Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.8% → 99.0%

Time: 11.7s

Alternatives: 16

Speedup: 0.5×

• 64.2% 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: 84.8% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x\_m \cdot \frac{y\_m}{x\_m} \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} + 0.5 \cdot \left(x\_m \cdot y\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{\cosh x\_m}{z}}{x\_m}\\ \end{array}\right) \end{array} \]

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 (<= (* (cosh x_m) (/ y_m x_m)) 2e+120)
     (/ (+ (/ y_m x_m) (* 0.5 (* x_m y_m))) 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 tmp;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+120) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / 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) :: tmp
    if ((cosh(x_m) * (y_m / x_m)) <= 2d+120) then
        tmp = ((y_m / x_m) + (0.5d0 * (x_m * y_m))) / 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 tmp;
	if ((Math.cosh(x_m) * (y_m / x_m)) <= 2e+120) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / 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):
	tmp = 0
	if (math.cosh(x_m) * (y_m / x_m)) <= 2e+120:
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / 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)
	tmp = 0.0
	if (Float64(cosh(x_m) * Float64(y_m / x_m)) <= 2e+120)
		tmp = Float64(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_m))) / 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)
	tmp = 0.0;
	if ((cosh(x_m) * (y_m / x_m)) <= 2e+120)
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_m))) / 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_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision], 2e+120], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] + N[(0.5 * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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)) < 2e120

   1. Initial program 97.1%

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

   3. Taylor expanded in x around 0 75.4%

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

   if 2e120 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 64.3%

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

      1. associate-*l/ 64.3%

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

   3. Simplified 64.3%

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

      1. expm1-log1p-u 30.6%

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

      2. expm1-udef 27.2%

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

      3. associate-*l/ 27.2%

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

      4. div-inv 27.2%

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

      5. associate-*l* 23.7%

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

      6. div-inv 23.7%

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

   6. Applied egg-rr 23.7%

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

      1. expm1-def 27.1%

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

      2. expm1-log1p 57.3%

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

      3. associate-*r/ 64.3%

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

      4. associate-*l/ 64.3%

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

      5. *-commutative 64.3%

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

      6. associate-*l/ 99.1%

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

      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 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 68.6% accurate, 3.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{y\_m}{x\_m \cdot z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-181}:\\ \;\;\;\;t\_0 + 0.5 \cdot \frac{y\_m}{\frac{z}{x\_m}}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \frac{z}{x\_m} + z \cdot \left(y\_m \cdot 0.5\right)}{z \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(y\_m \cdot \left(x\_m \cdot 0.5\right)\right) \cdot \left(x\_m \cdot z\right) + y\_m \cdot z}{z \cdot \left(x\_m \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z} + t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (/ y_m (* x_m z))))
   (*
    y_s
    (*
     x_s
     (if (<= z 1.7e-181)
       (+ t_0 (* 0.5 (/ y_m (/ z x_m))))
       (if (<= z 2.4e-46)
         (/ (+ (* (/ y_m x_m) (/ z x_m)) (* z (* y_m 0.5))) (* z (/ z x_m)))
         (if (<= z 1.8e+69)
           (/ (+ (* (* y_m (* x_m 0.5)) (* x_m z)) (* y_m z)) (* z (* x_m z)))
           (+ (* 0.5 (/ (* x_m y_m) z)) t_0))))))))

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 = y_m / (x_m * z);
	double tmp;
	if (z <= 1.7e-181) {
		tmp = t_0 + (0.5 * (y_m / (z / x_m)));
	} else if (z <= 2.4e-46) {
		tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5))) / (z * (z / x_m));
	} else if (z <= 1.8e+69) {
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	}
	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 = y_m / (x_m * z)
    if (z <= 1.7d-181) then
        tmp = t_0 + (0.5d0 * (y_m / (z / x_m)))
    else if (z <= 2.4d-46) then
        tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5d0))) / (z * (z / x_m))
    else if (z <= 1.8d+69) then
        tmp = (((y_m * (x_m * 0.5d0)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z))
    else
        tmp = (0.5d0 * ((x_m * y_m) / z)) + t_0
    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 = y_m / (x_m * z);
	double tmp;
	if (z <= 1.7e-181) {
		tmp = t_0 + (0.5 * (y_m / (z / x_m)));
	} else if (z <= 2.4e-46) {
		tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5))) / (z * (z / x_m));
	} else if (z <= 1.8e+69) {
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	}
	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 = y_m / (x_m * z)
	tmp = 0
	if z <= 1.7e-181:
		tmp = t_0 + (0.5 * (y_m / (z / x_m)))
	elif z <= 2.4e-46:
		tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5))) / (z * (z / x_m))
	elif z <= 1.8e+69:
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z))
	else:
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0
	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(y_m / Float64(x_m * z))
	tmp = 0.0
	if (z <= 1.7e-181)
		tmp = Float64(t_0 + Float64(0.5 * Float64(y_m / Float64(z / x_m))));
	elseif (z <= 2.4e-46)
		tmp = Float64(Float64(Float64(Float64(y_m / x_m) * Float64(z / x_m)) + Float64(z * Float64(y_m * 0.5))) / Float64(z * Float64(z / x_m)));
	elseif (z <= 1.8e+69)
		tmp = Float64(Float64(Float64(Float64(y_m * Float64(x_m * 0.5)) * Float64(x_m * z)) + Float64(y_m * z)) / Float64(z * Float64(x_m * z)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z)) + t_0);
	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 = y_m / (x_m * z);
	tmp = 0.0;
	if (z <= 1.7e-181)
		tmp = t_0 + (0.5 * (y_m / (z / x_m)));
	elseif (z <= 2.4e-46)
		tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5))) / (z * (z / x_m));
	elseif (z <= 1.8e+69)
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z));
	else
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	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[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, 1.7e-181], N[(t$95$0 + N[(0.5 * N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.4e-46], N[(N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(z * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e+69], N[(N[(N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < 1.7e-181

