Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.0% → 96.1%

Time: 7.6s

Alternatives: 16

Speedup: 1.0×

• 64.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% 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: 96.1% 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 85.5%

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

   1. associate-*r/ 98.0%

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

   2. associate-/l/ 85.7%

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

   3. associate-*l/ 85.2%

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

   4. *-commutative 85.2%

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

   5. *-commutative 85.2%

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

3. Simplified 85.2%

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

   1. *-commutative 85.2%

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

   2. associate-/r* 96.6%

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

   3. associate-*l/ 98.0%

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

5. Applied egg-rr 98.0%

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

6. Final simplification 98.0%

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

Alternative 2: 92.5% 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 97.8%

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

   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/ 59.4%

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

      3. associate-*l/ 59.4%

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

      4. *-commutative 59.4%

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

      5. *-commutative 59.4%

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

   3. Simplified 59.4%

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

4. Final simplification 93.0%

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

Alternative 3: 87.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+97} \lor \neg \left(z \leq 4.8 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2.8e+97) (not (<= z 4.8e+169)))
   (* (/ y x) (/ (cosh x) z))
   (* y (/ (cosh x) (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2.8e+97) || !(z <= 4.8e+169)) {
		tmp = (y / x) * (cosh(x) / z);
	} 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 <= (-2.8d+97)) .or. (.not. (z <= 4.8d+169))) then
        tmp = (y / x) * (cosh(x) / z)
    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 <= -2.8e+97) || !(z <= 4.8e+169)) {
		tmp = (y / x) * (Math.cosh(x) / z);
	} else {
		tmp = y * (Math.cosh(x) / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2.8e+97) or not (z <= 4.8e+169):
		tmp = (y / x) * (math.cosh(x) / z)
	else:
		tmp = y * (math.cosh(x) / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2.8e+97) || !(z <= 4.8e+169))
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	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 <= -2.8e+97) || ~((z <= 4.8e+169)))
		tmp = (y / x) * (cosh(x) / z);
	else
		tmp = y * (cosh(x) / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2.8e+97], N[Not[LessEqual[z, 4.8e+169]], $MachinePrecision]], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.7999999999999999e97 or 4.7999999999999997e169 < z

   1. Initial program 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-*r/ 80.1%

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

   3. Simplified 80.1%

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

   if -2.7999999999999999e97 < z < 4.7999999999999997e169

   1. Initial program 87.2%

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

      1. associate-*r/ 98.9%

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

      2. associate-/l/ 94.1%

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

      3. associate-*l/ 93.5%

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

      4. *-commutative 93.5%

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

      5. *-commutative 93.5%

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

   3. Simplified 93.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.8 \cdot 10^{+97} \lor \neg \left(z \leq 4.8 \cdot 10^{+169}\right):\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\cosh x}{x \cdot z}\\ \end{array} \]

