ 91.6%

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

      3. associate-*l/ 89.5%

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

      4. *-commutative 89.5%

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

      5. *-commutative 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around 0 89.4%

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

      1. div-inv 91.6%

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

      2. associate-/r* 93.5%

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

   6. Applied egg-rr 93.5%

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

   if 225 < x

   1. Initial program 72.9%

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

   2. Taylor expanded in x around 0 28.6%

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

      1. +-commutative 28.6%

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

      2. associate-/l* 33.2%

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

      3. *-commutative 33.2%

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

   4. Simplified 33.2%

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

      1. associate-*r/ 31.9%

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

      2. *-commutative 31.9%

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

      3. frac-add 30.6%

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

      4. *-commutative 30.6%

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

      5. *-commutative 30.6%

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

      6. *-commutative 30.6%

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

   6. Applied egg-rr 30.6%

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

   7. Taylor expanded in y around 0 37.4%

      \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{z}{x} + 0.5 \cdot \left(z \cdot x\right)\right)}{{z}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 40.9%

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

      2. unpow2 40.9%

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

   9. Simplified 40.9%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{x \cdot y}{\frac{z}{x}} \cdot \frac{z \cdot 0.5}{x \cdot z}\\ \mathbf{elif}\;x \leq 225:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z \cdot z}{\frac{z}{x} + 0.5 \cdot \left(x \cdot z\right)}}\\ \end{array} \]

Alternative 10: 63.8% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.7 \cdot 10^{-295}:\\ \;\;\;\;y \cdot \left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - x \cdot -0.5}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 7.7e-295)
   (* y (+ (/ 1.0 (* x z)) (* 0.5 (/ x z))))
   (/ (- (/ 1.0 x) (* x -0.5)) (/ z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.7e-295) {
		tmp = y * ((1.0 / (x * z)) + (0.5 * (x / z)));
	} else {
		tmp = ((1.0 / x) - (x * -0.5)) / (z / y);
	}
	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 <= 7.7d-295) then
        tmp = y * ((1.0d0 / (x * z)) + (0.5d0 * (x / z)))
    else
        tmp = ((1.0d0 / x) - (x * (-0.5d0))) / (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 7.7e-295) {
		tmp = y * ((1.0 / (x * z)) + (0.5 * (x / z)));
	} else {
		tmp = ((1.0 / x) - (x * -0.5)) / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 7.7e-295:
		tmp = y * ((1.0 / (x * z)) + (0.5 * (x / z)))
	else:
		tmp = ((1.0 / x) - (x * -0.5)) / (z / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 7.7e-295)
		tmp = Float64(y * Float64(Float64(1.0 / Float64(x * z)) + Float64(0.5 * Float64(x / z))));
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(x * -0.5)) / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 7.7e-295)
		tmp = y * ((1.0 / (x * z)) + (0.5 * (x / z)));
	else
		tmp = ((1.0 / x) - (x * -0.5)) / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 7.7e-295], N[(y * N[(N[(1.0 / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.70000000000000018e-295

   1. Initial program 83.5%

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

      1. associate-*r/ 96.3%

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

      2. associate-/l/ 89.0%

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

      3. associate-*l/ 88.6%

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

      4. *-commutative 88.6%

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

      5. *-commutative 88.6%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in x around 0 69.4%