   1. Initial program 82.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 62.0%

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

      1. expm1-log1p-u 38.8%

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

      2. expm1-udef 38.5%

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

      3. associate-/l* 37.4%

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

      4. div-inv 37.4%

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

      5. clear-num 37.4%

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

   7. Applied egg-rr 37.4%

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

      1. expm1-def 37.6%

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

      2. expm1-log1p 59.1%

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

      3. *-commutative 59.1%

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

      4. associate-*l/ 62.0%

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

      5. associate-/l* 62.7%

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

   9. Simplified 62.7%

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

   if 1.7e-181 < z < 2.40000000000000013e-46

   1. Initial program 91.3%

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

      1. associate-*l/ 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x around 0 82.9%

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

      1. expm1-log1p-u 52.4%

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

      2. expm1-udef 52.4%

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

      3. associate-/l* 52.4%

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

      4. div-inv 52.4%

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

      5. clear-num 52.4%

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

   7. Applied egg-rr 52.4%

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

      1. expm1-def 52.4%

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

      2. expm1-log1p 82.9%

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

      3. *-commutative 82.9%

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

      4. associate-*l/ 82.9%

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

      5. associate-/l* 91.0%

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

   9. Simplified 91.0%

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

       1. +-commutative 91.0%

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

       2. associate-/r* 91.8%

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

       3. associate-*r/ 91.8%

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

       4. frac-add 94.4%

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

   11. Applied egg-rr 94.4%

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

   if 2.40000000000000013e-46 < z < 1.8000000000000001e69

   1. Initial program 81.1%

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

      1. associate-*l/ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around 0 54.1%

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

      1. associate-*r/ 54.1%

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

      2. frac-add 58.1%

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

      3. associate-*r* 58.1%

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

   7. Applied egg-rr 58.1%

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

   if 1.8000000000000001e69 < z

   1. Initial program 78.7%

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

      1. associate-*l/ 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in x around 0 61.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \frac{z}{x} + z \cdot \left(y \cdot 0.5\right)}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot z\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.3% accurate, 3.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-181}:\\ \;\;\;\;\frac{y\_m \cdot \frac{-2}{x\_m} - z \cdot \left(x\_m \cdot \frac{y\_m}{z}\right)}{x\_m \cdot \left(z \cdot \frac{-2}{x\_m}\right)}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m} \cdot \frac{z}{x\_m} + z \cdot \left(y\_m \cdot 0.5\right)}{z \cdot \frac{z}{x\_m}}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(y\_m \cdot \left(x\_m \cdot 0.5\right)\right) \cdot \left(x\_m \cdot z\right) + y\_m \cdot z}{z \cdot \left(x\_m \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z} + \frac{y\_m}{x\_m \cdot z}\\ \end{array}\right) \end{array} \]

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 4.2e-181)
     (/
      (- (* y_m (/ -2.0 x_m)) (* z (* x_m (/ y_m z))))
      (* x_m (* z (/ -2.0 x_m))))
     (if (<= z 3.1e-46)
       (/ (+ (* (/ y_m x_m) (/ z x_m)) (* z (* y_m 0.5))) (* z (/ z x_m)))
       (if (<= z 5.7e+69)
         (/ (+ (* (* y_m (* x_m 0.5)) (* x_m z)) (* y_m z)) (* z (* x_m z)))
         (+ (* 0.5 (/ (* x_m y_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 (z <= 4.2e-181) {
		tmp = ((y_m * (-2.0 / x_m)) - (z * (x_m * (y_m / z)))) / (x_m * (z * (-2.0 / x_m)));
	} else if (z <= 3.1e-46) {
		tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5))) / (z * (z / x_m));
	} else if (z <= 5.7e+69) {
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + (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 <= 4.2d-181) then
        tmp = ((y_m * ((-2.0d0) / x_m)) - (z * (x_m * (y_m / z)))) / (x_m * (z * ((-2.0d0) / x_m)))
    else if (z <= 3.1d-46) then
        tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5d0))) / (z * (z / x_m))
    else if (z <= 5.7d+69) then
        tmp = (((y_m * (x_m * 0.5d0)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z))
    else
        tmp = (0.5d0 * ((x_m * y_m) / z)) + (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 <= 4.2e-181) {
		tmp = ((y_m * (-2.0 / x_m)) - (z * (x_m * (y_m / z)))) / (x_m * (z * (-2.0 / x_m)));
	} else if (z <= 3.1e-46) {
		tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5))) / (z * (z / x_m));
	} else if (z <= 5.7e+69) {
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + (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 <= 4.2e-181:
		tmp = ((y_m * (-2.0 / x_m)) - (z * (x_m * (y_m / z)))) / (x_m * (z * (-2.0 / x_m)))
	elif z <= 3.1e-46:
		tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5))) / (z * (z / x_m))
	elif z <= 5.7e+69:
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z))
	else:
		tmp = (0.5 * ((x_m * y_m) / z)) + (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 <= 4.2e-181)
		tmp = Float64(Float64(Float64(y_m * Float64(-2.0 / x_m)) - Float64(z * Float64(x_m * Float64(y_m / z)))) / Float64(x_m * Float64(z * Float64(-2.0 / x_m))));
	elseif (z <= 3.1e-46)
		tmp = Float64(Float64(Float64(Float64(y_m / x_m) * Float64(z / x_m)) + Float64(z * Float64(y_m * 0.5))) / Float64(z * Float64(z / x_m)));
	elseif (z <= 5.7e+69)
		tmp = Float64(Float64(Float64(Float64(y_m * Float64(x_m * 0.5)) * Float64(x_m * z)) + Float64(y_m * z)) / Float64(z * Float64(x_m * z)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z)) + 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 <= 4.2e-181)
		tmp = ((y_m * (-2.0 / x_m)) - (z * (x_m * (y_m / z)))) / (x_m * (z * (-2.0 / x_m)));
	elseif (z <= 3.1e-46)
		tmp = (((y_m / x_m) * (z / x_m)) + (z * (y_m * 0.5))) / (z * (z / x_m));
	elseif (z <= 5.7e+69)
		tmp = (((y_m * (x_m * 0.5)) * (x_m * z)) + (y_m * z)) / (z * (x_m * z));
	else
		tmp = (0.5 * ((x_m * y_m) / z)) + (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, 4.2e-181], N[(N[(N[(y$95$m * N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision] - N[(z * N[(x$95$m * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * N[(z * N[(-2.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e-46], N[(N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision] + N[(z * N[(y$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.7e+69], N[(N[(N[(N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision] * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(y$95$m * z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if z < 4.20000000000000006e-181

