Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 85.2% → 99.1%

Time: 9.5s

Alternatives: 14

Speedup: 1.0×

• 64.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z_m}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z_m}{\cosh x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.6e-113)
    (/ (/ (cosh x) (/ z_m y)) x)
    (/ y (* x (/ z_m (cosh x)))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.6e-113) {
		tmp = (cosh(x) / (z_m / y)) / x;
	} else {
		tmp = y / (x * (z_m / cosh(x)));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.6d-113) then
        tmp = (cosh(x) / (z_m / y)) / x
    else
        tmp = y / (x * (z_m / cosh(x)))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.6e-113) {
		tmp = (Math.cosh(x) / (z_m / y)) / x;
	} else {
		tmp = y / (x * (z_m / Math.cosh(x)));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 1.6e-113:
		tmp = (math.cosh(x) / (z_m / y)) / x
	else:
		tmp = y / (x * (z_m / math.cosh(x)))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 1.6e-113)
		tmp = Float64(Float64(cosh(x) / Float64(z_m / y)) / x);
	else
		tmp = Float64(y / Float64(x * Float64(z_m / cosh(x))));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 1.6e-113)
		tmp = (cosh(x) / (z_m / y)) / x;
	else
		tmp = y / (x * (z_m / cosh(x)));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 1.6e-113], N[(N[(N[Cosh[x], $MachinePrecision] / N[(z$95$m / y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(x * N[(z$95$m / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.6000000000000001e-113

   1. Initial program 83.4%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

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

      1. associate-/r/ 79.8%

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

      2. associate-/r/ 85.7%

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

      3. associate-/r* 91.7%

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

   6. Applied egg-rr 91.7%

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

   if 1.6000000000000001e-113 < z

   1. Initial program 81.4%

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

      1. associate-*l/ 81.3%

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

   3. Simplified 81.3%

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

      1. clear-num 81.3%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{\cosh x}{\frac{z}{y}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\cosh x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{z_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z_m} \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 1.3e-75) (/ y (* z_m x)) (* (/ (cosh x) z_m) (/ y x)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 1.3e-75) {
		tmp = y / (z_m * x);
	} else {
		tmp = (cosh(x) / z_m) * (y / x);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 1.3d-75) then
        tmp = y / (z_m * x)
    else
        tmp = (cosh(x) / z_m) * (y / x)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 1.3e-75) {
		tmp = y / (z_m * x);
	} else {
		tmp = (Math.cosh(x) / z_m) * (y / x);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 1.3e-75:
		tmp = y / (z_m * x)
	else:
		tmp = (math.cosh(x) / z_m) * (y / x)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 1.3e-75)
		tmp = Float64(y / Float64(z_m * x));
	else
		tmp = Float64(Float64(cosh(x) / z_m) * Float64(y / x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 1.3e-75)
		tmp = y / (z_m * x);
	else
		tmp = (cosh(x) / z_m) * (y / x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 1.3e-75], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cosh[x], $MachinePrecision] / z$95$m), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3e-75

   1. Initial program 81.8%

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

      1. associate-*l/ 81.8%

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

   3. Simplified 81.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 62.6%

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

   if 1.3e-75 < x

   1. Initial program 84.8%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cosh x}{z} \cdot \frac{y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z_m}{\cosh x}}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 2e-113)
    (/ (* y (/ (cosh x) x)) z_m)
    (/ y (* x (/ z_m (cosh x)))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2e-113) {
		tmp = (y * (cosh(x) / x)) / z_m;
	} else {
		tmp = y / (x * (z_m / cosh(x)));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 2d-113) then
        tmp = (y * (cosh(x) / x)) / z_m
    else
        tmp = y / (x * (z_m / cosh(x)))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 2e-113) {
		tmp = (y * (Math.cosh(x) / x)) / z_m;
	} else {
		tmp = y / (x * (z_m / Math.cosh(x)));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 2e-113:
		tmp = (y * (math.cosh(x) / x)) / z_m
	else:
		tmp = y / (x * (z_m / math.cosh(x)))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 2e-113)
		tmp = Float64(Float64(y * Float64(cosh(x) / x)) / z_m);
	else
		tmp = Float64(y / Float64(x * Float64(z_m / cosh(x))));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 2e-113)
		tmp = (y * (cosh(x) / x)) / z_m;
	else
		tmp = y / (x * (z_m / cosh(x)));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 2e-113], N[(N[(y * N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y / N[(x * N[(z$95$m / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999996e-113

