Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.2% → 97.8%

Time: 13.0s

Alternatives: 14

Speedup: 1.0×

• 64.6% 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: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 2e+133) (/ t_0 z) (/ (/ (* (cosh x) y) z) x))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 2e+133) {
		tmp = t_0 / 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) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) * (y / x)
    if (t_0 <= 2d+133) then
        tmp = t_0 / 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 t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if (t_0 <= 2e+133) {
		tmp = t_0 / z;
	} else {
		tmp = ((Math.cosh(x) * y) / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if t_0 <= 2e+133:
		tmp = t_0 / z
	else:
		tmp = ((math.cosh(x) * y) / z) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= 2e+133)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(Float64(cosh(x) * y) / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if (t_0 <= 2e+133)
		tmp = t_0 / z;
	else
		tmp = ((cosh(x) * y) / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e+133], N[(t$95$0 / z), $MachinePrecision], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2e133

   1. Initial program 99.8%

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

   if 2e133 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 64.4%

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

      1. associate-*r/ 56.5%

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

      2. associate-/r* 59.6%

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

   3. Simplified 59.6%

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

      1. associate-*r/ 72.8%

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

      2. *-commutative 72.8%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+133}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array} \]

Alternative 2: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{1 + 0.5 \cdot \left(x \cdot x\right)}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 INFINITY) (/ t_0 z) (/ (/ y (/ z (+ 1.0 (* 0.5 (* x x))))) x))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / z;
	} else {
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 / z
	else:
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(y / Float64(z / Float64(1.0 + Float64(0.5 * Float64(x * x))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 / z;
	else
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], N[(t$95$0 / z), $MachinePrecision], N[(N[(y / N[(z / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.4%

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

   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/ 0.0%

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

      2. associate-/r* 17.1%

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

   3. Simplified 17.1%

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

      1. associate-*r/ 51.4%

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

      2. *-commutative 51.4%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 74.8%

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

      1. *-commutative 74.8%

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

      2. unpow2 74.8%

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

   8. Simplified 74.8%

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

   9. Taylor expanded in y around 0 74.8%

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

       1. associate-/l* 80.3%

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

       2. unpow2 80.3%

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

   11. Simplified 80.3%

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

4. Final simplification 95.0%

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

Alternative 3: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{y}{\frac{2}{\frac{x \cdot x}{z}}}}{x}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+105}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{1 + 0.5 \cdot \left(x \cdot x\right)}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -4e+128)
   (/ (/ y (/ 2.0 (/ (* x x) z))) x)
   (if (<= x 8.6e+105)
     (* (cosh x) (/ y (* x z)))
     (/ (/ y (/ z (+ 1.0 (* 0.5 (* x x))))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e+128) {
		tmp = (y / (2.0 / ((x * x) / z))) / x;
	} else if (x <= 8.6e+105) {
		tmp = cosh(x) * (y / (x * z));
	} else {
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / 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 <= (-4d+128)) then
        tmp = (y / (2.0d0 / ((x * x) / z))) / x
    else if (x <= 8.6d+105) then
        tmp = cosh(x) * (y / (x * z))
    else
        tmp = (y / (z / (1.0d0 + (0.5d0 * (x * x))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -4e+128) {
		tmp = (y / (2.0 / ((x * x) / z))) / x;
	} else if (x <= 8.6e+105) {
		tmp = Math.cosh(x) * (y / (x * z));
	} else {
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -4e+128:
		tmp = (y / (2.0 / ((x * x) / z))) / x
	elif x <= 8.6e+105:
		tmp = math.cosh(x) * (y / (x * z))
	else:
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -4e+128)
		tmp = Float64(Float64(y / Float64(2.0 / Float64(Float64(x * x) / z))) / x);
	elseif (x <= 8.6e+105)
		tmp = Float64(cosh(x) * Float64(y / Float64(x * z)));
	else
		tmp = Float64(Float64(y / Float64(z / Float64(1.0 + Float64(0.5 * Float64(x * x))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -4e+128)
		tmp = (y / (2.0 / ((x * x) / z))) / x;
	elseif (x <= 8.6e+105)
		tmp = cosh(x) * (y / (x * z));
	else
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -4e+128], N[(N[(y / N[(2.0 / N[(N[(x * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 8.6e+105], N[(N[Cosh[x], $MachinePrecision] * N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(z / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0000000000000003e128