   1. Initial program 82.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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 62.0%

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

      1. *-commutative 62.0%

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

      2. *-commutative 62.0%

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

      3. associate-/r/ 62.0%

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

      4. associate-/l* 62.7%

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

      5. div-inv 62.7%

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

      6. *-un-lft-identity 62.7%

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

      7. times-frac 62.7%

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

      8. /-rgt-identity 62.7%

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

      9. metadata-eval 62.7%

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

   7. Applied egg-rr 62.7%

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

      1. +-commutative 62.7%

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

      2. frac-2neg 62.7%

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

      3. associate-/r* 59.1%

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

      4. frac-add 50.7%

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

      5. *-commutative 50.7%

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

      6. *-commutative 50.7%

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

   9. Applied egg-rr 50.7%

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

       1. *-commutative 50.7%

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

       2. distribute-lft-neg-out 50.7%

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

       3. unsub-neg 50.7%

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

       4. *-commutative 50.7%

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

       5. distribute-lft-neg-out 50.7%

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

       6. distribute-rgt-neg-in 50.7%

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

       7. distribute-neg-frac 50.7%

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

       8. metadata-eval 50.7%

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

       9. associate-*l* 56.4%

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

       10. distribute-lft-neg-out 56.4%

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

       11. *-commutative 56.4%

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

       12. associate-*r* 60.3%

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

       13. *-commutative 60.3%

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

       14. distribute-lft-neg-in 60.3%

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

       15. distribute-rgt-neg-in 60.3%

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

       16. distribute-neg-frac 60.3%

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

       17. metadata-eval 60.3%

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

   11. Simplified 60.3%

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

   if 4.20000000000000006e-181 < z < 3.1000000000000001e-46

   1. Initial program 91.3%

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

      1. associate-*l/ 91.2%

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

   3. Simplified 91.2%

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

   5. Taylor expanded in x around 0 82.9%

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

      1. expm1-log1p-u 52.4%

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

      2. expm1-udef 52.4%

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

      3. associate-/l* 52.4%

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

      4. div-inv 52.4%

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

      5. clear-num 52.4%

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

   7. Applied egg-rr 52.4%

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

      1. expm1-def 52.4%

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

      2. expm1-log1p 82.9%

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

      3. *-commutative 82.9%

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

      4. associate-*l/ 82.9%

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

      5. associate-/l* 91.0%

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

   9. Simplified 91.0%

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

       1. +-commutative 91.0%

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

       2. associate-/r* 91.8%

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

       3. associate-*r/ 91.8%

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

       4. frac-add 94.4%

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

   11. Applied egg-rr 94.4%

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

   if 3.1000000000000001e-46 < z < 5.7e69

   1. Initial program 81.1%

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

      1. associate-*l/ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in x around 0 54.1%

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

      1. associate-*r/ 54.1%

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

      2. frac-add 58.1%

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

      3. associate-*r* 58.1%

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

   7. Applied egg-rr 58.1%

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

   if 5.7e69 < z

   1. Initial program 78.7%

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

      1. associate-*l/ 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in x around 0 61.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{-181}:\\ \;\;\;\;\frac{y \cdot \frac{-2}{x} - z \cdot \left(x \cdot \frac{y}{z}\right)}{x \cdot \left(z \cdot \frac{-2}{x}\right)}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \frac{z}{x} + z \cdot \left(y \cdot 0.5\right)}{z \cdot \frac{z}{x}}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(y \cdot \left(x \cdot 0.5\right)\right) \cdot \left(x \cdot z\right) + y \cdot z}{z \cdot \left(x \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{y\_m}{x\_m \cdot z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-138}:\\ \;\;\;\;t\_0 + \frac{y\_m}{2} \cdot \frac{\frac{1}{z}}{\frac{1}{x\_m}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{z \cdot \frac{y\_m}{z} + x\_m \cdot \left(y\_m \cdot \left(x\_m \cdot 0.5\right)\right)}{x\_m \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z} + t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (/ y_m (* x_m z))))
   (*
    y_s
    (*
     x_s
     (if (<= z 5e-138)
       (+ t_0 (* (/ y_m 2.0) (/ (/ 1.0 z) (/ 1.0 x_m))))
       (if (<= z 2e+64)
         (/ (+ (* z (/ y_m z)) (* x_m (* y_m (* x_m 0.5)))) (* x_m z))
         (+ (* 0.5 (/ (* x_m y_m) z)) t_0)))))))