   1. Initial program 83.4%

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

      1. expm1-log1p-u 51.0%

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

      2. expm1-udef 40.1%

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

   4. Applied egg-rr 40.1%

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

      1. expm1-def 51.0%

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

      2. expm1-log1p 83.4%

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

      3. associate-*r/ 94.8%

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

      4. associate-*l/ 94.8%

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

      5. *-commutative 94.8%

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

   6. Simplified 94.8%

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

   if 1.99999999999999996e-113 < z

   1. Initial program 81.4%

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

      1. associate-*l/ 81.3%

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

   3. Simplified 81.3%

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

      1. clear-num 81.3%

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

      2. frac-times 99.8%

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

      3. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \frac{z}{\cosh x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \frac{y}{x \cdot \frac{z_m}{\cosh x}} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ y (* x (/ z_m (cosh x))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (x * (z_m / cosh(x))));
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (y / (x * (z_m / cosh(x))))
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (x * (z_m / Math.cosh(x))));
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (y / (x * (z_m / math.cosh(x))))

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(y / Float64(x * Float64(z_m / cosh(x)))))
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (y / (x * (z_m / cosh(x))));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(y / N[(x * N[(z$95$m / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. associate-*l/ 82.6%

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

3. Simplified 82.6%

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

   1. clear-num 82.6%

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

   2. frac-times 96.9%

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

   3. *-un-lft-identity 96.9%

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

6. Applied egg-rr 96.9%

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

7. Final simplification 96.9%

   \[\leadsto \frac{y}{x \cdot \frac{z}{\cosh x}} \]
8. Add Preprocessing

Alternative 5: 72.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{\frac{z_m}{x}}{y}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot t_0 + z_m \cdot 0.5}{z_m \cdot t_0}\\ \mathbf{elif}\;z_m \leq 6.9 \cdot 10^{+109}:\\ \;\;\;\;\frac{y + x \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{z_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z_m \cdot x} + 0.5 \cdot \frac{x \cdot y}{z_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ (/ z_m x) y)))
   (*
    z_s
    (if (<= z_m 1.6e-114)
      (/ (+ (* (/ y x) t_0) (* z_m 0.5)) (* z_m t_0))
      (if (<= z_m 6.9e+109)
        (/ (+ y (* x (* y (* x 0.5)))) (* z_m x))
        (+ (/ y (* z_m x)) (* 0.5 (/ (* x y) z_m))))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = (z_m / x) / y;
	double tmp;
	if (z_m <= 1.6e-114) {
		tmp = (((y / x) * t_0) + (z_m * 0.5)) / (z_m * t_0);
	} else if (z_m <= 6.9e+109) {
		tmp = (y + (x * (y * (x * 0.5)))) / (z_m * x);
	} else {
		tmp = (y / (z_m * x)) + (0.5 * ((x * y) / z_m));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z_m / x) / y
    if (z_m <= 1.6d-114) then
        tmp = (((y / x) * t_0) + (z_m * 0.5d0)) / (z_m * t_0)
    else if (z_m <= 6.9d+109) then
        tmp = (y + (x * (y * (x * 0.5d0)))) / (z_m * x)
    else
        tmp = (y / (z_m * x)) + (0.5d0 * ((x * y) / z_m))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = (z_m / x) / y;
	double tmp;
	if (z_m <= 1.6e-114) {
		tmp = (((y / x) * t_0) + (z_m * 0.5)) / (z_m * t_0);
	} else if (z_m <= 6.9e+109) {
		tmp = (y + (x * (y * (x * 0.5)))) / (z_m * x);
	} else {
		tmp = (y / (z_m * x)) + (0.5 * ((x * y) / z_m));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = (z_m / x) / y
	tmp = 0
	if z_m <= 1.6e-114:
		tmp = (((y / x) * t_0) + (z_m * 0.5)) / (z_m * t_0)
	elif z_m <= 6.9e+109:
		tmp = (y + (x * (y * (x * 0.5)))) / (z_m * x)
	else:
		tmp = (y / (z_m * x)) + (0.5 * ((x * y) / z_m))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(Float64(z_m / x) / y)
	tmp = 0.0
	if (z_m <= 1.6e-114)
		tmp = Float64(Float64(Float64(Float64(y / x) * t_0) + Float64(z_m * 0.5)) / Float64(z_m * t_0));
	elseif (z_m <= 6.9e+109)
		tmp = Float64(Float64(y + Float64(x * Float64(y * Float64(x * 0.5)))) / Float64(z_m * x));
	else
		tmp = Float64(Float64(y / Float64(z_m * x)) + Float64(0.5 * Float64(Float64(x * y) / z_m)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = (z_m / x) / y;
	tmp = 0.0;
	if (z_m <= 1.6e-114)
		tmp = (((y / x) * t_0) + (z_m * 0.5)) / (z_m * t_0);
	elseif (z_m <= 6.9e+109)
		tmp = (y + (x * (y * (x * 0.5)))) / (z_m * x);
	else
		tmp = (y / (z_m * x)) + (0.5 * ((x * y) / z_m));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(N[(z$95$m / x), $MachinePrecision] / y), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 1.6e-114], N[(N[(N[(N[(y / x), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(z$95$m * 0.5), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 6.9e+109], N[(N[(y + N[(x * N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 1.6000000000000001e-114