   1. Initial program 63.6%

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

      1. associate-*r/ 54.5%

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

      2. associate-/r* 36.4%

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

   3. Simplified 36.4%

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

      1. associate-*r/ 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 91.1%

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

      1. *-commutative 91.1%

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

      2. unpow2 91.1%

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

   8. Simplified 91.1%

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

   9. Taylor expanded in y around 0 91.1%

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

       1. associate-/l* 91.1%

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

       2. unpow2 91.1%

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

   11. Simplified 91.1%

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

   12. Taylor expanded in x around inf 91.1%

       \[\leadsto \frac{\frac{y}{\color{blue}{2 \cdot \frac{z}{{x}^{2}}}}}{x} \]
   13. Step-by-step derivation

       1. unpow2 91.1%

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

       2. associate-*r/ 91.1%

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

       3. associate-/l* 93.3%

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

   14. Simplified 93.3%

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

   if -4.0000000000000003e128 < x < 8.6000000000000003e105

   1. Initial program 95.4%

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

      1. associate-*r/ 92.5%

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

      2. associate-/r* 90.5%

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

   3. Simplified 90.5%

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

   if 8.6000000000000003e105 < x

   1. Initial program 58.5%

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

      1. associate-*r/ 51.2%

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

      2. associate-/r* 51.2%

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

   3. Simplified 51.2%

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

      1. associate-*r/ 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 90.4%

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

      1. *-commutative 90.4%

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

      2. unpow2 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in y around 0 90.4%

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

       1. associate-/l* 92.9%

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

       2. unpow2 92.9%

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

   11. Simplified 92.9%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{y}{\frac{2}{\frac{x \cdot x}{z}}}}{x}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{+105}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{1 + 0.5 \cdot \left(x \cdot x\right)}}}{x}\\ \end{array} \]

Alternative 4: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{y}{\frac{2}{\frac{x \cdot x}{z}}}}{x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{1 + 0.5 \cdot \left(x \cdot x\right)}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3.2e+124)
   (/ (/ y (/ 2.0 (/ (* x x) z))) x)
   (if (<= x 2.5e+105)
     (/ (cosh x) (/ z (/ y x)))
     (/ (/ y (/ z (+ 1.0 (* 0.5 (* x x))))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e+124) {
		tmp = (y / (2.0 / ((x * x) / z))) / x;
	} else if (x <= 2.5e+105) {
		tmp = cosh(x) / (z / (y / x));
	} else {
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / 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 <= (-3.2d+124)) then
        tmp = (y / (2.0d0 / ((x * x) / z))) / x
    else if (x <= 2.5d+105) then
        tmp = cosh(x) / (z / (y / x))
    else
        tmp = (y / (z / (1.0d0 + (0.5d0 * (x * x))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3.2e+124) {
		tmp = (y / (2.0 / ((x * x) / z))) / x;
	} else if (x <= 2.5e+105) {
		tmp = Math.cosh(x) / (z / (y / x));
	} else {
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3.2e+124:
		tmp = (y / (2.0 / ((x * x) / z))) / x
	elif x <= 2.5e+105:
		tmp = math.cosh(x) / (z / (y / x))
	else:
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3.2e+124)
		tmp = Float64(Float64(y / Float64(2.0 / Float64(Float64(x * x) / z))) / x);
	elseif (x <= 2.5e+105)
		tmp = Float64(cosh(x) / Float64(z / Float64(y / x)));
	else
		tmp = Float64(Float64(y / Float64(z / Float64(1.0 + Float64(0.5 * Float64(x * x))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3.2e+124)
		tmp = (y / (2.0 / ((x * x) / z))) / x;
	elseif (x <= 2.5e+105)
		tmp = cosh(x) / (z / (y / x));
	else
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3.2e+124], N[(N[(y / N[(2.0 / N[(N[(x * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 2.5e+105], N[(N[Cosh[x], $MachinePrecision] / N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(z / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.19999999999999993e124