Alternative 4: 83.1% 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 85.5%

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

   1. associate-*r/ 98.0%

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

   2. associate-/l/ 85.7%

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

   3. associate-*l/ 85.2%

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

   4. *-commutative 85.2%

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

   5. *-commutative 85.2%

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

3. Simplified 85.2%

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

4. Final simplification 85.2%

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

Alternative 5: 68.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-218}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+257}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e-218)
   (* y (/ (+ (* x 0.5) (/ 1.0 x)) z))
   (if (<= x 1.4)
     (/ (/ 1.0 z) (/ x y))
     (if (<= x 7.2e+257)
       (* (/ y x) (/ (* x (* x 0.5)) z))
       (* (/ 0.5 z) (* x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-218) {
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	} else if (x <= 1.4) {
		tmp = (1.0 / z) / (x / y);
	} else if (x <= 7.2e+257) {
		tmp = (y / x) * ((x * (x * 0.5)) / z);
	} else {
		tmp = (0.5 / z) * (x * 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 (x <= (-1.9d-218)) then
        tmp = y * (((x * 0.5d0) + (1.0d0 / x)) / z)
    else if (x <= 1.4d0) then
        tmp = (1.0d0 / z) / (x / y)
    else if (x <= 7.2d+257) then
        tmp = (y / x) * ((x * (x * 0.5d0)) / z)
    else
        tmp = (0.5d0 / z) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-218) {
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	} else if (x <= 1.4) {
		tmp = (1.0 / z) / (x / y);
	} else if (x <= 7.2e+257) {
		tmp = (y / x) * ((x * (x * 0.5)) / z);
	} else {
		tmp = (0.5 / z) * (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.9e-218:
		tmp = y * (((x * 0.5) + (1.0 / x)) / z)
	elif x <= 1.4:
		tmp = (1.0 / z) / (x / y)
	elif x <= 7.2e+257:
		tmp = (y / x) * ((x * (x * 0.5)) / z)
	else:
		tmp = (0.5 / z) * (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e-218)
		tmp = Float64(y * Float64(Float64(Float64(x * 0.5) + Float64(1.0 / x)) / z));
	elseif (x <= 1.4)
		tmp = Float64(Float64(1.0 / z) / Float64(x / y));
	elseif (x <= 7.2e+257)
		tmp = Float64(Float64(y / x) * Float64(Float64(x * Float64(x * 0.5)) / z));
	else
		tmp = Float64(Float64(0.5 / z) * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.9e-218)
		tmp = y * (((x * 0.5) + (1.0 / x)) / z);
	elseif (x <= 1.4)
		tmp = (1.0 / z) / (x / y);
	elseif (x <= 7.2e+257)
		tmp = (y / x) * ((x * (x * 0.5)) / z);
	else
		tmp = (0.5 / z) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e-218], N[(y * N[(N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4], N[(N[(1.0 / z), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e+257], N[(N[(y / x), $MachinePrecision] * N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / z), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.8999999999999999e-218

   1. Initial program 83.7%

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

      1. associate-*r/ 97.4%

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

      2. associate-/l/ 89.1%

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

      3. associate-*l/ 89.0%

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

      4. *-commutative 89.0%

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

      5. *-commutative 89.0%

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

   3. Simplified 89.0%

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

   4. Taylor expanded in x around 0 68.6%

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

   5. Taylor expanded in z around 0 68.6%

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

   if -1.8999999999999999e-218 < x < 1.3999999999999999

   1. Initial program 97.3%

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

      1. associate-*r/ 97.3%

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

      2. associate-/l/ 91.4%

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

      3. associate-*l/ 90.0%

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

      4. *-commutative 90.0%

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

      5. *-commutative 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around 0 88.0%

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

      1. div-inv 89.4%

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

      2. *-un-lft-identity 89.4%

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

      3. frac-times 95.2%

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

      4. clear-num 95.1%

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

      5. un-div-inv 96.1%

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

   6. Applied egg-rr 96.1%

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

   if 1.3999999999999999 < x < 7.19999999999999968e257

   1. Initial program 85.2%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 75.9%

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

      3. associate-*l/ 75.9%

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

      4. *-commutative 75.9%

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

      5. *-commutative 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in x around 0 44.8%

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

      1. associate-/r* 44.8%

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

      2. associate-*r/ 44.8%

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

      3. frac-add 49.6%

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

      4. *-commutative 49.6%

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

   6. Applied egg-rr 49.6%

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

   7. Taylor expanded in y around 0 47.9%

      \[\leadsto \color{blue}{\frac{\left(1 + 0.5 \cdot {x}^{2}\right) \cdot y}{z \cdot x}} \]
   8. Step-by-step derivation

      1. times-frac 58.5%

         \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot {x}^{2}}{z} \cdot \frac{y}{x}} \]

      2. unpow2 58.5%

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

   9. Simplified 58.5%

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

   10. Taylor expanded in x around inf 58.5%

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

       1. associate-*r/ 58.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{z}} \cdot \frac{y}{x} \]

       2. *-commutative 58.5%

          \[\leadsto \frac{\color{blue}{{x}^{2} \cdot 0.5}}{z} \cdot \frac{y}{x} \]

       3. unpow2 58.5%

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

       4. associate-*r* 58.5%

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

   12. Simplified 58.5%

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

   if 7.19999999999999968e257 < x

   1. Initial program 35.7%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 64.3%

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

      3. associate-*l/ 64.3%

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

      4. *-commutative 64.3%

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

      5. *-commutative 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in x around 0 59.0%

