Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 98.2%

Time: 7.5s

Alternatives: 13

Speedup: 1.0×

• 63.8% 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: 84.7% 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: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (<= t_0 2e+192) (/ 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+192) {
		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+192) 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+192) {
		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+192:
		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+192)
		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+192)
		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+192], 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)) < 2.00000000000000008e192

   1. Initial program 97.4%

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

   if 2.00000000000000008e192 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 66.0%

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

      1. associate-*l/ 66.0%

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

   3. Simplified 66.0%

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

      1. associate-*r/ 99.9%

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

      2. associate-*l/ 100.0%

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

   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 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \end{array} \]

Alternative 2: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.4e+211) (* (/ (cosh x) x) (/ y z)) (* (/ y x) (/ (cosh x) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.4e+211) {
		tmp = (cosh(x) / x) * (y / z);
	} else {
		tmp = (y / x) * (cosh(x) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 3.4d+211) then
        tmp = (cosh(x) / x) * (y / z)
    else
        tmp = (y / x) * (cosh(x) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.4e+211) {
		tmp = (Math.cosh(x) / x) * (y / z);
	} else {
		tmp = (y / x) * (Math.cosh(x) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.4e+211:
		tmp = (math.cosh(x) / x) * (y / z)
	else:
		tmp = (y / x) * (math.cosh(x) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.4e+211)
		tmp = Float64(Float64(cosh(x) / x) * Float64(y / z));
	else
		tmp = Float64(Float64(y / x) * Float64(cosh(x) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.4e+211)
		tmp = (cosh(x) / x) * (y / z);
	else
		tmp = (y / x) * (cosh(x) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.4e+211], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * N[(N[Cosh[x], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.3999999999999999e211

   1. Initial program 84.1%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 85.0%

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

      3. associate-*l/ 79.9%

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

      4. *-commutative 79.9%

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

   3. Simplified 79.9%

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

      1. associate-/l* 87.7%

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

      2. times-frac 94.6%

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

   5. Applied egg-rr 94.6%

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

   if 3.3999999999999999e211 < z

   1. Initial program 95.1%

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

      1. associate-*l/ 95.1%

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

   3. Simplified 95.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.4 \cdot 10^{+211}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \end{array} \]

Alternative 3: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{+212}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 1.1e+212) (* (/ (cosh x) x) (/ y z)) (/ (* (cosh x) (/ y x)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.1e+212) {
		tmp = (cosh(x) / x) * (y / z);
	} else {
		tmp = (cosh(x) * (y / x)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 1.1d+212) then
        tmp = (cosh(x) / x) * (y / z)
    else
        tmp = (cosh(x) * (y / x)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 1.1e+212) {
		tmp = (Math.cosh(x) / x) * (y / z);
	} else {
		tmp = (Math.cosh(x) * (y / x)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 1.1e+212:
		tmp = (math.cosh(x) / x) * (y / z)
	else:
		tmp = (math.cosh(x) * (y / x)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 1.1e+212)
		tmp = Float64(Float64(cosh(x) / x) * Float64(y / z));
	else
		tmp = Float64(Float64(cosh(x) * Float64(y / x)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 1.1e+212)
		tmp = (cosh(x) / x) * (y / z);
	else
		tmp = (cosh(x) * (y / x)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 1.1e+212], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.09999999999999998e212

   1. Initial program 84.1%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 85.0%

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

      3. associate-*l/ 79.9%

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

      4. *-commutative 79.9%

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

   3. Simplified 79.9%

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

      1. associate-/l* 87.7%

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

      2. times-frac 94.6%

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

   5. Applied egg-rr 94.6%

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

   if 1.09999999999999998e212 < z

   1. Initial program 95.1%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.1 \cdot 10^{+212}:\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \end{array} \]

Alternative 4: 90.0% accurate, 1.0× speedup

Math

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

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(cosh(x) / x) * Float64(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[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.9%

