Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 83.9% → 99.2%

Time: 9.1s

Alternatives: 11

Speedup: 0.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.9% 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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+266} \lor \neg \left(t_0 \leq 10^{+117}\right):\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (or (<= t_0 -4e+266) (not (<= t_0 1e+117)))
     (/ (/ (* (cosh x) y) z) x)
     (/ (+ (/ y x) (* 0.5 (* x y))) z))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if ((t_0 <= -4e+266) || !(t_0 <= 1e+117)) {
		tmp = ((cosh(x) * y) / 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) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) * (y / x)
    if ((t_0 <= (-4d+266)) .or. (.not. (t_0 <= 1d+117))) then
        tmp = ((cosh(x) * y) / 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 t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if ((t_0 <= -4e+266) || !(t_0 <= 1e+117)) {
		tmp = ((Math.cosh(x) * y) / z) / x;
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if (t_0 <= -4e+266) or not (t_0 <= 1e+117):
		tmp = ((math.cosh(x) * y) / z) / x
	else:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if ((t_0 <= -4e+266) || !(t_0 <= 1e+117))
		tmp = Float64(Float64(Float64(cosh(x) * y) / 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)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if ((t_0 <= -4e+266) || ~((t_0 <= 1e+117)))
		tmp = ((cosh(x) * y) / z) / x;
	else
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e+266], N[Not[LessEqual[t$95$0, 1e+117]], $MachinePrecision]], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / z), $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 (*.f64 (cosh.f64 x) (/.f64 y x)) < -4.0000000000000001e266 or 1.00000000000000005e117 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 78.3%

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

      1. associate-*r/ 67.7%

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

      2. associate-/r* 76.8%

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

   3. Simplified 76.8%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

      3. associate-/r* 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -4.0000000000000001e266 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.00000000000000005e117

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -4 \cdot 10^{+266} \lor \neg \left(\cosh x \cdot \frac{y}{x} \leq 10^{+117}\right):\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 2: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cosh x \cdot \frac{y}{x}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+266} \lor \neg \left(t_0 \leq 10^{+295}\right):\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (cosh x) (/ y x))))
   (if (or (<= t_0 -4e+266) (not (<= t_0 1e+295)))
     (* (/ (cosh x) x) (/ y z))
     (/ (+ (/ y x) (* 0.5 (* x y))) z))))

C

double code(double x, double y, double z) {
	double t_0 = cosh(x) * (y / x);
	double tmp;
	if ((t_0 <= -4e+266) || !(t_0 <= 1e+295)) {
		tmp = (cosh(x) / x) * (y / z);
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x) * (y / x)
    if ((t_0 <= (-4d+266)) .or. (.not. (t_0 <= 1d+295))) then
        tmp = (cosh(x) / x) * (y / z)
    else
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.cosh(x) * (y / x);
	double tmp;
	if ((t_0 <= -4e+266) || !(t_0 <= 1e+295)) {
		tmp = (Math.cosh(x) / x) * (y / z);
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.cosh(x) * (y / x)
	tmp = 0
	if (t_0 <= -4e+266) or not (t_0 <= 1e+295):
		tmp = (math.cosh(x) / x) * (y / z)
	else:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(cosh(x) * Float64(y / x))
	tmp = 0.0
	if ((t_0 <= -4e+266) || !(t_0 <= 1e+295))
		tmp = Float64(Float64(cosh(x) / x) * Float64(y / z));
	else
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = cosh(x) * (y / x);
	tmp = 0.0;
	if ((t_0 <= -4e+266) || ~((t_0 <= 1e+295)))
		tmp = (cosh(x) / x) * (y / z);
	else
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e+266], N[Not[LessEqual[t$95$0, 1e+295]], $MachinePrecision]], N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < -4.0000000000000001e266 or 9.9999999999999998e294 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 76.3%