   1. Initial program 83.4%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 71.3%

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

      1. div-inv 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-*l* 74.2%

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

   7. Applied egg-rr 74.2%

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

      1. +-commutative 74.2%

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

      2. associate-/r* 73.6%

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

      3. associate-*r* 73.6%

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

      4. un-div-inv 73.6%

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

      5. associate-*r/ 70.7%

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

      6. associate-*r* 70.7%

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

      7. associate-/l* 70.7%

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

      8. frac-add 51.9%

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

      9. *-commutative 51.9%

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

      10. associate-/r* 51.7%

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

      11. *-commutative 51.7%

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

      12. associate-/r* 57.3%

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

   9. Applied egg-rr 57.3%

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

   if 1.6000000000000001e-114 < z < 6.8999999999999999e109

   1. Initial program 84.6%

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

      1. associate-*l/ 84.5%

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

   3. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 58.1%

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

      1. +-commutative 58.1%

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

      2. associate-/r* 56.1%

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

      3. *-un-lft-identity 56.1%

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

      4. associate-*l/ 56.1%

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

      5. associate-*r/ 58.0%

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

      6. associate-*r/ 58.0%

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

      7. frac-add 70.6%

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

      8. associate-*l/ 70.6%

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

      9. *-un-lft-identity 70.6%

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

      10. associate-*r* 70.6%

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

      11. *-commutative 70.6%

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

   7. Applied egg-rr 70.6%

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

   8. Taylor expanded in y around 0 70.6%

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

   if 6.8999999999999999e109 < z

   1. Initial program 78.1%

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

      1. associate-*l/ 78.0%

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

   3. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.8%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \frac{\frac{z}{x}}{y} + z \cdot 0.5}{z \cdot \frac{\frac{z}{x}}{y}}\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{+109}:\\ \;\;\;\;\frac{y + x \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{y}{z_m} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \mathbf{elif}\;z_m \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{y + x \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{z_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z_m \cdot x} + 0.5 \cdot \frac{x \cdot y}{z_m}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.6e-113)
    (* (/ y z_m) (+ (* x 0.5) (/ 1.0 x)))
    (if (<= z_m 2.2e+109)
      (/ (+ y (* x (* y (* x 0.5)))) (* z_m x))
      (+ (/ y (* z_m x)) (* 0.5 (/ (* x y) z_m)))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.6e-113) {
		tmp = (y / z_m) * ((x * 0.5) + (1.0 / x));
	} else if (z_m <= 2.2e+109) {
		tmp = (y + (x * (y * (x * 0.5)))) / (z_m * x);
	} else {
		tmp = (y / (z_m * x)) + (0.5 * ((x * y) / z_m));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.6d-113) then
        tmp = (y / z_m) * ((x * 0.5d0) + (1.0d0 / x))
    else if (z_m <= 2.2d+109) then
        tmp = (y + (x * (y * (x * 0.5d0)))) / (z_m * x)
    else
        tmp = (y / (z_m * x)) + (0.5d0 * ((x * y) / z_m))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.6e-113) {
		tmp = (y / z_m) * ((x * 0.5) + (1.0 / x));
	} else if (z_m <= 2.2e+109) {
		tmp = (y + (x * (y * (x * 0.5)))) / (z_m * x);
	} else {
		tmp = (y / (z_m * x)) + (0.5 * ((x * y) / z_m));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 1.6e-113:
		tmp = (y / z_m) * ((x * 0.5) + (1.0 / x))
	elif z_m <= 2.2e+109:
		tmp = (y + (x * (y * (x * 0.5)))) / (z_m * x)
	else:
		tmp = (y / (z_m * x)) + (0.5 * ((x * y) / z_m))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 1.6e-113)
		tmp = Float64(Float64(y / z_m) * Float64(Float64(x * 0.5) + Float64(1.0 / x)));
	elseif (z_m <= 2.2e+109)
		tmp = Float64(Float64(y + Float64(x * Float64(y * Float64(x * 0.5)))) / Float64(z_m * x));
	else
		tmp = Float64(Float64(y / Float64(z_m * x)) + Float64(0.5 * Float64(Float64(x * y) / z_m)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 1.6e-113)
		tmp = (y / z_m) * ((x * 0.5) + (1.0 / x));
	elseif (z_m <= 2.2e+109)
		tmp = (y + (x * (y * (x * 0.5)))) / (z_m * x);
	else
		tmp = (y / (z_m * x)) + (0.5 * ((x * y) / z_m));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 1.6e-113], N[(N[(y / z$95$m), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.2e+109], N[(N[(y + N[(x * N[(y * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(x * y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.6000000000000001e-113