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

   5. Taylor expanded in x around inf 59.0%

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

      1. associate-*r* 59.0%

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

      2. clear-num 59.0%

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

      3. un-div-inv 59.0%

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

      4. *-commutative 59.0%

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

   7. Applied egg-rr 59.0%

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

      1. div-inv 59.0%

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

      2. frac-times 65.5%

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

      3. associate-/l* 65.5%

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

      4. /-rgt-identity 65.5%

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

      5. *-commutative 65.5%

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

   9. Applied egg-rr 65.5%

      \[\leadsto \color{blue}{\frac{0.5}{z} \cdot \left(x \cdot y\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-218}:\\ \;\;\;\;y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+257}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 6: 69.4% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-201} \lor \neg \left(x \leq 1.25 \cdot 10^{+274}\right):\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.1e-201) (not (<= x 1.25e+274)))
   (* y (/ (+ 1.0 (* x (* x 0.5))) (* x z)))
   (* (/ y x) (/ (+ 1.0 (* 0.5 (* x x))) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.1e-201) || !(x <= 1.25e+274)) {
		tmp = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	} else {
		tmp = (y / x) * ((1.0 + (0.5 * (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 ((x <= (-3.1d-201)) .or. (.not. (x <= 1.25d+274))) then
        tmp = y * ((1.0d0 + (x * (x * 0.5d0))) / (x * z))
    else
        tmp = (y / x) * ((1.0d0 + (0.5d0 * (x * x))) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.1e-201) || !(x <= 1.25e+274)) {
		tmp = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	} else {
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.1e-201) or not (x <= 1.25e+274):
		tmp = y * ((1.0 + (x * (x * 0.5))) / (x * z))
	else:
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.1e-201) || !(x <= 1.25e+274))
		tmp = Float64(y * Float64(Float64(1.0 + Float64(x * Float64(x * 0.5))) / Float64(x * z)));
	else
		tmp = Float64(Float64(y / x) * Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.1e-201) || ~((x <= 1.25e+274)))
		tmp = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	else
		tmp = (y / x) * ((1.0 + (0.5 * (x * x))) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.1e-201], N[Not[LessEqual[x, 1.25e+274]], $MachinePrecision]], N[(y * N[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0999999999999999e-201 or 1.24999999999999995e274 < x

   1. Initial program 78.0%

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

      1. associate-*r/ 97.4%

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

      2. associate-/l/ 87.6%

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

      3. associate-*l/ 87.6%

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

      4. *-commutative 87.6%

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

      5. *-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in x around 0 66.9%

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

      1. associate-/r* 66.8%

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

      2. associate-*r/ 66.8%

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

      3. frac-add 74.2%

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

      4. *-commutative 74.2%

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

   6. Applied egg-rr 74.2%

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

   7. Taylor expanded in z around 0 74.2%

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

   if -3.0999999999999999e-201 < x < 1.24999999999999995e274

   1. Initial program 91.5%

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

      1. associate-*r/ 98.5%

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

      2. associate-/l/ 84.1%

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

      3. associate-*l/ 83.3%

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

      4. *-commutative 83.3%

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

      5. *-commutative 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in x around 0 71.2%

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

      1. associate-/r* 71.2%

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

      2. associate-*r/ 71.2%

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

      3. frac-add 72.8%

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

      4. *-commutative 72.8%

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

   6. Applied egg-rr 72.8%

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

   7. Taylor expanded in y around 0 73.0%

      \[\leadsto \color{blue}{\frac{\left(1 + 0.5 \cdot {x}^{2}\right) \cdot y}{z \cdot x}} \]
   8. Step-by-step derivation

      1. times-frac 80.8%

         \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot {x}^{2}}{z} \cdot \frac{y}{x}} \]

      2. unpow2 80.8%

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

   9. Simplified 80.8%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-201} \lor \neg \left(x \leq 1.25 \cdot 10^{+274}\right):\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{z}\\ \end{array} \]