   1. Initial program 63.6%

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

      1. associate-*r/ 54.5%

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

      2. associate-/r* 36.4%

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

   3. Simplified 36.4%

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

      1. associate-*r/ 50.0%

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

      2. *-commutative 50.0%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 91.1%

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

      1. *-commutative 91.1%

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

      2. unpow2 91.1%

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

   8. Simplified 91.1%

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

   9. Taylor expanded in y around 0 91.1%

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

       1. associate-/l* 91.1%

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

       2. unpow2 91.1%

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

   11. Simplified 91.1%

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

   12. Taylor expanded in x around inf 91.1%

       \[\leadsto \frac{\frac{y}{\color{blue}{2 \cdot \frac{z}{{x}^{2}}}}}{x} \]
   13. Step-by-step derivation

       1. unpow2 91.1%

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

       2. associate-*r/ 91.1%

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

       3. associate-/l* 93.3%

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

   14. Simplified 93.3%

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

   if -3.19999999999999993e124 < x < 2.50000000000000023e105

   1. Initial program 95.4%

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

      1. associate-/l* 92.3%

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

   3. Simplified 92.3%

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

   if 2.50000000000000023e105 < x

   1. Initial program 58.5%

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

      1. associate-*r/ 51.2%

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

      2. associate-/r* 51.2%

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

   3. Simplified 51.2%

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

      1. associate-*r/ 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 90.4%

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

      1. *-commutative 90.4%

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

      2. unpow2 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in y around 0 90.4%

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

       1. associate-/l* 92.9%

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

       2. unpow2 92.9%

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

   11. Simplified 92.9%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{y}{\frac{2}{\frac{x \cdot x}{z}}}}{x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\cosh x}{\frac{z}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{1 + 0.5 \cdot \left(x \cdot x\right)}}}{x}\\ \end{array} \]

Alternative 5: 80.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -400000000:\\ \;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{z}}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{1 + 0.5 \cdot \left(x \cdot x\right)}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -400000000.0)
   (/ (* 0.5 (/ (* y (* x x)) z)) x)
   (if (<= x 7.5e-280)
     (/ (+ (/ y x) (* 0.5 (* x y))) z)
     (/ (/ y (/ z (+ 1.0 (* 0.5 (* x x))))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -400000000.0) {
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	} else if (x <= 7.5e-280) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / 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 <= (-400000000.0d0)) then
        tmp = (0.5d0 * ((y * (x * x)) / z)) / x
    else if (x <= 7.5d-280) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else
        tmp = (y / (z / (1.0d0 + (0.5d0 * (x * x))))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -400000000.0) {
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	} else if (x <= 7.5e-280) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -400000000.0:
		tmp = (0.5 * ((y * (x * x)) / z)) / x
	elif x <= 7.5e-280:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	else:
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -400000000.0)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * Float64(x * x)) / z)) / x);
	elseif (x <= 7.5e-280)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	else
		tmp = Float64(Float64(y / Float64(z / Float64(1.0 + Float64(0.5 * Float64(x * x))))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -400000000.0)
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	elseif (x <= 7.5e-280)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	else
		tmp = (y / (z / (1.0 + (0.5 * (x * x))))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -400000000.0], N[(N[(0.5 * N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 7.5e-280], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(z / N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4e8

   1. Initial program 73.0%

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

      1. associate-*r/ 63.5%

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

      2. associate-/r* 49.2%

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

   3. Simplified 49.2%

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

      1. associate-*r/ 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. unpow2 76.9%

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

   8. Simplified 76.9%

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

   9. Taylor expanded in y around 0 76.9%

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

       1. associate-/l* 72.4%

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

       2. unpow2 72.4%

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

   11. Simplified 72.4%

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

   12. Taylor expanded in x around inf 76.9%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{{x}^{2} \cdot y}{z}}}{x} \]
   13. Step-by-step derivation

       1. unpow2 76.9%

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

       2. *-commutative 76.9%

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

   14. Simplified 76.9%

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

   if -4e8 < x < 7.4999999999999999e-280

   1. Initial program 97.1%

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

   2. Taylor expanded in x around 0 94.6%

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

   if 7.4999999999999999e-280 < x

   1. Initial program 82.1%

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

      1. associate-*r/ 77.2%

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

      2. associate-/r* 77.7%

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

   3. Simplified 77.7%

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

      1. associate-*r/ 82.6%

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

      2. *-commutative 82.6%

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

      3. associate-/r* 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in x around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. unpow2 83.2%