   1. Initial program 83.4%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 71.3%

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

      1. div-inv 71.3%

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

      2. *-commutative 71.3%

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

      3. associate-*l* 74.2%

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

   7. Applied egg-rr 74.2%

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

      1. un-div-inv 74.2%

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

      2. clear-num 74.2%

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

   9. Applied egg-rr 74.2%

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

   10. Taylor expanded in y around 0 73.6%

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

       1. distribute-lft-in 73.6%

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

       2. *-commutative 73.6%

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

       3. associate-*r/ 74.2%

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

       4. *-rgt-identity 74.2%

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

       5. *-commutative 74.2%

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

       6. associate-*r* 74.2%

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

       7. associate-*r/ 71.3%

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

       8. associate-*l/ 69.1%

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

       9. +-commutative 69.1%

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

       10. *-rgt-identity 69.1%

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

       11. times-frac 70.0%

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

       12. associate-*l* 70.0%

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

       13. distribute-lft-out 70.0%

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

   12. Simplified 70.0%

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

   if 1.6000000000000001e-113 < z < 2.1999999999999999e109

   1. Initial program 84.6%

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

      1. associate-*l/ 84.5%

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

   3. Simplified 84.5%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 58.1%

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

      1. +-commutative 58.1%

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

      2. associate-/r* 56.1%

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

      3. *-un-lft-identity 56.1%

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

      4. associate-*l/ 56.1%

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

      5. associate-*r/ 58.0%

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

      6. associate-*r/ 58.0%

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

      7. frac-add 70.6%

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

      8. associate-*l/ 70.6%

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

      9. *-un-lft-identity 70.6%

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

      10. associate-*r* 70.6%

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

      11. *-commutative 70.6%

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

   7. Applied egg-rr 70.6%

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

   8. Taylor expanded in y around 0 70.6%

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

   if 2.1999999999999999e109 < z

   1. Initial program 78.1%

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

      1. associate-*l/ 78.0%

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

   3. Simplified 78.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 75.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{y}{z} \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{y + x \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 7.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z_m \cdot x} + 0.5 \cdot \left(x \cdot \frac{y}{z_m}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 7.1e+71)
    (/ (* y (+ (* x 0.5) (/ 1.0 x))) z_m)
    (+ (/ y (* z_m x)) (* 0.5 (* x (/ y z_m)))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 7.1e+71) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z_m;