Alternative 7: 68.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5000000:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+256}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 5000000.0)
   (/ (+ (/ y x) (* x (* y 0.5))) z)
   (if (<= x 7.5e+256)
     (* (/ y x) (/ (* x (* x 0.5)) z))
     (* (/ 0.5 z) (* x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 5000000.0) {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	} else if (x <= 7.5e+256) {
		tmp = (y / x) * ((x * (x * 0.5)) / z);
	} else {
		tmp = (0.5 / z) * (x * 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 (x <= 5000000.0d0) then
        tmp = ((y / x) + (x * (y * 0.5d0))) / z
    else if (x <= 7.5d+256) then
        tmp = (y / x) * ((x * (x * 0.5d0)) / z)
    else
        tmp = (0.5d0 / z) * (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 5000000.0) {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	} else if (x <= 7.5e+256) {
		tmp = (y / x) * ((x * (x * 0.5)) / z);
	} else {
		tmp = (0.5 / z) * (x * y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 5000000.0:
		tmp = ((y / x) + (x * (y * 0.5))) / z
	elif x <= 7.5e+256:
		tmp = (y / x) * ((x * (x * 0.5)) / z)
	else:
		tmp = (0.5 / z) * (x * y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 5000000.0)
		tmp = Float64(Float64(Float64(y / x) + Float64(x * Float64(y * 0.5))) / z);
	elseif (x <= 7.5e+256)
		tmp = Float64(Float64(y / x) * Float64(Float64(x * Float64(x * 0.5)) / z));
	else
		tmp = Float64(Float64(0.5 / z) * Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 5000000.0)
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	elseif (x <= 7.5e+256)
		tmp = (y / x) * ((x * (x * 0.5)) / z);
	else
		tmp = (0.5 / z) * (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 5000000.0], N[(N[(N[(y / x), $MachinePrecision] + N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 7.5e+256], N[(N[(y / x), $MachinePrecision] * N[(N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / z), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 5e6

   1. Initial program 89.5%

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

   2. Taylor expanded in x around 0 78.8%

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

      1. associate-*r* 78.8%

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

   4. Simplified 78.8%

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

   if 5e6 < x < 7.4999999999999999e256

   1. Initial program 84.6%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 75.0%

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

      3. associate-*l/ 75.0%

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

      4. *-commutative 75.0%

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

      5. *-commutative 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in x around 0 46.4%

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

      1. associate-/r* 46.4%

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

      2. associate-*r/ 46.4%

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

      3. frac-add 51.3%

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

      4. *-commutative 51.3%

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

   6. Applied egg-rr 51.3%

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

   7. Taylor expanded in y around 0 49.6%

      \[\leadsto \color{blue}{\frac{\left(1 + 0.5 \cdot {x}^{2}\right) \cdot y}{z \cdot x}} \]
   8. Step-by-step derivation

      1. times-frac 60.5%

         \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot {x}^{2}}{z} \cdot \frac{y}{x}} \]

      2. unpow2 60.5%

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

   9. Simplified 60.5%

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

   10. Taylor expanded in x around inf 60.5%

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

       1. associate-*r/ 60.5%

          \[\leadsto \color{blue}{\frac{0.5 \cdot {x}^{2}}{z}} \cdot \frac{y}{x} \]

       2. *-commutative 60.5%

          \[\leadsto \frac{\color{blue}{{x}^{2} \cdot 0.5}}{z} \cdot \frac{y}{x} \]

       3. unpow2 60.5%

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

       4. associate-*r* 60.5%

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

   12. Simplified 60.5%

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

   if 7.4999999999999999e256 < x

   1. Initial program 35.7%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 64.3%

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

      3. associate-*l/ 64.3%

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

      4. *-commutative 64.3%

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

      5. *-commutative 64.3%

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

   3. Simplified 64.3%

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

   4. Taylor expanded in x around 0 59.0%

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

   5. Taylor expanded in x around inf 59.0%

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

      1. associate-*r* 59.0%

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

      2. clear-num 59.0%

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

      3. un-div-inv 59.0%

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

      4. *-commutative 59.0%

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

   7. Applied egg-rr 59.0%

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

      1. div-inv 59.0%

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

      2. frac-times 65.5%

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

      3. associate-/l* 65.5%

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

      4. /-rgt-identity 65.5%

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

      5. *-commutative 65.5%

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

   9. Applied egg-rr 65.5%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5000000:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+256}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{x \cdot \left(x \cdot 0.5\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 8: 68.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+171}:\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 4.5e+171)
   (* y (/ (+ 1.0 (* x (* x 0.5))) (* x z)))
   (/ (+ (/ y x) (* x (* y 0.5))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+171) {
		tmp = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	} else {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 4.5d+171) then
        tmp = y * ((1.0d0 + (x * (x * 0.5d0))) / (x * z))
    else
        tmp = ((y / x) + (x * (y * 0.5d0))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 4.5e+171) {
		tmp = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	} else {
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 4.5e+171:
		tmp = y * ((1.0 + (x * (x * 0.5))) / (x * z))
	else:
		tmp = ((y / x) + (x * (y * 0.5))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 4.5e+171)
		tmp = Float64(y * Float64(Float64(1.0 + Float64(x * Float64(x * 0.5))) / Float64(x * z)));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(x * Float64(y * 0.5))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 4.5e+171)
		tmp = y * ((1.0 + (x * (x * 0.5))) / (x * z));
	else
		tmp = ((y / x) + (x * (y * 0.5))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 4.5e+171], N[(y * N[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(x * N[(y * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.49999999999999969e171