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

   8. Simplified 83.2%

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

   9. Taylor expanded in y around 0 83.2%

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

       1. associate-/l* 83.2%

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

       2. unpow2 83.2%

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

   11. Simplified 83.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -400000000:\\ \;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{z}}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-280}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{z}{1 + 0.5 \cdot \left(x \cdot x\right)}}}{x}\\ \end{array} \]

Alternative 6: 74.9% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{x \cdot x}{\frac{z}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.45)))
   (/ (* 0.5 (/ (* x x) (/ z y))) x)
   (/ (/ y x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.45)) {
		tmp = (0.5 * ((x * x) / (z / y))) / 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 <= (-1.4d0)) .or. (.not. (x <= 1.45d0))) then
        tmp = (0.5d0 * ((x * x) / (z / y))) / 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 <= -1.4) || !(x <= 1.45)) {
		tmp = (0.5 * ((x * x) / (z / y))) / x;
	} else {
		tmp = (y / x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.45):
		tmp = (0.5 * ((x * x) / (z / y))) / x
	else:
		tmp = (y / x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.45))
		tmp = Float64(Float64(0.5 * Float64(Float64(x * x) / Float64(z / y))) / 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 <= -1.4) || ~((x <= 1.45)))
		tmp = (0.5 * ((x * x) / (z / y))) / x;
	else
		tmp = (y / x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4], N[Not[LessEqual[x, 1.45]], $MachinePrecision]], N[(N[(0.5 * N[(N[(x * x), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.44999999999999996 < x

   1. Initial program 72.4%

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

      1. associate-*r/ 63.0%

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

      2. associate-/r* 55.9%

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

   3. Simplified 55.9%

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

      1. associate-*r/ 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 73.4%

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

      1. *-commutative 73.4%

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

      2. unpow2 73.4%

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

   8. Simplified 73.4%

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

   9. Taylor expanded in x around inf 73.4%

      \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{{x}^{2} \cdot y}{z}}}{x} \]
   10. Step-by-step derivation

       1. associate-/l* 61.0%

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

       2. unpow2 61.0%

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

   11. Simplified 61.0%

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

   if -1.3999999999999999 < x < 1.44999999999999996

   1. Initial program 95.5%

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

   2. Taylor expanded in x around 0 94.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.45\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{x \cdot x}{\frac{z}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \end{array} \]

Alternative 7: 80.2% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.3999999999999999 < x

   1. Initial program 72.4%

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

      1. associate-*r/ 63.0%

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

      2. associate-/r* 55.9%

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

   3. Simplified 55.9%

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

      1. associate-*r/ 67.7%

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

      2. *-commutative 67.7%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 73.4%

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

      1. *-commutative 73.4%

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

      2. unpow2 73.4%

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

   8. Simplified 73.4%

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

   9. Taylor expanded in y around 0 73.4%

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

       1. associate-/l* 71.2%

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

       2. unpow2 71.2%

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

   11. Simplified 71.2%

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

   12. Taylor expanded in x around inf 73.4%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{{x}^{2} \cdot y}{z}}}{x} \]
   13. Step-by-step derivation

       1. unpow2 73.4%

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

       2. *-commutative 73.4%

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

   14. Simplified 73.4%

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

   if -1.3999999999999999 < x < 1.3999999999999999

   1. Initial program 95.5%

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

   2. Taylor expanded in x around 0 94.7%

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

4. Final simplification 84.1%

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

Alternative 8: 80.3% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2000000000 \lor \neg \left(x \leq 2 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2000000000.0) (not (<= x 2e+18)))
   (/ (* 0.5 (/ (* y (* x x)) z)) x)
   (/ (+ (/ y x) (* 0.5 (* x y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2000000000.0) || !(x <= 2e+18)) {
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-2000000000.0d0)) .or. (.not. (x <= 2d+18))) then
        tmp = (0.5d0 * ((y * (x * x)) / z)) / x
    else
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2000000000.0) || !(x <= 2e+18)) {
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2000000000.0) or not (x <= 2e+18):
		tmp = (0.5 * ((y * (x * x)) / z)) / x
	else:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2000000000.0) || !(x <= 2e+18))
		tmp = Float64(Float64(0.5 * Float64(Float64(y * Float64(x * x)) / z)) / x);
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2000000000.0) || ~((x <= 2e+18)))
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	else
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2000000000.0], N[Not[LessEqual[x, 2e+18]], $MachinePrecision]], N[(N[(0.5 * N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2e9 or 2e18 < x