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 = y_m / (x_m * z);
	double tmp;
	if (z <= 5e-138) {
		tmp = t_0 + ((y_m / 2.0) * ((1.0 / z) / (1.0 / x_m)));
	} else if (z <= 2e+64) {
		tmp = ((z * (y_m / z)) + (x_m * (y_m * (x_m * 0.5)))) / (x_m * z);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	}
	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 = y_m / (x_m * z)
    if (z <= 5d-138) then
        tmp = t_0 + ((y_m / 2.0d0) * ((1.0d0 / z) / (1.0d0 / x_m)))
    else if (z <= 2d+64) then
        tmp = ((z * (y_m / z)) + (x_m * (y_m * (x_m * 0.5d0)))) / (x_m * z)
    else
        tmp = (0.5d0 * ((x_m * y_m) / z)) + t_0
    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 = y_m / (x_m * z);
	double tmp;
	if (z <= 5e-138) {
		tmp = t_0 + ((y_m / 2.0) * ((1.0 / z) / (1.0 / x_m)));
	} else if (z <= 2e+64) {
		tmp = ((z * (y_m / z)) + (x_m * (y_m * (x_m * 0.5)))) / (x_m * z);
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	}
	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 = y_m / (x_m * z)
	tmp = 0
	if z <= 5e-138:
		tmp = t_0 + ((y_m / 2.0) * ((1.0 / z) / (1.0 / x_m)))
	elif z <= 2e+64:
		tmp = ((z * (y_m / z)) + (x_m * (y_m * (x_m * 0.5)))) / (x_m * z)
	else:
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0
	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(y_m / Float64(x_m * z))
	tmp = 0.0
	if (z <= 5e-138)
		tmp = Float64(t_0 + Float64(Float64(y_m / 2.0) * Float64(Float64(1.0 / z) / Float64(1.0 / x_m))));
	elseif (z <= 2e+64)
		tmp = Float64(Float64(Float64(z * Float64(y_m / z)) + Float64(x_m * Float64(y_m * Float64(x_m * 0.5)))) / Float64(x_m * z));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z)) + t_0);
	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 = y_m / (x_m * z);
	tmp = 0.0;
	if (z <= 5e-138)
		tmp = t_0 + ((y_m / 2.0) * ((1.0 / z) / (1.0 / x_m)));
	elseif (z <= 2e+64)
		tmp = ((z * (y_m / z)) + (x_m * (y_m * (x_m * 0.5)))) / (x_m * z);
	else
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	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[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, 5e-138], N[(t$95$0 + N[(N[(y$95$m / 2.0), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e+64], N[(N[(N[(z * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(y$95$m * N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 4.99999999999999989e-138

   1. Initial program 83.5%

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

      1. associate-*l/ 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in x around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. *-commutative 63.3%

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

      3. associate-/r/ 63.3%

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

      4. associate-/l* 64.0%

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

      5. div-inv 64.0%

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

      6. *-un-lft-identity 64.0%

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

      7. times-frac 64.0%

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

      8. /-rgt-identity 64.0%

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

      9. metadata-eval 64.0%

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

   7. Applied egg-rr 64.0%

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

      1. associate-/r* 60.6%

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

      2. div-inv 60.6%

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

      3. div-inv 60.6%

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

      4. times-frac 64.5%

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

   9. Applied egg-rr 64.5%

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

   if 4.99999999999999989e-138 < z < 2.00000000000000004e64

   1. Initial program 81.9%

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

      1. associate-*l/ 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 65.3%

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

      1. div-inv 65.3%

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

      2. associate-/l/ 65.2%

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

      3. +-commutative 65.2%

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

      4. associate-*r/ 65.2%

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

      5. div-inv 65.2%

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

      6. associate-*r/ 65.2%

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

      7. frac-add 76.6%

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

      8. associate-*r* 76.6%

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

   7. Applied egg-rr 76.6%

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

   if 2.00000000000000004e64 < z

   1. Initial program 79.5%

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

      1. associate-*l/ 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around 0 61.2%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{y}{2} \cdot \frac{\frac{1}{z}}{\frac{1}{x}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+64}:\\ \;\;\;\;\frac{z \cdot \frac{y}{z} + x \cdot \left(y \cdot \left(x \cdot 0.5\right)\right)}{x \cdot z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{y\_m}{x\_m \cdot z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-52}:\\ \;\;\;\;t\_0 + \frac{y\_m}{2} \cdot \frac{\frac{1}{z}}{\frac{1}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z} + t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (/ y_m (* x_m z))))
   (*
    y_s
    (*
     x_s
     (if (<= z 5e-52)
       (+ t_0 (* (/ y_m 2.0) (/ (/ 1.0 z) (/ 1.0 x_m))))
       (+ (* 0.5 (/ (* x_m y_m) z)) t_0))))))

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 = y_m / (x_m * z);
	double tmp;
	if (z <= 5e-52) {
		tmp = t_0 + ((y_m / 2.0) * ((1.0 / z) / (1.0 / x_m)));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	}
	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 = y_m / (x_m * z)
    if (z <= 5d-52) then
        tmp = t_0 + ((y_m / 2.0d0) * ((1.0d0 / z) / (1.0d0 / x_m)))
    else
        tmp = (0.5d0 * ((x_m * y_m) / z)) + t_0
    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 = y_m / (x_m * z);
	double tmp;
	if (z <= 5e-52) {
		tmp = t_0 + ((y_m / 2.0) * ((1.0 / z) / (1.0 / x_m)));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	}
	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 = y_m / (x_m * z)
	tmp = 0
	if z <= 5e-52:
		tmp = t_0 + ((y_m / 2.0) * ((1.0 / z) / (1.0 / x_m)))
	else:
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0
	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(y_m / Float64(x_m * z))
	tmp = 0.0
	if (z <= 5e-52)
		tmp = Float64(t_0 + Float64(Float64(y_m / 2.0) * Float64(Float64(1.0 / z) / Float64(1.0 / x_m))));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z)) + t_0);
	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 = y_m / (x_m * z);
	tmp = 0.0;
	if (z <= 5e-52)
		tmp = t_0 + ((y_m / 2.0) * ((1.0 / z) / (1.0 / x_m)));
	else
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	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[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[z, 5e-52], N[(t$95$0 + N[(N[(y$95$m / 2.0), $MachinePrecision] * N[(N[(1.0 / z), $MachinePrecision] / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5e-52