   1. Initial program 83.8%

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

      1. associate-*r/ 98.2%

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

      2. associate-/l/ 85.3%

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

      3. associate-*l/ 84.8%

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

      4. *-commutative 84.8%

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

      5. *-commutative 84.8%

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

   3. Simplified 84.8%

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

   4. Taylor expanded in x around 0 67.6%

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

      1. associate-/r* 67.6%

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

      2. associate-*r/ 67.6%

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

      3. frac-add 72.5%

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

      4. *-commutative 72.5%

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

   6. Applied egg-rr 72.5%

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

   7. Taylor expanded in z around 0 72.5%

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

   if 4.49999999999999969e171 < y

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 94.3%

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

      1. associate-*r* 94.3%

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

   4. Simplified 94.3%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+171}:\\ \;\;\;\;y \cdot \frac{1 + x \cdot \left(x \cdot 0.5\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + x \cdot \left(y \cdot 0.5\right)}{z}\\ \end{array} \]

Alternative 9: 65.0% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.42) (not (<= x 1.4))) (* 0.5 (/ y (/ z x))) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.42) || !(x <= 1.4)) {
		tmp = 0.5 * (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 ((x <= (-1.42d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = 0.5d0 * (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 ((x <= -1.42) || !(x <= 1.4)) {
		tmp = 0.5 * (y / (z / x));
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.42) or not (x <= 1.4):
		tmp = 0.5 * (y / (z / x))
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.42) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(y / Float64(z / x)));
	else
		tmp = Float64(Float64(y / x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.42) || ~((x <= 1.4)))
		tmp = 0.5 * (y / (z / x));
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.42], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4199999999999999 or 1.3999999999999999 < x

   1. Initial program 73.3%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 75.8%

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

      3. associate-*l/ 75.8%

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

      4. *-commutative 75.8%

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

      5. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in x around 0 42.5%

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

   5. Taylor expanded in x around inf 40.0%

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

      1. associate-/l* 38.5%

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

   7. Simplified 38.5%

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

   if -1.4199999999999999 < x < 1.3999999999999999

   1. Initial program 96.3%

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

      1. associate-*r/ 96.3%

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

      2. associate-/l/ 94.3%

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

      3. associate-*l/ 93.5%

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

      4. *-commutative 93.5%

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

      5. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 93.2%

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

      1. associate-/l/ 95.2%

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

   6. Simplified 95.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 10: 65.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.42) (not (<= x 1.4))) (* y (* 0.5 (/ x z))) (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.42) || !(x <= 1.4)) {
		tmp = y * (0.5 * (x / z));
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.42d0)) .or. (.not. (x <= 1.4d0))) then
        tmp = y * (0.5d0 * (x / z))
    else
        tmp = (y / x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.42) || !(x <= 1.4)) {
		tmp = y * (0.5 * (x / z));
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.42) or not (x <= 1.4):
		tmp = y * (0.5 * (x / z))
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.42) || !(x <= 1.4))
		tmp = Float64(y * Float64(0.5 * 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 ((x <= -1.42) || ~((x <= 1.4)))
		tmp = y * (0.5 * (x / z));
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.42], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(y * N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4199999999999999 or 1.3999999999999999 < x