	} else {
		tmp = (y / (z_m * x)) + (0.5 * (x * (y / z_m)));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 7.1d+71) then
        tmp = (y * ((x * 0.5d0) + (1.0d0 / x))) / z_m
    else
        tmp = (y / (z_m * x)) + (0.5d0 * (x * (y / z_m)))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 7.1e+71) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z_m;
	} else {
		tmp = (y / (z_m * x)) + (0.5 * (x * (y / z_m)));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 7.1e+71:
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z_m
	else:
		tmp = (y / (z_m * x)) + (0.5 * (x * (y / z_m)))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 7.1e+71)
		tmp = Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z_m);
	else
		tmp = Float64(Float64(y / Float64(z_m * x)) + Float64(0.5 * Float64(x * Float64(y / z_m))));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 7.1e+71)
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z_m;
	else
		tmp = (y / (z_m * x)) + (0.5 * (x * (y / z_m)));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 7.1e+71], N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(x * N[(y / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.09999999999999986e71

   1. Initial program 83.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 68.5%

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

   4. Taylor expanded in y around 0 68.5%

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

   if 7.09999999999999986e71 < z

   1. Initial program 78.9%

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

      1. associate-*l/ 78.8%

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

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 72.9%

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

      1. div-inv 72.9%

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

      2. associate-*l* 65.2%

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

      3. *-commutative 65.2%

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

      4. associate-*l/ 65.2%

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

      5. *-un-lft-identity 65.2%

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

   7. Applied egg-rr 65.2%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \left(x \cdot \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.4% accurate, 7.1× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := \frac{y}{z_m \cdot x}\\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2 \cdot 10^{-49}:\\ \;\;\;\;t_0 + 0.5 \cdot \frac{y}{\frac{z_m}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + 0.5 \cdot \frac{x \cdot y}{z_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (/ y (* z_m x))))
   (*
    z_s
    (if (<= z_m 2e-49)
      (+ t_0 (* 0.5 (/ y (/ z_m x))))
      (+ t_0 (* 0.5 (/ (* x y) z_m)))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = y / (z_m * x);
	double tmp;
	if (z_m <= 2e-49) {
		tmp = t_0 + (0.5 * (y / (z_m / x)));
	} else {
		tmp = t_0 + (0.5 * ((x * y) / z_m));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (z_m * x)
    if (z_m <= 2d-49) then
        tmp = t_0 + (0.5d0 * (y / (z_m / x)))
    else
        tmp = t_0 + (0.5d0 * ((x * y) / z_m))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = y / (z_m * x);
	double tmp;
	if (z_m <= 2e-49) {
		tmp = t_0 + (0.5 * (y / (z_m / x)));
	} else {
		tmp = t_0 + (0.5 * ((x * y) / z_m));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = y / (z_m * x)
	tmp = 0
	if z_m <= 2e-49:
		tmp = t_0 + (0.5 * (y / (z_m / x)))
	else:
		tmp = t_0 + (0.5 * ((x * y) / z_m))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(y / Float64(z_m * x))
	tmp = 0.0
	if (z_m <= 2e-49)
		tmp = Float64(t_0 + Float64(0.5 * Float64(y / Float64(z_m / x))));
	else
		tmp = Float64(t_0 + Float64(0.5 * Float64(Float64(x * y) / z_m)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = y / (z_m * x);
	tmp = 0.0;
	if (z_m <= 2e-49)
		tmp = t_0 + (0.5 * (y / (z_m / x)));
	else
		tmp = t_0 + (0.5 * ((x * y) / z_m));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[z$95$m, 2e-49], N[(t$95$0 + N[(0.5 * N[(y / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(0.5 * N[(N[(x * y), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999987e-49