   1. Initial program 71.1%

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

      1. associate-*r/ 61.2%

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

      2. associate-/r* 53.7%

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

   3. Simplified 53.7%

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

      1. associate-*r/ 66.1%

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

      2. *-commutative 66.1%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. unpow2 75.9%

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

   8. Simplified 75.9%

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

   9. Taylor expanded in y around 0 75.9%

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

       1. associate-/l* 73.6%

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

       2. unpow2 73.6%

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

   11. Simplified 73.6%

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

   12. Taylor expanded in x around inf 75.9%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{{x}^{2} \cdot y}{z}}}{x} \]
   13. Step-by-step derivation

       1. unpow2 75.9%

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

       2. *-commutative 75.9%

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

   14. Simplified 75.9%

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

   if -2e9 < x < 2e18

   1. Initial program 95.7%

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

   2. Taylor expanded in x around 0 92.0%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000000000 \lor \neg \left(x \leq 2 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 9: 80.0% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{z}}{x}\\ \mathbf{elif}\;x \leq 10^{+35}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{2}{\frac{x \cdot x}{z}}}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2000000000.0)
   (/ (* 0.5 (/ (* y (* x x)) z)) x)
   (if (<= x 1e+35)
     (/ (+ (/ y x) (* 0.5 (* x y))) z)
     (/ (/ y (/ 2.0 (/ (* x x) z))) x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2000000000.0) {
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	} else if (x <= 1e+35) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = (y / (2.0 / ((x * x) / 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 <= (-2000000000.0d0)) then
        tmp = (0.5d0 * ((y * (x * x)) / z)) / x
    else if (x <= 1d+35) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else
        tmp = (y / (2.0d0 / ((x * x) / z))) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2000000000.0) {
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	} else if (x <= 1e+35) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = (y / (2.0 / ((x * x) / z))) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2000000000.0:
		tmp = (0.5 * ((y * (x * x)) / z)) / x
	elif x <= 1e+35:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	else:
		tmp = (y / (2.0 / ((x * x) / z))) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2000000000.0)
		tmp = Float64(Float64(0.5 * Float64(Float64(y * Float64(x * x)) / z)) / x);
	elseif (x <= 1e+35)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	else
		tmp = Float64(Float64(y / Float64(2.0 / Float64(Float64(x * x) / z))) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2000000000.0)
		tmp = (0.5 * ((y * (x * x)) / z)) / x;
	elseif (x <= 1e+35)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	else
		tmp = (y / (2.0 / ((x * x) / z))) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2000000000.0], N[(N[(0.5 * N[(N[(y * N[(x * x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1e+35], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / N[(2.0 / N[(N[(x * x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e9

   1. Initial program 73.0%

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

      1. associate-*r/ 63.5%

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

      2. associate-/r* 49.2%

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

   3. Simplified 49.2%

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

      1. associate-*r/ 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 76.9%

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

      1. *-commutative 76.9%

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

      2. unpow2 76.9%

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

   8. Simplified 76.9%

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

   9. Taylor expanded in y around 0 76.9%

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

       1. associate-/l* 72.4%

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

       2. unpow2 72.4%

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

   11. Simplified 72.4%

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

   12. Taylor expanded in x around inf 76.9%

       \[\leadsto \frac{\color{blue}{0.5 \cdot \frac{{x}^{2} \cdot y}{z}}}{x} \]
   13. Step-by-step derivation