   1. Initial program 83.7%

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

      1. associate-*l/ 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in x around 0 64.6%

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

      1. *-commutative 64.6%

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

      2. *-commutative 64.6%

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

      3. associate-/r/ 64.6%

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

      4. associate-/l* 66.2%

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

      5. div-inv 66.2%

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

      6. *-un-lft-identity 66.2%

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

      7. times-frac 66.2%

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

      8. /-rgt-identity 66.2%

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

      9. metadata-eval 66.2%

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

   7. Applied egg-rr 66.2%

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

      1. associate-/r* 62.1%

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

      2. div-inv 62.1%

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

      3. div-inv 62.1%

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

      4. times-frac 66.7%

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

   9. Applied egg-rr 66.7%

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

   if 5e-52 < z

   1. Initial program 79.4%

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

      1. associate-*l/ 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around 0 59.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{-52}:\\ \;\;\;\;\frac{y}{x \cdot z} + \frac{y}{2} \cdot \frac{\frac{1}{z}}{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{-199}:\\ \;\;\;\;y\_m \cdot \frac{\frac{1}{z}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \left(\frac{1}{x\_m} + x\_m \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{0.5}{z}\right)\\ \end{array}\right) \end{array} \]

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 1.4e-199)
     (* y_m (/ (/ 1.0 z) x_m))
     (if (<= x_m 4e+16)
       (* (/ y_m z) (+ (/ 1.0 x_m) (* x_m 0.5)))
       (* y_m (* x_m (/ 0.5 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 <= 1.4e-199) {
		tmp = y_m * ((1.0 / z) / x_m);
	} else if (x_m <= 4e+16) {
		tmp = (y_m / z) * ((1.0 / x_m) + (x_m * 0.5));
	} else {
		tmp = y_m * (x_m * (0.5 / 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 <= 1.4d-199) then
        tmp = y_m * ((1.0d0 / z) / x_m)
    else if (x_m <= 4d+16) then
        tmp = (y_m / z) * ((1.0d0 / x_m) + (x_m * 0.5d0))
    else
        tmp = y_m * (x_m * (0.5d0 / 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 <= 1.4e-199) {
		tmp = y_m * ((1.0 / z) / x_m);
	} else if (x_m <= 4e+16) {
		tmp = (y_m / z) * ((1.0 / x_m) + (x_m * 0.5));
	} else {
		tmp = y_m * (x_m * (0.5 / 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 <= 1.4e-199:
		tmp = y_m * ((1.0 / z) / x_m)
	elif x_m <= 4e+16:
		tmp = (y_m / z) * ((1.0 / x_m) + (x_m * 0.5))
	else:
		tmp = y_m * (x_m * (0.5 / 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 <= 1.4e-199)
		tmp = Float64(y_m * Float64(Float64(1.0 / z) / x_m));
	elseif (x_m <= 4e+16)
		tmp = Float64(Float64(y_m / z) * Float64(Float64(1.0 / x_m) + Float64(x_m * 0.5)));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(0.5 / 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 <= 1.4e-199)
		tmp = y_m * ((1.0 / z) / x_m);
	elseif (x_m <= 4e+16)
		tmp = (y_m / z) * ((1.0 / x_m) + (x_m * 0.5));
	else
		tmp = y_m * (x_m * (0.5 / 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, 1.4e-199], N[(y$95$m * N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 4e+16], N[(N[(y$95$m / z), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] + N[(x$95$m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.40000000000000009e-199

   1. Initial program 86.3%

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

      1. associate-*l/ 86.3%

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

   3. Simplified 86.3%

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

      1. expm1-log1p-u 51.2%

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

      2. expm1-udef 35.3%

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

      3. associate-*l/ 35.3%

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

      4. div-inv 35.3%

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

      5. associate-*l* 31.9%

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

      6. div-inv 31.9%

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

   6. Applied egg-rr 31.9%

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

      1. expm1-def 47.8%

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

      2. expm1-log1p 80.2%

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

      3. associate-*r/ 86.3%

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

      4. associate-*l/ 86.3%

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

      5. *-commutative 86.3%

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

      6. associate-*l/ 94.6%

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

      7. associate-*r/ 95.9%

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

   8. Simplified 95.9%

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

   9. Taylor expanded in x around 0 53.3%

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

   if 1.40000000000000009e-199 < x < 4e16

   1. Initial program 93.4%

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

      1. associate-*l/ 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in x around 0 80.3%

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

   6. Taylor expanded in y around 0 80.2%

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

      1. associate-/l/ 80.2%

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

      2. +-commutative 80.2%

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

      3. distribute-rgt-in 80.2%

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

      4. associate-/l/ 80.2%

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

      5. associate-/r* 80.2%

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

      6. associate-*l/ 80.3%

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

      7. associate-*r/ 84.0%

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

      8. associate-*r/ 84.0%

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

      9. associate-*l/ 84.0%

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

      10. associate-*r* 84.0%

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

      11. associate-*r/ 84.0%

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

      12. associate-*r/ 84.0%

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

      13. associate-*r* 84.0%

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

      14. distribute-rgt-out 84.0%

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

      15. *-commutative 84.0%

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

   8. Simplified 84.0%

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

   if 4e16 < x

   1. Initial program 66.7%

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

      1. associate-*l/ 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in x around 0 41.6%

