Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.7% → 99.8%

Time: 8.6s

Alternatives: 10

Speedup: 1.0×

• 64.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x_m \cdot \frac{y_m}{x_m}\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 10^{+245}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x_m}{z}}{x_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    y_s
    (* x_s (if (<= t_0 1e+245) (/ t_0 z) (/ (* y_m (/ (cosh x_m) z)) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 1e+245) {
		tmp = t_0 / z;
	} else {
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 1d+245) then
        tmp = t_0 / z
    else
        tmp = (y_m * (cosh(x_m) / z)) / x_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 1e+245) {
		tmp = t_0 / z;
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 1e+245:
		tmp = t_0 / z
	else:
		tmp = (y_m * (math.cosh(x_m) / z)) / x_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 1e+245)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z)) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 1e+245)
		tmp = t_0 / z;
	else
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 1e+245], N[(t$95$0 / z), $MachinePrecision], N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.00000000000000004e245

   1. Initial program 94.5%

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

   if 1.00000000000000004e245 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.1%

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

      1. associate-*l/ 62.1%

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

   3. Simplified 62.1%

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

      1. associate-*r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+245}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x_m \cdot \frac{y_m}{x_m}\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{\frac{\cosh x_m}{z}}{x_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    y_s
    (* x_s (if (<= t_0 5e+271) (/ t_0 z) (* y_m (/ (/ (cosh x_m) z) x_m)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 5e+271) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * ((cosh(x_m) / z) / x_m);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 5d+271) then
        tmp = t_0 / z
    else
        tmp = y_m * ((cosh(x_m) / z) / x_m)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 5e+271) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * ((Math.cosh(x_m) / z) / x_m);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 5e+271:
		tmp = t_0 / z
	else:
		tmp = y_m * ((math.cosh(x_m) / z) / x_m)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 5e+271)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(y_m * Float64(Float64(cosh(x_m) / z) / x_m));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 5e+271)
		tmp = t_0 / z;
	else
		tmp = y_m * ((cosh(x_m) / z) / x_m);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 5e+271], N[(t$95$0 / z), $MachinePrecision], N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.0000000000000003e271

   1. Initial program 94.6%

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

   if 5.0000000000000003e271 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 60.5%

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

      1. associate-*l/ 60.5%

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

   3. Simplified 60.5%

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

      1. expm1-log1p-u 33.3%

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

      2. expm1-udef 33.3%

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

      3. associate-*l/ 33.3%

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

      4. div-inv 33.3%

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

      5. associate-*l* 31.2%

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

      6. div-inv 31.2%

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

   6. Applied egg-rr 31.2%

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

      1. expm1-def 31.2%

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

      2. expm1-log1p 54.0%

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

      3. associate-*r/ 60.5%

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

      4. associate-*l/ 60.5%

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

      5. *-commutative 60.5%

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

      6. associate-*l/ 100.0%

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

      7. associate-*r/ 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+271}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (* y_m (/ (/ (cosh x_m) z) x_m)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m * ((cosh(x_m) / z) / x_m)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (y_m * ((cosh(x_m) / z) / x_m)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m * ((Math.cosh(x_m) / z) / x_m)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (y_m * ((math.cosh(x_m) / z) / x_m)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(y_m * Float64(Float64(cosh(x_m) / z) / x_m))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (y_m * ((cosh(x_m) / z) / x_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(y$95$m * N[(N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

   1. associate-*l/ 82.1%

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

3. Simplified 82.1%

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

   1. expm1-log1p-u 46.2%

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

   2. expm1-udef 35.8%

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

   3. associate-*l/ 35.8%

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

   4. div-inv 35.8%

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

   5. associate-*l* 34.2%

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

   6. div-inv 34.2%

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

6. Applied egg-rr 34.2%

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

   1. expm1-def 44.4%

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

   2. expm1-log1p 77.7%

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

   3. associate-*r/ 82.2%

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

   4. associate-*l/ 82.1%

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

   5. *-commutative 82.1%

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

   6. associate-*l/ 95.3%

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

   7. associate-*r/ 96.6%

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 4: 67.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.46 \cdot 10^{-10}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{z + \left(x_m \cdot z\right) \cdot \left(x_m \cdot 0.5\right)}{z \cdot \left(x_m \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= x_m 1.46e-10)
     (/ y_m (* x_m z))
     (* y_m (/ (+ z (* (* x_m z) (* x_m 0.5))) (* z (* x_m z))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.46e-10) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = y_m * ((z + ((x_m * z) * (x_m * 0.5))) / (z * (x_m * z)));
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.46d-10) then
        tmp = y_m / (x_m * z)
    else
        tmp = y_m * ((z + ((x_m * z) * (x_m * 0.5d0))) / (z * (x_m * z)))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.46e-10) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = y_m * ((z + ((x_m * z) * (x_m * 0.5))) / (z * (x_m * z)));
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.46e-10:
		tmp = y_m / (x_m * z)
	else:
		tmp = y_m * ((z + ((x_m * z) * (x_m * 0.5))) / (z * (x_m * z)))
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.46e-10)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(y_m * Float64(Float64(z + Float64(Float64(x_m * z) * Float64(x_m * 0.5))) / Float64(z * Float64(x_m * z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.46e-10)
		tmp = y_m / (x_m * z);
	else
		tmp = y_m * ((z + ((x_m * z) * (x_m * 0.5))) / (z * (x_m * z)));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.46e-10], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(z + N[(N[(x$95$m * z), $MachinePrecision] * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4600000000000001e-10