   1. Initial program 73.3%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 75.8%

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

      3. associate-*l/ 75.8%

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

      4. *-commutative 75.8%

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

      5. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in x around 0 42.5%

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

   5. Taylor expanded in x around inf 42.5%

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

   if -1.4199999999999999 < x < 1.3999999999999999

   1. Initial program 96.3%

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

      1. associate-*r/ 96.3%

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

      2. associate-/l/ 94.3%

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

      3. associate-*l/ 93.5%

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

      4. *-commutative 93.5%

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

      5. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 93.2%

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

      1. associate-/l/ 95.2%

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

   6. Simplified 95.2%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \lor \neg \left(x \leq 1.4\right):\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 11: 65.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.42)
   (* (/ 0.5 z) (* x y))
   (if (<= x 1.4) (/ (/ y x) z) (* y (* 0.5 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.42) {
		tmp = (0.5 / z) * (x * y);
	} else if (x <= 1.4) {
		tmp = (y / x) / z;
	} else {
		tmp = y * (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 <= (-1.42d0)) then
        tmp = (0.5d0 / z) * (x * y)
    else if (x <= 1.4d0) then
        tmp = (y / x) / z
    else
        tmp = y * (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 <= -1.42) {
		tmp = (0.5 / z) * (x * y);
	} else if (x <= 1.4) {
		tmp = (y / x) / z;
	} else {
		tmp = y * (0.5 * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.42:
		tmp = (0.5 / z) * (x * y)
	elif x <= 1.4:
		tmp = (y / x) / z
	else:
		tmp = y * (0.5 * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.42)
		tmp = Float64(Float64(0.5 / z) * Float64(x * y));
	elseif (x <= 1.4)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(y * Float64(0.5 * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.42)
		tmp = (0.5 / z) * (x * y);
	elseif (x <= 1.4)
		tmp = (y / x) / z;
	else
		tmp = y * (0.5 * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.42], N[(N[(0.5 / z), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4199999999999999

   1. Initial program 71.2%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 78.8%

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

      3. associate-*l/ 78.8%

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

      4. *-commutative 78.8%

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

      5. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in x around 0 35.6%

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

   5. Taylor expanded in x around inf 35.6%

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

      1. associate-*r* 35.6%

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

      2. clear-num 35.6%

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

      3. un-div-inv 30.1%

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

      4. *-commutative 30.1%

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

   7. Applied egg-rr 30.1%

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

      1. div-inv 30.1%

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

      2. frac-times 37.1%

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

      3. associate-/l* 37.1%

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

      4. /-rgt-identity 37.1%

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

      5. *-commutative 37.1%

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

   9. Applied egg-rr 37.1%

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

   if -1.4199999999999999 < x < 1.3999999999999999

   1. Initial program 96.3%

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

      1. associate-*r/ 96.3%

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

      2. associate-/l/ 94.3%

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

      3. associate-*l/ 93.5%

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

      4. *-commutative 93.5%

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

      5. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 93.2%

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

      1. associate-/l/ 95.2%

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

   6. Simplified 95.2%

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

   if 1.3999999999999999 < x

   1. Initial program 75.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 73.5%

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

      3. associate-*l/ 73.5%

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

      4. *-commutative 73.5%

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

      5. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in x around 0 47.8%

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

   5. Taylor expanded in x around inf 47.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \]

Alternative 12: 65.9% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.42)
   (* (/ 0.5 z) (* x y))
   (if (<= x 1.4) (/ (/ 1.0 z) (/ x y)) (* y (* 0.5 (/ x z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.42) {
		tmp = (0.5 / z) * (x * y);
	} else if (x <= 1.4) {
		tmp = (1.0 / z) / (x / y);
	} else {
		tmp = y * (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 <= (-1.42d0)) then
        tmp = (0.5d0 / z) * (x * y)
    else if (x <= 1.4d0) then
        tmp = (1.0d0 / z) / (x / y)
    else
        tmp = y * (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 <= -1.42) {
		tmp = (0.5 / z) * (x * y);
	} else if (x <= 1.4) {
		tmp = (1.0 / z) / (x / y);
	} else {
		tmp = y * (0.5 * (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.42:
		tmp = (0.5 / z) * (x * y)
	elif x <= 1.4:
		tmp = (1.0 / z) / (x / y)
	else:
		tmp = y * (0.5 * (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.42)
		tmp = Float64(Float64(0.5 / z) * Float64(x * y));
	elseif (x <= 1.4)
		tmp = Float64(Float64(1.0 / z) / Float64(x / y));
	else
		tmp = Float64(y * Float64(0.5 * Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.42)
		tmp = (0.5 / z) * (x * y);
	elseif (x <= 1.4)
		tmp = (1.0 / z) / (x / y);
	else
		tmp = y * (0.5 * (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.42], N[(N[(0.5 / z), $MachinePrecision] * N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4], N[(N[(1.0 / z), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4199999999999999