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

   6. Taylor expanded in x around inf 41.6%

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

      1. associate-*r/ 41.6%

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

      2. *-commutative 41.6%

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

      3. associate-*r/ 41.6%

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

      4. *-commutative 41.6%

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

      5. associate-*r* 40.3%

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

   8. Simplified 40.3%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{+16}:\\ \;\;\;\;\frac{y}{z} \cdot \left(\frac{1}{x} + x \cdot 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \frac{y\_m}{x\_m \cdot z}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-86}:\\ \;\;\;\;t\_0 + 0.5 \cdot \frac{y\_m}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x\_m \cdot y\_m}{z} + t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (/ y_m (* x_m z))))
   (*
    y_s
    (*
     x_s
     (if (<= y_m 5e-86)
       (+ t_0 (* 0.5 (/ y_m (/ z x_m))))
       (+ (* 0.5 (/ (* x_m y_m) z)) t_0))))))

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 = y_m / (x_m * z);
	double tmp;
	if (y_m <= 5e-86) {
		tmp = t_0 + (0.5 * (y_m / (z / x_m)));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	}
	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 = y_m / (x_m * z)
    if (y_m <= 5d-86) then
        tmp = t_0 + (0.5d0 * (y_m / (z / x_m)))
    else
        tmp = (0.5d0 * ((x_m * y_m) / z)) + t_0
    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 = y_m / (x_m * z);
	double tmp;
	if (y_m <= 5e-86) {
		tmp = t_0 + (0.5 * (y_m / (z / x_m)));
	} else {
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	}
	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 = y_m / (x_m * z)
	tmp = 0
	if y_m <= 5e-86:
		tmp = t_0 + (0.5 * (y_m / (z / x_m)))
	else:
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0
	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(y_m / Float64(x_m * z))
	tmp = 0.0
	if (y_m <= 5e-86)
		tmp = Float64(t_0 + Float64(0.5 * Float64(y_m / Float64(z / x_m))));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x_m * y_m) / z)) + t_0);
	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 = y_m / (x_m * z);
	tmp = 0.0;
	if (y_m <= 5e-86)
		tmp = t_0 + (0.5 * (y_m / (z / x_m)));
	else
		tmp = (0.5 * ((x_m * y_m) / z)) + t_0;
	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[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 5e-86], N[(t$95$0 + N[(0.5 * N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x$95$m * y$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999999e-86

   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 59.3%

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

      1. expm1-log1p-u 45.3%

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

      2. expm1-udef 45.0%

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

      3. associate-/l* 43.9%

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

      4. div-inv 43.9%

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

      5. clear-num 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. expm1-def 44.1%

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

      2. expm1-log1p 57.0%

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

      3. *-commutative 57.0%

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

      4. associate-*l/ 59.3%

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

      5. associate-/l* 61.4%

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

   9. Simplified 61.4%

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

   if 4.9999999999999999e-86 < y

   1. Initial program 90.8%

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

      1. associate-*l/ 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in x around 0 71.2%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\frac{y}{x \cdot z} + 0.5 \cdot \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{elif}\;x\_m \leq 1.42:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{0.5}{z}\right)\\ \end{array}\right) \end{array} \]

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 3.8e-199)
     (/ y_m (* x_m z))
     (if (<= x_m 1.42) (/ (/ y_m z) x_m) (* x_m (* y_m (/ 0.5 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 <= 3.8e-199) {
		tmp = y_m / (x_m * z);
	} else if (x_m <= 1.42) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = x_m * (y_m * (0.5 / 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 <= 3.8d-199) then
        tmp = y_m / (x_m * z)
    else if (x_m <= 1.42d0) then
        tmp = (y_m / z) / x_m
    else
        tmp = x_m * (y_m * (0.5d0 / 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 <= 3.8e-199) {
		tmp = y_m / (x_m * z);
	} else if (x_m <= 1.42) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = x_m * (y_m * (0.5 / 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 <= 3.8e-199:
		tmp = y_m / (x_m * z)
	elif x_m <= 1.42:
		tmp = (y_m / z) / x_m
	else:
		tmp = x_m * (y_m * (0.5 / 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 <= 3.8e-199)
		tmp = Float64(y_m / Float64(x_m * z));
	elseif (x_m <= 1.42)
		tmp = Float64(Float64(y_m / z) / x_m);
	else
		tmp = Float64(x_m * Float64(y_m * Float64(0.5 / 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 <= 3.8e-199)
		tmp = y_m / (x_m * z);
	elseif (x_m <= 1.42)
		tmp = (y_m / z) / x_m;
	else
		tmp = x_m * (y_m * (0.5 / 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, 3.8e-199], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.42], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision], N[(x$95$m * N[(y$95$m * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.7999999999999998e-199

   1. Initial program 86.3%

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

      1. associate-*l/ 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in x around 0 54.8%

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

   if 3.7999999999999998e-199 < x < 1.4199999999999999

   1. Initial program 92.1%

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

      1. associate-*l/ 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in x around 0 91.4%

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

      1. associate-*r/ 96.1%

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

      2. associate-*l/ 96.1%

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

      3. *-un-lft-identity 96.1%

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

   7. Applied egg-rr 96.1%

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

   if 1.4199999999999999 < x

   1. Initial program 69.9%

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

   3. Taylor expanded in x around 0 39.3%

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

   4. Taylor expanded in x around inf 39.3%

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

      1. associate-*r/ 39.3%

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

      2. *-commutative 39.3%

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

      3. *-commutative 39.3%

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

      4. associate-/l* 39.3%

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

   6. Simplified 39.3%

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

      1. div-inv 39.3%

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

      2. *-commutative 39.3%

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

      3. clear-num 39.3%

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

      4. associate-*l* 33.0%

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

   8. Applied egg-rr 33.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \mathbf{elif}\;x\_m \leq 1.42:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{0.5}{z}\right)\\ \end{array}\right) \end{array} \]