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

      1. associate-/l* 64.8%

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

   3. Simplified 64.8%

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

      1. associate-/r/ 76.3%

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

      2. frac-times 80.4%

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

      3. *-commutative 80.4%

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

      4. times-frac 92.5%

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

   5. Applied egg-rr 92.5%

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

   if -4.0000000000000001e266 < (*.f64 (cosh.f64 x) (/.f64 y x)) < 9.9999999999999998e294

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq -4 \cdot 10^{+266} \lor \neg \left(\cosh x \cdot \frac{y}{x} \leq 10^{+295}\right):\\ \;\;\;\;\frac{\cosh x}{x} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 3: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+242} \lor \neg \left(x \leq 9 \cdot 10^{+223}\right):\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.22e+242) (not (<= x 9e+223)))
   (* 0.5 (/ (* x y) z))
   (* (cosh x) (/ y (* x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.22e+242) || !(x <= 9e+223)) {
		tmp = 0.5 * ((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 ((x <= (-1.22d+242)) .or. (.not. (x <= 9d+223))) then
        tmp = 0.5d0 * ((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 ((x <= -1.22e+242) || !(x <= 9e+223)) {
		tmp = 0.5 * ((x * y) / z);
	} else {
		tmp = Math.cosh(x) * (y / (x * z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.22e+242) or not (x <= 9e+223):
		tmp = 0.5 * ((x * y) / z)
	else:
		tmp = math.cosh(x) * (y / (x * z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.22e+242) || !(x <= 9e+223))
		tmp = Float64(0.5 * Float64(Float64(x * y) / z));
	else
		tmp = Float64(cosh(x) * Float64(y / Float64(x * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.22e+242) || ~((x <= 9e+223)))
		tmp = 0.5 * ((x * y) / z);
	else
		tmp = cosh(x) * (y / (x * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.22e+242], N[Not[LessEqual[x, 9e+223]], $MachinePrecision]], N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2199999999999999e242 or 9e223 < x

   1. Initial program 70.7%

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

   2. Taylor expanded in x around 0 71.9%

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

   3. Taylor expanded in x around inf 71.9%

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

   if -1.2199999999999999e242 < x < 9e223

   1. Initial program 86.3%

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

      1. associate-*r/ 81.7%

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

      2. associate-/r* 87.8%

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

   3. Simplified 87.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+242} \lor \neg \left(x \leq 9 \cdot 10^{+223}\right):\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 4: 65.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 200000000000:\\ \;\;\;\;\frac{-y}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y 200000000000.0)
   (/ (- y) (/ z (+ (* x -0.5) (/ -1.0 x))))
   (/ (+ (/ y x) (* 0.5 (* x y))) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= 200000000000.0) {
		tmp = -y / (z / ((x * -0.5) + (-1.0 / 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 (y <= 200000000000.0d0) then
        tmp = -y / (z / ((x * (-0.5d0)) + ((-1.0d0) / 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 (y <= 200000000000.0) {
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)));
	} else {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= 200000000000.0:
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)))
	else:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= 200000000000.0)
		tmp = Float64(Float64(-y) / Float64(z / Float64(Float64(x * -0.5) + Float64(-1.0 / 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 (y <= 200000000000.0)
		tmp = -y / (z / ((x * -0.5) + (-1.0 / x)));
	else
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, 200000000000.0], N[((-y) / N[(z / N[(N[(x * -0.5), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2e11

   1. Initial program 79.2%

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

   2. Taylor expanded in x around 0 59.1%

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

   3. Taylor expanded in y around -inf 59.0%

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

      1. mul-1-neg 59.0%

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

      2. associate-/l* 65.9%

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

      3. *-commutative 65.9%

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

   5. Simplified 65.9%

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

   if 2e11 < y

   1. Initial program 97.1%

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

   2. Taylor expanded in x around 0 88.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 200000000000:\\ \;\;\;\;\frac{-y}{\frac{z}{x \cdot -0.5 + \frac{-1}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]