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

   if 7.70000000000000018e-295 < y

   1. Initial program 81.5%

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

   2. Taylor expanded in x around 0 61.4%

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

      1. +-commutative 61.4%

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

      2. associate-/l* 62.9%

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

      3. *-commutative 62.9%

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

   4. Simplified 62.9%

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

      1. +-commutative 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*r/ 62.3%

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

      4. frac-add 41.0%

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

      5. *-commutative 41.0%

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

      6. *-commutative 41.0%

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

      7. *-commutative 41.0%

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

   6. Applied egg-rr 41.0%

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

      1. +-commutative 41.0%

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

      2. *-commutative 41.0%

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

      3. associate-*l* 41.0%

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

      4. distribute-lft-out 41.0%

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

      5. *-commutative 41.0%

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

      6. *-commutative 41.0%

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

      7. associate-*l* 46.9%

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

   8. Simplified 46.9%

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

   9. Taylor expanded in z around -inf 65.7%

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

       1. mul-1-neg 65.7%

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

       2. associate-/l* 66.2%

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

       3. *-commutative 66.2%

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

   11. Simplified 66.2%

       \[\leadsto \color{blue}{-\frac{x \cdot -0.5 - \frac{1}{x}}{\frac{z}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.7 \cdot 10^{-295}:\\ \;\;\;\;y \cdot \left(\frac{1}{x \cdot z} + 0.5 \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - x \cdot -0.5}{\frac{z}{y}}\\ \end{array} \]

Alternative 11: 63.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-290}:\\ \;\;\;\;\frac{y}{x \cdot z} + y \cdot \frac{0.5}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - x \cdot -0.5}{\frac{z}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 3.1e-290)
   (+ (/ y (* x z)) (* y (/ 0.5 (/ z x))))
   (/ (- (/ 1.0 x) (* x -0.5)) (/ z y))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.1e-290) {
		tmp = (y / (x * z)) + (y * (0.5 / (z / x)));
	} else {
		tmp = ((1.0 / x) - (x * -0.5)) / (z / y);
	}
	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 <= 3.1d-290) then
        tmp = (y / (x * z)) + (y * (0.5d0 / (z / x)))
    else
        tmp = ((1.0d0 / x) - (x * (-0.5d0))) / (z / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 3.1e-290) {
		tmp = (y / (x * z)) + (y * (0.5 / (z / x)));
	} else {
		tmp = ((1.0 / x) - (x * -0.5)) / (z / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 3.1e-290:
		tmp = (y / (x * z)) + (y * (0.5 / (z / x)))
	else:
		tmp = ((1.0 / x) - (x * -0.5)) / (z / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 3.1e-290)
		tmp = Float64(Float64(y / Float64(x * z)) + Float64(y * Float64(0.5 / Float64(z / x))));
	else
		tmp = Float64(Float64(Float64(1.0 / x) - Float64(x * -0.5)) / Float64(z / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 3.1e-290)
		tmp = (y / (x * z)) + (y * (0.5 / (z / x)));
	else
		tmp = ((1.0 / x) - (x * -0.5)) / (z / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 3.1e-290], N[(N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision] + N[(y * N[(0.5 / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] - N[(x * -0.5), $MachinePrecision]), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.0999999999999999e-290

   1. Initial program 83.5%

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

   2. Taylor expanded in x around 0 67.8%

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

      1. +-commutative 67.8%

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

      2. associate-/l* 69.2%

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

      3. *-commutative 69.2%

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

   4. Simplified 69.2%

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

      1. clear-num 69.2%

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

      2. un-div-inv 69.2%

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

   6. Applied egg-rr 69.2%

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

      1. associate-/r/ 69.9%

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

   8. Applied egg-rr 69.9%

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

   if 3.0999999999999999e-290 < y

   1. Initial program 81.5%

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

   2. Taylor expanded in x around 0 61.4%

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

      1. +-commutative 61.4%

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

      2. associate-/l* 62.9%

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

      3. *-commutative 62.9%

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

   4. Simplified 62.9%

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

      1. +-commutative 62.9%

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

      2. *-commutative 62.9%

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

      3. associate-*r/ 62.3%

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

      4. frac-add 41.0%

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

      5. *-commutative 41.0%

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

      6. *-commutative 41.0%

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

      7. *-commutative 41.0%

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

   6. Applied egg-rr 41.0%

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

      1. +-commutative 41.0%

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

      2. *-commutative 41.0%

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

      3. associate-*l* 41.0%

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

      4. distribute-lft-out 41.0%

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

      5. *-commutative 41.0%

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

      6. *-commutative 41.0%

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

      7. associate-*l* 46.9%

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

   8. Simplified 46.9%

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

   9. Taylor expanded in z around -inf 65.7%

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

       1. mul-1-neg 65.7%

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

       2. associate-/l* 66.2%

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

       3. *-commutative 66.2%

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

   11. Simplified 66.2%

       \[\leadsto \color{blue}{-\frac{x \cdot -0.5 - \frac{1}{x}}{\frac{z}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-290}:\\ \;\;\;\;\frac{y}{x \cdot z} + y \cdot \frac{0.5}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x} - x \cdot -0.5}{\frac{z}{y}}\\ \end{array} \]