   1. Initial program 71.2%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 78.8%

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

      3. associate-*l/ 78.8%

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

      4. *-commutative 78.8%

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

      5. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in x around 0 35.6%

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

   5. Taylor expanded in x around inf 35.6%

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

      1. associate-*r* 35.6%

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

      2. clear-num 35.6%

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

      3. un-div-inv 30.1%

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

      4. *-commutative 30.1%

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

   7. Applied egg-rr 30.1%

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

      1. div-inv 30.1%

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

      2. frac-times 37.1%

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

      3. associate-/l* 37.1%

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

      4. /-rgt-identity 37.1%

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

      5. *-commutative 37.1%

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

   9. Applied egg-rr 37.1%

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

   if -1.4199999999999999 < x < 1.3999999999999999

   1. Initial program 96.3%

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

      1. associate-*r/ 96.3%

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

      2. associate-/l/ 94.3%

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

      3. associate-*l/ 93.5%

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

      4. *-commutative 93.5%

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

      5. *-commutative 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around 0 92.3%

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

      1. div-inv 93.2%

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

      2. *-un-lft-identity 93.2%

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

      3. frac-times 95.0%

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

      4. clear-num 94.9%

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

      5. un-div-inv 95.5%

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

   6. Applied egg-rr 95.5%

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

   if 1.3999999999999999 < x

   1. Initial program 75.0%

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

      1. associate-*r/ 100.0%

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

      2. associate-/l/ 73.5%

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

      3. associate-*l/ 73.5%

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

      4. *-commutative 73.5%

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

      5. *-commutative 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in x around 0 47.8%

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

   5. Taylor expanded in x around inf 47.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{\frac{1}{z}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \]

Alternative 13: 65.4% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. associate-*r/ 98.0%

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

   2. associate-/l/ 85.7%

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

   3. associate-*l/ 85.2%

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

   4. *-commutative 85.2%

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

   5. *-commutative 85.2%

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

3. Simplified 85.2%

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

4. Taylor expanded in x around 0 69.3%

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

5. Taylor expanded in z around 0 69.3%

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

6. Final simplification 69.3%

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

Alternative 14: 52.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2e-56) (/ (/ y x) z) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-56) {
		tmp = (y / x) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2d-56) then
        tmp = (y / x) / z
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2e-56) {
		tmp = (y / x) / z;
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2e-56:
		tmp = (y / x) / z
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2e-56)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2e-56)
		tmp = (y / x) / z;
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2e-56], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000001e-56

   1. Initial program 80.8%

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

      1. associate-*r/ 98.8%

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

      2. associate-/l/ 83.3%

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

      3. associate-*l/ 82.6%

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

      4. *-commutative 82.6%

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

      5. *-commutative 82.6%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in x around 0 52.1%

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

      1. associate-/l/ 55.3%

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

   6. Simplified 55.3%

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

   if 2.0000000000000001e-56 < y

   1. Initial program 95.4%

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

      1. associate-*r/ 96.5%

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

      2. associate-/l/ 90.4%

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

      3. associate-*l/ 90.4%

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

      4. *-commutative 90.4%

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

      5. *-commutative 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in x around 0 55.0%

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

      1. div-inv 55.0%

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

      2. associate-/r* 65.2%

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

   6. Applied egg-rr 65.2%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 15: 48.5% 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 85.5%

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

   1. associate-*r/ 98.0%

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

   2. associate-/l/ 85.7%

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

   3. associate-*l/ 85.2%

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

   4. *-commutative 85.2%

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

   5. *-commutative 85.2%

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

3. Simplified 85.2%

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

4. Taylor expanded in x around 0 53.0%

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

5. Final simplification 53.0%

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

Alternative 16: 48.6% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

   1. associate-*r/ 98.0%

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

   2. associate-/l/ 85.7%

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

   3. associate-*l/ 85.2%

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

   4. *-commutative 85.2%

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

   5. *-commutative 85.2%

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

3. Simplified 85.2%

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

4. Taylor expanded in x around 0 53.0%

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

   1. associate-/l/ 54.1%

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

6. Simplified 54.1%

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

7. Final simplification 54.1%

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

Developer target: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023238 
(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))