Alternative 5: 65.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z}{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.2e-72)
   (/ (+ (/ y x) (* 0.5 (* x y))) z)
   (if (<= x 1.4) (* y (/ (/ 1.0 z) x)) (* y (/ x (/ z 0.5))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-72) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else if (x <= 1.4) {
		tmp = y * ((1.0 / z) / x);
	} else {
		tmp = y * (x / (z / 0.5));
	}
	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 <= (-6.2d-72)) then
        tmp = ((y / x) + (0.5d0 * (x * y))) / z
    else if (x <= 1.4d0) then
        tmp = y * ((1.0d0 / z) / x)
    else
        tmp = y * (x / (z / 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e-72) {
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	} else if (x <= 1.4) {
		tmp = y * ((1.0 / z) / x);
	} else {
		tmp = y * (x / (z / 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.2e-72:
		tmp = ((y / x) + (0.5 * (x * y))) / z
	elif x <= 1.4:
		tmp = y * ((1.0 / z) / x)
	else:
		tmp = y * (x / (z / 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.2e-72)
		tmp = Float64(Float64(Float64(y / x) + Float64(0.5 * Float64(x * y))) / z);
	elseif (x <= 1.4)
		tmp = Float64(y * Float64(Float64(1.0 / z) / x));
	else
		tmp = Float64(y * Float64(x / Float64(z / 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.2e-72)
		tmp = ((y / x) + (0.5 * (x * y))) / z;
	elseif (x <= 1.4)
		tmp = y * ((1.0 / z) / x);
	else
		tmp = y * (x / (z / 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.2e-72], N[(N[(N[(y / x), $MachinePrecision] + N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[x, 1.4], N[(y * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.1999999999999996e-72

   1. Initial program 84.5%

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

   2. Taylor expanded in x around 0 54.0%

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

   if -6.1999999999999996e-72 < x < 1.3999999999999999

   1. Initial program 90.0%

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

      1. associate-*r/ 90.0%

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

      2. associate-/r* 97.2%

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

   3. Simplified 97.2%

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

   4. Taylor expanded in x around 0 97.1%

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

      1. clear-num 95.8%

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

      2. associate-/r/ 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-/r* 97.1%

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

   6. Applied egg-rr 97.1%

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

   if 1.3999999999999999 < x

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0 40.9%

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

   3. Taylor expanded in x around inf 40.9%

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

      1. expm1-log1p-u 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{x \cdot y}{z}\right)\right)} \]

      2. expm1-udef 23.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{x \cdot y}{z}\right)} - 1} \]

      3. associate-*r/ 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}}\right)} - 1 \]

      4. *-commutative 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z}\right)} - 1 \]

      5. associate-*r* 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(0.5 \cdot y\right) \cdot x}}{z}\right)} - 1 \]

   5. Applied egg-rr 23.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{z}\right)\right)} \]

      2. expm1-log1p 40.9%

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

      3. *-commutative 40.9%

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

      4. associate-*r* 40.9%

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

      5. *-commutative 40.9%

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

      6. associate-*l/ 42.4%

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

      7. *-commutative 42.4%

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

      8. *-commutative 42.4%

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

      9. associate-/l* 42.4%

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

   7. Simplified 42.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z}{0.5}}\\ \end{array} \]

Alternative 6: 61.9% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.45) (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 <= -1.45) || !(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 <= (-1.45d0)) .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 <= -1.45) || !(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 <= -1.45) 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 <= -1.45) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(x * Float64(y / z)));
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.45) || ~((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, -1.45], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1.3999999999999999 < x

   1. Initial program 77.3%

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

   2. Taylor expanded in x around 0 43.0%

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

   3. Taylor expanded in x around inf 43.0%

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

      1. *-un-lft-identity 43.0%

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

      2. times-frac 32.0%

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

      3. /-rgt-identity 32.0%

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

   5. Applied egg-rr 32.0%

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

   if -1.44999999999999996 < x < 1.3999999999999999

   1. Initial program 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-/r* 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in x around 0 96.7%