Alternative 12: 66.0% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.3999999999999999 < x

   1. Initial program 70.9%

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

   2. Taylor expanded in x around 0 32.8%

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

      1. associate-*r* 32.8%

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

   4. Simplified 32.8%

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

   5. Taylor expanded in x around inf 32.8%

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

      1. associate-*r/ 36.7%

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

      2. *-commutative 36.7%

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

      3. *-commutative 36.7%

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

      4. associate-*l* 36.0%

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

   7. Simplified 36.0%

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

   if -1.3999999999999999 < x < 1.3999999999999999

   1. Initial program 92.3%

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

      1. associate-*r/ 92.3%

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

      2. associate-/l/ 91.6%

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

      3. associate-*l/ 89.5%

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

      4. *-commutative 89.5%

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

      5. *-commutative 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in x around 0 89.4%

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

      1. div-inv 91.6%

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

      2. associate-/r* 93.5%

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

   6. Applied egg-rr 93.5%

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

4. Final simplification 67.2%

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

Alternative 13: 65.9% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

2. Taylor expanded in x around 0 65.1%

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

   1. associate-*r* 65.1%

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

4. Simplified 65.1%

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

5. Final simplification 65.1%

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

Alternative 14: 53.1% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+21} \lor \neg \left(z \leq 1.85 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.8e+21) (not (<= z 1.85e-19))) (/ y (* x z)) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.8e+21) || !(z <= 1.85e-19)) {
		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 ((z <= (-1.8d+21)) .or. (.not. (z <= 1.85d-19))) 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 ((z <= -1.8e+21) || !(z <= 1.85e-19)) {
		tmp = y / (x * z);
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.8e+21) or not (z <= 1.85e-19):
		tmp = y / (x * z)
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.8e+21) || !(z <= 1.85e-19))
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.8e+21) || ~((z <= 1.85e-19)))
		tmp = y / (x * z);
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.8e+21], N[Not[LessEqual[z, 1.85e-19]], $MachinePrecision]], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e21 or 1.85000000000000003e-19 < z

   1. Initial program 76.3%

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

   2. Taylor expanded in x around 0 50.8%

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

   if -1.8e21 < z < 1.85000000000000003e-19

   1. Initial program 87.7%

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

   2. Taylor expanded in x around 0 60.4%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+21} \lor \neg \left(z \leq 1.85 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 15: 55.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 7e-18) (/ (/ y z) x) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7e-18) {
		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 <= 7d-18) 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 <= 7e-18) {
		tmp = (y / z) / x;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7e-18:
		tmp = (y / z) / x
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7e-18)
		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 <= 7e-18)
		tmp = (y / z) / x;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7e-18], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 6.9999999999999997e-18

   1. Initial program 86.1%

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

      1. associate-*r/ 97.5%

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

      2. associate-/l/ 85.3%

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

      3. associate-*l/ 83.8%

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

      4. *-commutative 83.8%

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

      5. *-commutative 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in x around 0 49.2%

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

      1. div-inv 50.7%

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

      2. associate-/r* 57.9%

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

   6. Applied egg-rr 57.9%

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

   if 6.9999999999999997e-18 < z

   1. Initial program 69.6%

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

   2. Taylor expanded in x around 0 55.3%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 16: 49.5% 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 82.5%

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

2. Taylor expanded in x around 0 51.7%

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

3. Final simplification 51.7%

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

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 2023257 
(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))