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 1.9e-198)
     (/ y_m (* x_m z))
     (if (<= x_m 1.42) (/ (/ y_m z) x_m) (* y_m (* x_m (/ 0.5 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 <= 1.9e-198) {
		tmp = y_m / (x_m * z);
	} else if (x_m <= 1.42) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = y_m * (x_m * (0.5 / 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 <= 1.9d-198) then
        tmp = y_m / (x_m * z)
    else if (x_m <= 1.42d0) then
        tmp = (y_m / z) / x_m
    else
        tmp = y_m * (x_m * (0.5d0 / 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 <= 1.9e-198) {
		tmp = y_m / (x_m * z);
	} else if (x_m <= 1.42) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = y_m * (x_m * (0.5 / 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 <= 1.9e-198:
		tmp = y_m / (x_m * z)
	elif x_m <= 1.42:
		tmp = (y_m / z) / x_m
	else:
		tmp = y_m * (x_m * (0.5 / 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 <= 1.9e-198)
		tmp = Float64(y_m / Float64(x_m * z));
	elseif (x_m <= 1.42)
		tmp = Float64(Float64(y_m / z) / x_m);
	else
		tmp = Float64(y_m * Float64(x_m * Float64(0.5 / 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 <= 1.9e-198)
		tmp = y_m / (x_m * z);
	elseif (x_m <= 1.42)
		tmp = (y_m / z) / x_m;
	else
		tmp = y_m * (x_m * (0.5 / 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, 1.9e-198], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.42], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 1.9000000000000001e-198

   1. Initial program 86.3%

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

      1. associate-*l/ 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in x around 0 54.8%

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

   if 1.9000000000000001e-198 < x < 1.4199999999999999

   1. Initial program 92.1%

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

      1. associate-*l/ 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in x around 0 91.4%

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

      1. associate-*r/ 96.1%

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

      2. associate-*l/ 96.1%

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

      3. *-un-lft-identity 96.1%

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

   7. Applied egg-rr 96.1%

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

   if 1.4199999999999999 < x

   1. Initial program 69.9%

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

      1. associate-*l/ 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in x around 0 39.3%

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

   6. Taylor expanded in x around inf 39.3%

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

      1. associate-*r/ 39.3%

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

      2. *-commutative 39.3%

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

      3. associate-*r/ 39.3%

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

      4. *-commutative 39.3%

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

      5. associate-*r* 38.1%

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

   8. Simplified 38.1%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.9 \cdot 10^{-198}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 6.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.7 \cdot 10^{-199}:\\ \;\;\;\;y\_m \cdot \frac{\frac{1}{z}}{x\_m}\\ \mathbf{elif}\;x\_m \leq 1.42:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{0.5}{z}\right)\\ \end{array}\right) \end{array} \]

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 3.7e-199)
     (* y_m (/ (/ 1.0 z) x_m))
     (if (<= x_m 1.42) (/ (/ y_m z) x_m) (* y_m (* x_m (/ 0.5 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 <= 3.7e-199) {
		tmp = y_m * ((1.0 / z) / x_m);
	} else if (x_m <= 1.42) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = y_m * (x_m * (0.5 / 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 <= 3.7d-199) then
        tmp = y_m * ((1.0d0 / z) / x_m)
    else if (x_m <= 1.42d0) then
        tmp = (y_m / z) / x_m
    else
        tmp = y_m * (x_m * (0.5d0 / 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 <= 3.7e-199) {
		tmp = y_m * ((1.0 / z) / x_m);
	} else if (x_m <= 1.42) {
		tmp = (y_m / z) / x_m;
	} else {
		tmp = y_m * (x_m * (0.5 / 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 <= 3.7e-199:
		tmp = y_m * ((1.0 / z) / x_m)
	elif x_m <= 1.42:
		tmp = (y_m / z) / x_m
	else:
		tmp = y_m * (x_m * (0.5 / 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 <= 3.7e-199)
		tmp = Float64(y_m * Float64(Float64(1.0 / z) / x_m));
	elseif (x_m <= 1.42)
		tmp = Float64(Float64(y_m / z) / x_m);
	else
		tmp = Float64(y_m * Float64(x_m * Float64(0.5 / 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 <= 3.7e-199)
		tmp = y_m * ((1.0 / z) / x_m);
	elseif (x_m <= 1.42)
		tmp = (y_m / z) / x_m;
	else
		tmp = y_m * (x_m * (0.5 / 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, 3.7e-199], N[(y$95$m * N[(N[(1.0 / z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$95$m, 1.42], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 3.69999999999999999e-199

   1. Initial program 86.3%

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

      1. associate-*l/ 86.3%

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

   3. Simplified 86.3%

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

      1. expm1-log1p-u 51.2%

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

      2. expm1-udef 35.3%

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

      3. associate-*l/ 35.3%

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

      4. div-inv 35.3%

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

      5. associate-*l* 31.9%

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

      6. div-inv 31.9%

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

   6. Applied egg-rr 31.9%

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

      1. expm1-def 47.8%

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

      2. expm1-log1p 80.2%

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

      3. associate-*r/ 86.3%

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

      4. associate-*l/ 86.3%

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

      5. *-commutative 86.3%

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

      6. associate-*l/ 94.6%

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

      7. associate-*r/ 95.9%

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

   8. Simplified 95.9%

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

   9. Taylor expanded in x around 0 53.3%

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

   if 3.69999999999999999e-199 < x < 1.4199999999999999

   1. Initial program 92.1%

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

      1. associate-*l/ 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in x around 0 91.4%