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

4. Final simplification 64.3%

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

Alternative 7: 66.1% accurate, 9.6× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.45) (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 <= -1.45) || !(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 <= (-1.45d0)) .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 <= -1.45) || !(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 <= -1.45) 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 <= -1.45) || !(x <= 1.4))
		tmp = Float64(0.5 * Float64(Float64(x * y) / z));
	else
		tmp = Float64(y / Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.45) || ~((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, -1.45], N[Not[LessEqual[x, 1.4]], $MachinePrecision]], N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.44999999999999996 or 1.3999999999999999 < x

   1. Initial program 77.3%

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

   2. Taylor expanded in x around 0 43.0%

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

   3. Taylor expanded in x around inf 43.0%

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

   if -1.44999999999999996 < x < 1.3999999999999999

   1. Initial program 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-/r* 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in x around 0 96.7%

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

4. Final simplification 69.9%

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

Alternative 8: 66.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z}{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45)
   (* 0.5 (/ (* x y) z))
   (if (<= x 1.4) (/ y (* x z)) (* y (/ x (/ z 0.5))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.5 * ((x * y) / z);
	} else if (x <= 1.4) {
		tmp = y / (x * z);
	} else {
		tmp = y * (x / (z / 0.5));
	}
	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 <= (-1.45d0)) then
        tmp = 0.5d0 * ((x * y) / z)
    else if (x <= 1.4d0) then
        tmp = y / (x * z)
    else
        tmp = y * (x / (z / 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.5 * ((x * y) / z);
	} else if (x <= 1.4) {
		tmp = y / (x * z);
	} else {
		tmp = y * (x / (z / 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45:
		tmp = 0.5 * ((x * y) / z)
	elif x <= 1.4:
		tmp = y / (x * z)
	else:
		tmp = y * (x / (z / 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45)
		tmp = Float64(0.5 * Float64(Float64(x * y) / z));
	elseif (x <= 1.4)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(y * Float64(x / Float64(z / 0.5)));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45], N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996

   1. Initial program 82.5%

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

   2. Taylor expanded in x around 0 45.2%

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

   3. Taylor expanded in x around inf 45.2%

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

   if -1.44999999999999996 < x < 1.3999999999999999

   1. Initial program 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-/r* 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in x around 0 96.7%

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

   if 1.3999999999999999 < x

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0 40.9%

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

   3. Taylor expanded in x around inf 40.9%

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

      1. expm1-log1p-u 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{x \cdot y}{z}\right)\right)} \]

      2. expm1-udef 23.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{x \cdot y}{z}\right)} - 1} \]

      3. associate-*r/ 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}}\right)} - 1 \]

      4. *-commutative 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z}\right)} - 1 \]

      5. associate-*r* 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(0.5 \cdot y\right) \cdot x}}{z}\right)} - 1 \]

   5. Applied egg-rr 23.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{z}\right)\right)} \]

      2. expm1-log1p 40.9%

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

      3. *-commutative 40.9%

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

      4. associate-*r* 40.9%

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

      5. *-commutative 40.9%

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

      6. associate-*l/ 42.4%

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

      7. *-commutative 42.4%

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

      8. *-commutative 42.4%

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

      9. associate-/l* 42.4%

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

   7. Simplified 42.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z}{0.5}}\\ \end{array} \]

Alternative 9: 65.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z}{0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.45)
   (* 0.5 (/ (* x y) z))
   (if (<= x 1.4) (* y (/ (/ 1.0 z) x)) (* y (/ x (/ z 0.5))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.5 * ((x * y) / z);
	} else if (x <= 1.4) {
		tmp = y * ((1.0 / z) / x);
	} else {
		tmp = y * (x / (z / 0.5));
	}
	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 <= (-1.45d0)) then
        tmp = 0.5d0 * ((x * y) / z)
    else if (x <= 1.4d0) then
        tmp = y * ((1.0d0 / z) / x)
    else
        tmp = y * (x / (z / 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.45) {
		tmp = 0.5 * ((x * y) / z);
	} else if (x <= 1.4) {
		tmp = y * ((1.0 / z) / x);
	} else {
		tmp = y * (x / (z / 0.5));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.45:
		tmp = 0.5 * ((x * y) / z)
	elif x <= 1.4:
		tmp = y * ((1.0 / z) / x)
	else:
		tmp = y * (x / (z / 0.5))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.45)
		tmp = Float64(0.5 * Float64(Float64(x * y) / z));
	elseif (x <= 1.4)
		tmp = Float64(y * Float64(Float64(1.0 / z) / x));
	else
		tmp = Float64(y * Float64(x / Float64(z / 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.45)
		tmp = 0.5 * ((x * y) / z);
	elseif (x <= 1.4)
		tmp = y * ((1.0 / z) / x);
	else
		tmp = y * (x / (z / 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.45], N[(0.5 * N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4], N[(y * N[(N[(1.0 / z), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / N[(z / 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.44999999999999996