   1. Initial program 84.9%

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

      1. associate-*l/ 84.8%

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

   3. Simplified 84.8%

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

   5. Taylor expanded in x around 0 62.6%

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

   if 1.4600000000000001e-10 < x

   1. Initial program 74.6%

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

      1. associate-*l/ 74.6%

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

   3. Simplified 74.6%

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

      1. expm1-log1p-u 46.1%

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

      2. expm1-udef 44.7%

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

      3. associate-*l/ 44.7%

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

      4. div-inv 44.7%

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

      5. associate-*l* 41.7%

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

      6. div-inv 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. expm1-def 42.2%

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

      2. expm1-log1p 67.7%

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

      3. associate-*r/ 74.6%

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

      4. associate-*l/ 74.6%

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

      5. *-commutative 74.6%

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

      6. associate-*l/ 100.0%

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

      7. associate-*r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 41.9%

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

       1. associate-*r/ 41.9%

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

       2. frac-add 51.0%

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

       3. *-commutative 51.0%

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

       4. *-un-lft-identity 51.0%

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

   11. Applied egg-rr 51.0%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.46 \cdot 10^{-10}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z + \left(x \cdot z\right) \cdot \left(x \cdot 0.5\right)}{z \cdot \left(x \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.4:\\ \;\;\;\;\frac{y_m}{x_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= x_m 1.4) (/ y_m (* x_m z)) (* 0.5 (/ (* x_m y_m) z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = 0.5 * ((x_m * y_m) / z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = 0.5d0 * ((x_m * y_m) / z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = 0.5 * ((x_m * y_m) / z);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.4:
		tmp = y_m / (x_m * z)
	else:
		tmp = 0.5 * ((x_m * y_m) / z)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(0.5 * Float64(Float64(x_m * y_m) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = y_m / (x_m * z);
	else
		tmp = 0.5 * ((x_m * y_m) / z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.2%

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

      1. associate-*l/ 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in x around 0 62.7%

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

   if 1.3999999999999999 < x

   1. Initial program 73.4%

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

      1. associate-*l/ 73.4%

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

   3. Simplified 73.4%

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

      1. expm1-log1p-u 43.8%

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

      2. expm1-udef 42.3%

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

      3. associate-*l/ 42.3%

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

      4. div-inv 42.3%

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

      5. associate-*l* 39.2%

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

      6. div-inv 39.2%

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

   6. Applied egg-rr 39.2%

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

      1. expm1-def 39.6%

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

      2. expm1-log1p 66.2%

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

      3. associate-*r/ 73.4%

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

      4. associate-*l/ 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*l/ 100.0%

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

      7. associate-*r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 40.3%

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

   10. Taylor expanded in x around inf 31.5%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.8% accurate, 8.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 1.4:\\ \;\;\;\;\frac{y_m}{x_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(0.5 \cdot \frac{x_m}{z}\right)\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= x_m 1.4) (/ y_m (* x_m z)) (* y_m (* 0.5 (/ x_m z)))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = y_m * (0.5 * (x_m / z));
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.4d0) then
        tmp = y_m / (x_m * z)
    else
        tmp = y_m * (0.5d0 * (x_m / z))
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.4) {
		tmp = y_m / (x_m * z);
	} else {
		tmp = y_m * (0.5 * (x_m / z));
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 1.4:
		tmp = y_m / (x_m * z)
	else:
		tmp = y_m * (0.5 * (x_m / z))
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.4)
		tmp = Float64(y_m / Float64(x_m * z));
	else
		tmp = Float64(y_m * Float64(0.5 * Float64(x_m / z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 1.4)
		tmp = y_m / (x_m * z);
	else
		tmp = y_m * (0.5 * (x_m / z));
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.4], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(0.5 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 85.2%

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

      1. associate-*l/ 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in x around 0 62.7%

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

   if 1.3999999999999999 < x

   1. Initial program 73.4%

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

      1. associate-*l/ 73.4%

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

   3. Simplified 73.4%

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

      1. expm1-log1p-u 43.8%

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

      2. expm1-udef 42.3%

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

      3. associate-*l/ 42.3%

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

      4. div-inv 42.3%

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

      5. associate-*l* 39.2%

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

      6. div-inv 39.2%

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

   6. Applied egg-rr 39.2%

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

      1. expm1-def 39.6%

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

      2. expm1-log1p 66.2%

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

      3. associate-*r/ 73.4%

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

      4. associate-*l/ 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-*l/ 100.0%

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

      7. associate-*r/ 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in x around 0 40.3%