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

      1. associate-*r/ 96.1%

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

      2. associate-*l/ 96.1%

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

      3. *-un-lft-identity 96.1%

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

   7. Applied egg-rr 96.1%

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

   if 1.4199999999999999 < x

   1. Initial program 69.9%

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

      1. associate-*l/ 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in x around 0 39.3%

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

   6. Taylor expanded in x around inf 39.3%

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

      1. associate-*r/ 39.3%

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

      2. *-commutative 39.3%

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

      3. associate-*r/ 39.3%

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

      4. *-commutative 39.3%

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

      5. associate-*r* 38.1%

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

   8. Simplified 38.1%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.7 \cdot 10^{-199}:\\ \;\;\;\;y \cdot \frac{\frac{1}{z}}{x}\\ \mathbf{elif}\;x \leq 1.42:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.0% accurate, 8.2× speedup

Math

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

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 (+ (* 0.5 (/ x_m z)) (/ 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 * ((0.5 * (x_m / z)) + (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 * ((0.5d0 * (x_m / z)) + (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 * ((0.5 * (x_m / z)) + (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 * ((0.5 * (x_m / z)) + (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(0.5 * Float64(x_m / z)) + Float64(1.0 / 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 * ((0.5 * (x_m / z)) + (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[(0.5 * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

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

   1. associate-*l/ 82.4%

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

3. Simplified 82.4%

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

   1. expm1-log1p-u 45.0%

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

   2. expm1-udef 31.2%

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

   3. associate-*l/ 31.2%

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

   4. div-inv 31.2%

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

   5. associate-*l* 27.7%

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

   6. div-inv 27.7%

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

6. Applied egg-rr 27.7%

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

   1. expm1-def 41.6%

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

   2. expm1-log1p 75.8%

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

   3. associate-*r/ 82.5%

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

   4. associate-*l/ 82.4%

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

   5. *-commutative 82.4%

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

   6. associate-*l/ 96.4%

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

   7. associate-*r/ 96.5%

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

8. Simplified 96.5%

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

9. Taylor expanded in x around 0 62.7%

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

10. Final simplification 62.7%

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

Alternative 12: 64.9% accurate, 8.2× speedup

Math

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

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 (* x_m z)) (* 0.5 (/ y_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 / (x_m * z)) + (0.5 * (y_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 / (x_m * z)) + (0.5d0 * (y_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 / (x_m * z)) + (0.5 * (y_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 / (x_m * z)) + (0.5 * (y_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(Float64(y_m / Float64(x_m * z)) + Float64(0.5 * Float64(y_m / Float64(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 / (x_m * z)) + (0.5 * (y_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[(N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(y$95$m / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

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

   1. associate-*l/ 82.4%

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

3. Simplified 82.4%

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

5. Taylor expanded in x around 0 63.1%

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

   1. expm1-log1p-u 44.1%

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

   2. expm1-udef 43.9%

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

   3. associate-/l* 42.4%

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

   4. div-inv 42.4%

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

   5. clear-num 42.4%

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

7. Applied egg-rr 42.4%

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

   1. expm1-def 42.6%

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

   2. expm1-log1p 60.2%

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

   3. *-commutative 60.2%

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

   4. associate-*l/ 63.1%

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

   5. associate-/l* 63.2%

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

9. Simplified 63.2%

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

10. Final simplification 63.2%

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

Alternative 13: 66.0% accurate, 9.7× speedup

Math

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

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) (* 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) {
	return y_s * (x_s * (((y_m / x_m) + (0.5 * (x_m * y_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 * (x_m * y_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 * (x_m * y_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 * (x_m * y_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(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_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 * (x_m * y_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[(N[(N[(y$95$m / x$95$m), $MachinePrecision] + N[(0.5 * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

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

3. Taylor expanded in x around 0 62.7%

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

4. Final simplification 62.7%

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

Alternative 14: 50.2% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y\_m}{x\_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= z 3.5e-153) (/ (/ 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 (z <= 3.5e-153) {
		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 (z <= 3.5d-153) 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 (z <= 3.5e-153) {
		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 z <= 3.5e-153:
		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 (z <= 3.5e-153)
		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 (z <= 3.5e-153)
		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[z, 3.5e-153], 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 z < 3.49999999999999981e-153

   1. Initial program 83.2%

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

   3. Taylor expanded in x around 0 46.3%

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

   if 3.49999999999999981e-153 < 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 48.4%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{x\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{x\_m \cdot z}\\ \end{array}\right) \end{array} \]

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 2e-75) (/ (/ 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 <= 2e-75) {
		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 <= 2d-75) 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 <= 2e-75) {
		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 <= 2e-75:
		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 <= 2e-75)
		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 <= 2e-75)
		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, 2e-75], 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 < 1.9999999999999999e-75

   1. Initial program 83.8%

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

      1. associate-*l/ 83.8%

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

   3. Simplified 83.8%

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

   5. Taylor expanded in x around 0 45.6%

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

      1. associate-*r/ 51.9%

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

      2. associate-*l/ 51.9%

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

      3. *-un-lft-identity 51.9%

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

   7. Applied egg-rr 51.9%

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

   if 1.9999999999999999e-75 < z

   1. Initial program 79.4%

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

      1. associate-*l/ 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around 0 50.3%

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

4. Final simplification 51.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 48.7% 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 82.5%

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

   1. associate-*l/ 82.4%

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

3. Simplified 82.4%

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

5. Taylor expanded in x around 0 46.0%

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

6. Final simplification 46.0%

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

Developer target: 97.2% 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 2024027 
(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))