   1. Initial program 82.5%

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

   2. Taylor expanded in x around 0 45.2%

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

   3. Taylor expanded in x around inf 45.2%

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

   if -1.44999999999999996 < x < 1.3999999999999999

   1. Initial program 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-/r* 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in x around 0 96.7%

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

      1. clear-num 95.5%

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

      2. associate-/r/ 96.6%

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

      3. *-commutative 96.6%

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

      4. associate-/r* 96.7%

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

   6. Applied egg-rr 96.7%

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

   if 1.3999999999999999 < x

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0 40.9%

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

   3. Taylor expanded in x around inf 40.9%

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

      1. expm1-log1p-u 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot \frac{x \cdot y}{z}\right)\right)} \]

      2. expm1-udef 23.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot \frac{x \cdot y}{z}\right)} - 1} \]

      3. associate-*r/ 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{0.5 \cdot \left(x \cdot y\right)}{z}}\right)} - 1 \]

      4. *-commutative 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{0.5 \cdot \color{blue}{\left(y \cdot x\right)}}{z}\right)} - 1 \]

      5. associate-*r* 23.1%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\left(0.5 \cdot y\right) \cdot x}}{z}\right)} - 1 \]

   5. Applied egg-rr 23.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{z}\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(0.5 \cdot y\right) \cdot x}{z}\right)\right)} \]

      2. expm1-log1p 40.9%

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

      3. *-commutative 40.9%

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

      4. associate-*r* 40.9%

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

      5. *-commutative 40.9%

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

      6. associate-*l/ 42.4%

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

      7. *-commutative 42.4%

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

      8. *-commutative 42.4%

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

      9. associate-/l* 42.4%

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

   7. Simplified 42.4%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z}{0.5}}\\ \end{array} \]

Alternative 10: 55.1% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.9e+30) (/ y (* x z)) (/ (/ y z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+30) {
		tmp = y / (x * z);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.9d+30)) then
        tmp = y / (x * z)
    else
        tmp = (y / z) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.9e+30) {
		tmp = y / (x * z);
	} else {
		tmp = (y / z) / x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.9e+30:
		tmp = y / (x * z)
	else:
		tmp = (y / z) / x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.9e+30)
		tmp = Float64(y / Float64(x * z));
	else
		tmp = Float64(Float64(y / z) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.9e+30)
		tmp = y / (x * z);
	else
		tmp = (y / z) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.9e+30], N[(y / N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9000000000000001e30

   1. Initial program 84.8%

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

      1. associate-*r/ 67.6%

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

      2. associate-/r* 72.2%

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

   3. Simplified 72.2%

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

   4. Taylor expanded in x around 0 55.7%

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

   if -1.9000000000000001e30 < z

   1. Initial program 83.6%

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

      1. associate-*r/ 78.5%

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

      2. associate-/r* 83.8%

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

   3. Simplified 83.8%

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

      1. associate-*r/ 87.9%

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

      2. *-commutative 87.9%

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

      3. associate-/r* 98.5%

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

   5. Applied egg-rr 98.5%

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

   6. Taylor expanded in x around 0 55.2%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+30}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 11: 49.5% accurate, 21.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

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

   1. associate-*r/ 76.0%

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

   2. associate-/r* 81.2%

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

3. Simplified 81.2%

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

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: 97.3% 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 2023285 
(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))