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

   10. Taylor expanded in x around inf 40.3%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.5% accurate, 9.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (* y_m (/ (+ (* x_m 0.5) (/ 1.0 x_m)) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m * (((x_m * 0.5) + (1.0 / x_m)) / z)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (y_m * (((x_m * 0.5d0) + (1.0d0 / x_m)) / z)))
end function

Java

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

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (y_m * (((x_m * 0.5) + (1.0 / x_m)) / z)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(y_m * Float64(Float64(Float64(x_m * 0.5) + Float64(1.0 / x_m)) / z))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (y_m * (((x_m * 0.5) + (1.0 / x_m)) / z)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(y$95$m * N[(N[(N[(x$95$m * 0.5), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

   1. associate-*l/ 82.1%

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

3. Simplified 82.1%

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

   1. expm1-log1p-u 46.2%

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

   2. expm1-udef 35.8%

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

   3. associate-*l/ 35.8%

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

   4. div-inv 35.8%

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

   5. associate-*l* 34.2%

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

   6. div-inv 34.2%

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

6. Applied egg-rr 34.2%

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

   1. expm1-def 44.4%

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

   2. expm1-log1p 77.7%

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

   3. associate-*r/ 82.2%

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

   4. associate-*l/ 82.1%

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

   5. *-commutative 82.1%

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

   6. associate-*l/ 95.3%

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

   7. associate-*r/ 96.6%

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

8. Simplified 96.6%

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

9. Taylor expanded in x around 0 65.7%

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

10. Taylor expanded in z around 0 65.7%

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

11. Final simplification 65.7%

    \[\leadsto y \cdot \frac{x \cdot 0.5 + \frac{1}{x}}{z} \]
12. Add Preprocessing

Alternative 8: 52.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 10^{-68}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= y_m 1e-68) (/ (/ y_m x_m) z) (/ y_m (* x_m z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1e-68) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (x_m * z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 1d-68) then
        tmp = (y_m / x_m) / z
    else
        tmp = y_m / (x_m * z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 1e-68) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (x_m * z);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 1e-68:
		tmp = (y_m / x_m) / z
	else:
		tmp = y_m / (x_m * z)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 1e-68)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(y_m / Float64(x_m * z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 1e-68)
		tmp = (y_m / x_m) / z;
	else
		tmp = y_m / (x_m * z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 1e-68], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000007e-68

   1. Initial program 79.5%

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

   3. Taylor expanded in x around 0 45.7%

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

   if 1.00000000000000007e-68 < y

   1. Initial program 90.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 60.4%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-68}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 53.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y_m}{z}}{x_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= z 5.2e-92) (/ (/ y_m z) x_m) (/ y_m (* x_m z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 5.2e-92) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = y_m / (x_m * z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 5.2d-92) then
        tmp = (y_m / z) / x_m
    else
        tmp = y_m / (x_m * z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= 5.2e-92) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = y_m / (x_m * z);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if z <= 5.2e-92:
		tmp = (y_m / z) / x_m
	else:
		tmp = y_m / (x_m * z)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= 5.2e-92)
		tmp = Float64(Float64(y_m / z) / x_m);
	else
		tmp = Float64(y_m / Float64(x_m * z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (z <= 5.2e-92)
		tmp = (y_m / z) / x_m;
	else
		tmp = y_m / (x_m * z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[z, 5.2e-92], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.2e-92

   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. expm1-log1p-u 47.2%

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

      2. expm1-udef 37.5%

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

      3. associate-*l/ 37.5%

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

      4. div-inv 37.5%

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

      5. associate-*l* 36.3%

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

      6. div-inv 36.3%

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

   6. Applied egg-rr 36.3%

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

      1. expm1-def 46.1%

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

      2. expm1-log1p 79.8%

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

      3. associate-*r/ 83.4%

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

      4. associate-*l/ 83.3%

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

      5. *-commutative 83.3%

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

      6. associate-*l/ 97.2%

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

      7. associate-*r/ 94.9%

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

   8. Simplified 94.9%

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

   9. Taylor expanded in x around 0 49.7%

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

       1. associate-/l/ 56.6%

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

   11. Simplified 56.6%

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

   if 5.2e-92 < z

   1. Initial program 80.0%

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

      1. associate-*l/ 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in x around 0 46.1%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.2 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.8% accurate, 21.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (/ y_m (* x_m z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m / (x_m * z)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = y_s * (x_s * (y_m / (x_m * z)))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	return y_s * (x_s * (y_m / (x_m * z)));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	return y_s * (x_s * (y_m / (x_m * z)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	return Float64(y_s * Float64(x_s * Float64(y_m / Float64(x_m * z))))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp = code(y_s, x_s, x_m, y_m, z)
	tmp = y_s * (x_s * (y_m / (x_m * z)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.2%

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

   1. associate-*l/ 82.1%

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

3. Simplified 82.1%

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

5. Taylor expanded in x around 0 48.5%

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

6. Final simplification 48.5%

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

Developer target: 97.3% accurate, 0.9× 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 2024021 
(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))