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

   1. associate-/l* 78.1%

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

   2. associate-/r/ 83.2%

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

   3. associate-*l/ 78.8%

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

   4. *-commutative 78.8%

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

3. Simplified 78.8%

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

   1. associate-/l* 86.3%

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

   2. times-frac 92.5%

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

5. Applied egg-rr 92.5%

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

6. Final simplification 92.5%

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

Alternative 5: 65.7% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+48} \lor \neg \left(x \leq 10^{+14}\right):\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.5e+48) (not (<= x 1e+14)))
   (* y (* x (/ 0.5 z)))
   (* (/ y z) (+ (* x 0.5) (/ 1.0 x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e+48) || !(x <= 1e+14)) {
		tmp = y * (x * (0.5 / z));
	} else {
		tmp = (y / z) * ((x * 0.5) + (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.5d+48)) .or. (.not. (x <= 1d+14))) then
        tmp = y * (x * (0.5d0 / z))
    else
        tmp = (y / z) * ((x * 0.5d0) + (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.5e+48) || !(x <= 1e+14)) {
		tmp = y * (x * (0.5 / z));
	} else {
		tmp = (y / z) * ((x * 0.5) + (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.5e+48) or not (x <= 1e+14):
		tmp = y * (x * (0.5 / z))
	else:
		tmp = (y / z) * ((x * 0.5) + (1.0 / x))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.5e+48) || !(x <= 1e+14))
		tmp = Float64(y * Float64(x * Float64(0.5 / z)));
	else
		tmp = Float64(Float64(y / z) * Float64(Float64(x * 0.5) + Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.5e+48) || ~((x <= 1e+14)))
		tmp = y * (x * (0.5 / z));
	else
		tmp = (y / z) * ((x * 0.5) + (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.5e+48], N[Not[LessEqual[x, 1e+14]], $MachinePrecision]], N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e48 or 1e14 < x

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0 33.6%

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

   3. Taylor expanded in x around inf 33.6%

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

      1. associate-/l* 24.5%

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

      2. associate-*r/ 24.5%

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

      3. *-commutative 24.5%

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

   5. Simplified 24.5%

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

      1. associate-/r/ 37.2%

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

      2. *-un-lft-identity 37.2%

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

      3. times-frac 37.2%

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

      4. /-rgt-identity 37.2%

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

   7. Applied egg-rr 37.2%

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

   if -1.5e48 < x < 1e14

   1. Initial program 93.6%

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

      1. associate-/l* 91.5%

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

      2. associate-/r/ 92.1%

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

      3. associate-*l/ 91.3%

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

      4. *-commutative 91.3%

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

   3. Simplified 91.3%

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

      1. associate-/l* 93.4%

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

      2. times-frac 93.3%

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

   5. Applied egg-rr 93.3%

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

   6. Taylor expanded in x around 0 86.4%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+48} \lor \neg \left(x \leq 10^{+14}\right):\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \end{array} \]

Alternative 6: 67.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.3e-183)
   (/ (+ (/ y x) (* 0.5 (* x y))) z)
   (+ (* 0.5 (/ (* x y) z)) (/ y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e-183) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.3d-183) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else
        tmp = (0.5d0 * ((x * y) / z)) + (y / (x * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e-183) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else {
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.3e-183:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	else:
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.3e-183)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x * y) / z)) + Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.3e-183)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	else
		tmp = (0.5 * ((x * y) / z)) + (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.3e-183], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.30000000000000016e-183

   1. Initial program 80.3%

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

   2. Taylor expanded in x around 0 64.4%

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

   if 2.30000000000000016e-183 < y

   1. Initial program 91.6%

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

      1. associate-*l/ 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in x around 0 70.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 7: 61.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;x \cdot \frac{0.5}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.3))) (* x (/ 0.5 (/ z y))) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.3)) {
		tmp = x * (0.5 / (z / y));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d0)) .or. (.not. (x <= 1.3d0))) then
        tmp = x * (0.5d0 / (z / y))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.3)) {
		tmp = x * (0.5 / (z / y));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.3):
		tmp = x * (0.5 / (z / y))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.3))
		tmp = Float64(x * Float64(0.5 / Float64(z / y)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.3)))
		tmp = x * (0.5 / (z / y));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4], N[Not[LessEqual[x, 1.3]], $MachinePrecision]], N[(x * N[(0.5 / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.30000000000000004 < x