   1. Initial program 83.4%

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

      1. associate-*l/ 83.3%

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

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 71.0%

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

      1. div-inv 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*l* 73.6%

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

   7. Applied egg-rr 73.6%

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

      1. associate-*r* 71.0%

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

      2. div-inv 71.0%

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

      3. associate-/l* 73.6%

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

   9. Applied egg-rr 73.6%

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

   if 1.99999999999999987e-49 < z

   1. Initial program 80.9%

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

      1. associate-*l/ 80.8%

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

   3. Simplified 80.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x} + 0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 8.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z_m \cdot x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.8e+72) (/ (* y (+ (* x 0.5) (/ 1.0 x))) z_m) (/ y (* z_m x)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.8e+72) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z_m;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.8d+72) then
        tmp = (y * ((x * 0.5d0) + (1.0d0 / x))) / z_m
    else
        tmp = y / (z_m * x)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.8e+72) {
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z_m;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 1.8e+72:
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z_m
	else:
		tmp = y / (z_m * x)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 1.8e+72)
		tmp = Float64(Float64(y * Float64(Float64(x * 0.5) + Float64(1.0 / x))) / z_m);
	else
		tmp = Float64(y / Float64(z_m * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 1.8e+72)
		tmp = (y * ((x * 0.5) + (1.0 / x))) / z_m;
	else
		tmp = y / (z_m * x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 1.8e+72], N[(N[(y * N[(N[(x * 0.5), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.80000000000000017e72

   1. Initial program 83.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 68.2%

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

   4. Taylor expanded in y around 0 68.2%

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

   if 1.80000000000000017e72 < z

   1. Initial program 78.5%

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

      1. associate-*l/ 78.4%

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

   3. Simplified 78.4%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.0%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+72}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(\frac{y}{z_m \cdot x} + 0.5 \cdot \frac{y}{\frac{z_m}{x}}\right) \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (+ (/ y (* z_m x)) (* 0.5 (/ y (/ z_m x))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * ((y / (z_m * x)) + (0.5 * (y / (z_m / x))));
}

Fortran

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

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * ((y / (z_m * x)) + (0.5 * (y / (z_m / x))));
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * ((y / (z_m * x)) + (0.5 * (y / (z_m / x))))

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(Float64(y / Float64(z_m * x)) + Float64(0.5 * Float64(y / Float64(z_m / x)))))
end

MATLAB

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

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y / N[(z$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. associate-*l/ 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 69.8%

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

   1. div-inv 69.8%

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

   2. *-commutative 69.8%

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

   3. associate-*l* 70.1%

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

7. Applied egg-rr 70.1%

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

   1. associate-*r* 69.8%

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

   2. div-inv 69.8%

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

   3. associate-/l* 70.1%

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

9. Applied egg-rr 70.1%

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

10. Final simplification 70.1%

    \[\leadsto \frac{y}{z \cdot x} + 0.5 \cdot \frac{y}{\frac{z}{x}} \]
11. Add Preprocessing

Alternative 11: 56.7% accurate, 11.8× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y}{z_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z_m}\right)\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 1.45) (/ y (* z_m x)) (* 0.5 (* y (/ x z_m))))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 1.45) {
		tmp = y / (z_m * x);
	} else {
		tmp = 0.5 * (y * (x / z_m));
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 1.45d0) then
        tmp = y / (z_m * x)
    else
        tmp = 0.5d0 * (y * (x / z_m))
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 1.45) {
		tmp = y / (z_m * x);
	} else {
		tmp = 0.5 * (y * (x / z_m));
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 1.45:
		tmp = y / (z_m * x)
	else:
		tmp = 0.5 * (y * (x / z_m))
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 1.45)
		tmp = Float64(y / Float64(z_m * x));
	else
		tmp = Float64(0.5 * Float64(y * Float64(x / z_m)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 1.45)
		tmp = y / (z_m * x);
	else
		tmp = 0.5 * (y * (x / z_m));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 1.45], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(y * N[(x / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 83.2%