       1. unpow2 76.9%

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

       2. *-commutative 76.9%

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

   14. Simplified 76.9%

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

   if -2e9 < x < 9.9999999999999997e34

   1. Initial program 95.8%

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

   2. Taylor expanded in x around 0 90.8%

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

   if 9.9999999999999997e34 < x

   1. Initial program 66.7%

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

      1. associate-*r/ 55.6%

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

      2. associate-/r* 55.6%

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

   3. Simplified 55.6%

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

      1. associate-*r/ 66.7%

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

      2. *-commutative 66.7%

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

      3. associate-/r* 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 76.5%

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

      1. *-commutative 76.5%

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

      2. unpow2 76.5%

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

   8. Simplified 76.5%

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

   9. Taylor expanded in y around 0 76.5%

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

       1. associate-/l* 78.4%

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

       2. unpow2 78.4%

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

   11. Simplified 78.4%

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

   12. Taylor expanded in x around inf 78.4%

       \[\leadsto \frac{\frac{y}{\color{blue}{2 \cdot \frac{z}{{x}^{2}}}}}{x} \]
   13. Step-by-step derivation

       1. unpow2 78.4%

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

       2. associate-*r/ 78.4%

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

       3. associate-/l* 78.4%

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

   14. Simplified 78.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2000000000:\\ \;\;\;\;\frac{0.5 \cdot \frac{y \cdot \left(x \cdot x\right)}{z}}{x}\\ \mathbf{elif}\;x \leq 10^{+35}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\frac{2}{\frac{x \cdot x}{z}}}}{x}\\ \end{array} \]

Alternative 10: 80.5% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. associate-*r/ 79.4%

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

   2. associate-/r* 74.9%

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

3. Simplified 74.9%

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

   1. associate-*r/ 80.8%

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

   2. *-commutative 80.8%

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

   3. associate-/r* 95.9%

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

5. Applied egg-rr 95.9%

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

6. Taylor expanded in x around 0 82.6%

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

   1. *-commutative 82.6%

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

   2. unpow2 82.6%

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

8. Simplified 82.6%

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

9. Final simplification 82.6%

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

Alternative 11: 65.4% accurate, 9.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.3999999999999999 < x

   1. Initial program 72.4%

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

   2. Taylor expanded in x around 0 43.9%

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

   3. Taylor expanded in x around inf 43.9%

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

   if -1.3999999999999999 < x < 1.3999999999999999

   1. Initial program 95.5%

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

   2. Taylor expanded in x around 0 94.7%

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

4. Final simplification 69.5%

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

Alternative 12: 50.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 9.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 9.8e-64) (/ (/ y x) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 9.8e-64) {
		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 <= 9.8d-64) 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 <= 9.8e-64) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 9.8e-64:
		tmp = (y / x) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 9.8e-64)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 9.8e-64)
		tmp = (y / x) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 9.8e-64], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 9.8000000000000003e-64

   1. Initial program 87.5%

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

   2. Taylor expanded in x around 0 59.9%

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

   if 9.8000000000000003e-64 < z

   1. Initial program 77.1%

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

      1. associate-*r/ 66.7%

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

      2. associate-/r* 60.4%

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

   3. Simplified 60.4%

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

   4. Taylor expanded in x around 0 39.5%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 13: 54.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 7e-61) (/ (/ y z) x) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 7e-61) {
		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-61) 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-61) {
		tmp = (y / z) / x;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 7e-61:
		tmp = (y / z) / x
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 7e-61)
		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-61)
		tmp = (y / z) / x;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 7e-61], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 7.0000000000000006e-61

   1. Initial program 87.6%

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

      1. associate-*r/ 85.9%

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

      2. associate-/r* 82.4%

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

   3. Simplified 82.4%

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

      1. associate-*r/ 85.9%

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

      2. *-commutative 85.9%

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

      3. associate-/r* 97.8%

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

   5. Applied egg-rr 97.8%

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

   6. Taylor expanded in x around 0 62.0%

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

   if 7.0000000000000006e-61 < z

   1. Initial program 76.9%

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

      1. associate-*r/ 66.3%

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

      2. associate-/r* 59.9%

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

   3. Simplified 59.9%

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

   4. Taylor expanded in x around 0 39.9%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 14: 49.1% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.1%

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

   1. associate-*r/ 79.4%

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

   2. associate-/r* 74.9%

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

3. Simplified 74.9%

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

4. Taylor expanded in x around 0 50.4%

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

5. Final simplification 50.4%

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

Developer target: 97.1% 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 2023275 
(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))