   1. Initial program 75.2%

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

   2. Taylor expanded in x around 0 31.5%

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

   3. Taylor expanded in x around inf 31.5%

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

      1. associate-/l* 23.4%

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

      2. associate-*r/ 23.4%

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

      3. *-commutative 23.4%

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

   5. Simplified 23.4%

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

      1. *-un-lft-identity 23.4%

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

      2. times-frac 23.4%

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

      3. /-rgt-identity 23.4%

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

   7. Applied egg-rr 23.4%

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

   if -1.3999999999999999 < x < 1.30000000000000004

   1. Initial program 93.0%

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

      1. associate-*l/ 92.9%

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

   3. Simplified 92.9%

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

      1. associate-*r/ 93.3%

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

      2. associate-*l/ 93.4%

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

   5. Applied egg-rr 93.4%

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

   6. Taylor expanded in x around 0 92.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;x \cdot \frac{0.5}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 8: 65.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.4) (not (<= x 1.3))) (* y (* x (/ 0.5 z))) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.3)) {
		tmp = y * (x * (0.5 / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.4d0)) .or. (.not. (x <= 1.3d0))) then
        tmp = y * (x * (0.5d0 / z))
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.4) || !(x <= 1.3)) {
		tmp = y * (x * (0.5 / z));
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.4) or not (x <= 1.3):
		tmp = y * (x * (0.5 / z))
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.4) || !(x <= 1.3))
		tmp = Float64(y * Float64(x * Float64(0.5 / z)));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.4) || ~((x <= 1.3)))
		tmp = y * (x * (0.5 / z));
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.4], N[Not[LessEqual[x, 1.3]], $MachinePrecision]], N[(y * N[(x * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999 or 1.30000000000000004 < x

   1. Initial program 75.2%

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

   2. Taylor expanded in x around 0 31.5%

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

   3. Taylor expanded in x around inf 31.5%

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

      1. associate-/l* 23.4%

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

      2. associate-*r/ 23.4%

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

      3. *-commutative 23.4%

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

   5. Simplified 23.4%

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

      1. associate-/r/ 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. times-frac 34.6%

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

      4. /-rgt-identity 34.6%

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

   7. Applied egg-rr 34.6%

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

   if -1.3999999999999999 < x < 1.30000000000000004

   1. Initial program 93.0%

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

      1. associate-*l/ 92.9%

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

   3. Simplified 92.9%

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

      1. associate-*r/ 93.3%

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

      2. associate-*l/ 93.4%

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

   5. Applied egg-rr 93.4%

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

   6. Taylor expanded in x around 0 92.6%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \lor \neg \left(x \leq 1.3\right):\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 9: 66.1% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in x around 0 64.8%

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

3. Taylor expanded in y around 0 64.7%

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

4. Final simplification 64.7%

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

Alternative 10: 66.2% accurate, 9.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

2. Taylor expanded in x around 0 64.8%

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

3. Final simplification 64.8%

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

Alternative 11: 50.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 2.3e-183) (/ (/ y x) z) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e-183) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 2.3d-183) then
        tmp = (y / x) / z
    else
        tmp = y / (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= 2.3e-183) {
		tmp = (y / x) / z;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 2.3e-183:
		tmp = (y / x) / z
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 2.3e-183)
		tmp = Float64(Float64(y / x) / z);
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= 2.3e-183)
		tmp = (y / x) / z;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 2.3e-183], N[(N[(y / x), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.30000000000000016e-183

   1. Initial program 80.3%

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

   2. Taylor expanded in x around 0 51.2%

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

   if 2.30000000000000016e-183 < y

   1. Initial program 91.6%

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

      1. associate-*l/ 91.6%

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

   3. Simplified 91.6%

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

   4. Taylor expanded in x around 0 61.5%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.3 \cdot 10^{-183}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 12: 55.2% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 2e+34) (/ (/ y z) x) (/ y (* x z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 2e+34) {
		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 <= 2d+34) 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 <= 2e+34) {
		tmp = (y / z) / x;
	} else {
		tmp = y / (x * z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 2e+34:
		tmp = (y / z) / x
	else:
		tmp = y / (x * z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 2e+34)
		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 <= 2e+34)
		tmp = (y / z) / x;
	else
		tmp = y / (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 2e+34], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999989e34

   1. Initial program 84.3%

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

      1. associate-*l/ 84.2%

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

   3. Simplified 84.2%

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

      1. associate-*r/ 98.8%

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

      2. associate-*l/ 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in x around 0 57.9%

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

   if 1.99999999999999989e34 < z

   1. Initial program 87.2%

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

      1. associate-*l/ 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around 0 60.0%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 13: 49.2% 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 84.9%

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

   1. associate-*l/ 84.8%

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

3. Simplified 84.8%

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

4. Taylor expanded in x around 0 53.1%

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

5. Final simplification 53.1%

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

Developer target: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023310 
(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))