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

      1. associate-*l/ 83.1%

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

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 64.4%

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

   if 1.44999999999999996 < x

   1. Initial program 81.0%

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

      1. associate-*l/ 81.0%

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

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 42.0%

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

   6. Taylor expanded in x around inf 42.0%

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

      1. associate-*l/ 41.9%

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

      2. *-commutative 41.9%

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

   8. Simplified 41.9%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y \cdot \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 50.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x}}{z_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z_m \cdot x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= y 1.15e-63) (/ (/ y x) z_m) (/ y (* z_m x)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.15e-63) {
		tmp = (y / x) / z_m;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (y <= 1.15d-63) then
        tmp = (y / x) / z_m
    else
        tmp = y / (z_m * x)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 1.15e-63) {
		tmp = (y / x) / z_m;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if y <= 1.15e-63:
		tmp = (y / x) / z_m
	else:
		tmp = y / (z_m * x)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 1.15e-63)
		tmp = Float64(Float64(y / x) / z_m);
	else
		tmp = Float64(y / Float64(z_m * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (y <= 1.15e-63)
		tmp = (y / x) / z_m;
	else
		tmp = y / (z_m * x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 1.15e-63], N[(N[(y / x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.15e-63

   1. Initial program 79.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 53.2%

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

   if 1.15e-63 < y

   1. Initial program 89.9%

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

      1. associate-*l/ 89.9%

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

   3. Simplified 89.9%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 44.2%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.3% accurate, 15.2× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 100000000000:\\ \;\;\;\;\frac{\frac{y}{z_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z_m \cdot x}\\ \end{array} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= z_m 100000000000.0) (/ (/ y z_m) x) (/ y (* z_m x)))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 100000000000.0) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 100000000000.0d0) then
        tmp = (y / z_m) / x
    else
        tmp = y / (z_m * x)
    end if
    code = z_s * tmp
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 100000000000.0) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y / (z_m * x);
	}
	return z_s * tmp;
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 100000000000.0:
		tmp = (y / z_m) / x
	else:
		tmp = y / (z_m * x)
	return z_s * tmp

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 100000000000.0)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(y / Float64(z_m * x));
	end
	return Float64(z_s * tmp)
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 100000000000.0)
		tmp = (y / z_m) / x;
	else
		tmp = y / (z_m * x);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 100000000000.0], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e11

   1. Initial program 83.2%

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

      1. associate-*l/ 83.1%

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

   3. Simplified 83.1%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 47.8%

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

      1. associate-*r/ 56.4%

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

      2. associate-*l/ 56.4%

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

      3. *-un-lft-identity 56.4%

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

   7. Applied egg-rr 56.4%

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

   if 1e11 < z

   1. Initial program 81.1%

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

      1. associate-*l/ 81.0%

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

   3. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 61.9%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 100000000000:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.6% accurate, 21.4× speedup

Math

\[\begin{array}{l} z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \frac{y}{z_m \cdot x} \end{array} \]

FPCore

z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ y (* z_m x))))

C

z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (z_m * x));
}

Fortran

z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (y / (z_m * x))
end function

Java

z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (z_m * x));
}

Python

z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (y / (z_m * x))

Julia

z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(y / Float64(z_m * x)))
end

MATLAB

z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (y / (z_m * x));
end

Wolfram

z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

   1. associate-*l/ 82.6%

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

3. Simplified 82.6%

   \[\leadsto \color{blue}{\frac{\cosh x}{z} \cdot \frac{y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 51.6%

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

6. Final simplification 51.6%

   \[\leadsto \frac{y}{z \cdot x} \]
7. Add Preprocessing

Developer target: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(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))