Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 99.8%

Time: 7.4s

Alternatives: 14

Speedup: 0.5×

• 64.1% 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: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x_m \cdot \frac{y_m}{x_m}\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq 10^{+300}:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot \frac{\cosh x_m}{z}}{x_m}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (*
    y_s
    (* x_s (if (<= t_0 1e+300) (/ t_0 z) (/ (* y_m (/ (cosh x_m) z)) x_m))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 1e+300) {
		tmp = t_0 / z;
	} else {
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cosh(x_m) * (y_m / x_m)
    if (t_0 <= 1d+300) then
        tmp = t_0 / z
    else
        tmp = (y_m * (cosh(x_m) / z)) / x_m
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= 1e+300) {
		tmp = t_0 / z;
	} else {
		tmp = (y_m * (Math.cosh(x_m) / z)) / x_m;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= 1e+300:
		tmp = t_0 / z
	else:
		tmp = (y_m * (math.cosh(x_m) / z)) / x_m
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= 1e+300)
		tmp = Float64(t_0 / z);
	else
		tmp = Float64(Float64(y_m * Float64(cosh(x_m) / z)) / x_m);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= 1e+300)
		tmp = t_0 / z;
	else
		tmp = (y_m * (cosh(x_m) / z)) / x_m;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, 1e+300], N[(t$95$0 / z), $MachinePrecision], N[(N[(y$95$m * N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 1.0000000000000001e300

   1. Initial program 96.4%

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

   if 1.0000000000000001e300 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 62.9%

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

      1. associate-*l/ 62.9%

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

   3. Simplified 62.9%

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

      1. associate-*r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 10^{+300}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{\cosh x}{z}}{x}\\ \end{array} \]

Alternative 2: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ \begin{array}{l} t_0 := \cosh x_m \cdot \frac{y_m}{x_m}\\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_0 \leq \infty:\\ \;\;\;\;\frac{t_0}{z}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(x_m \cdot \frac{0.5}{z}\right)\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (cosh x_m) (/ y_m x_m))))
   (* y_s (* x_s (if (<= t_0 INFINITY) (/ t_0 z) (* 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 t_0 = cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0 / z;
	} else {
		tmp = y_m * (x_m * (0.5 / z));
	}
	return y_s * (x_s * tmp);
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = Math.cosh(x_m) * (y_m / x_m);
	double tmp;
	if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = t_0 / z;
	} 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):
	t_0 = math.cosh(x_m) * (y_m / x_m)
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0 / z
	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)
	t_0 = Float64(cosh(x_m) * Float64(y_m / x_m))
	tmp = 0.0
	if (t_0 <= Inf)
		tmp = Float64(t_0 / z);
	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)
	t_0 = cosh(x_m) * (y_m / x_m);
	tmp = 0.0;
	if (t_0 <= Inf)
		tmp = t_0 / z;
	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_] := Block[{t$95$0 = N[(N[Cosh[x$95$m], $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, Infinity], N[(t$95$0 / z), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < +inf.0

   1. Initial program 96.2%

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

   if +inf.0 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 21.8%

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

   3. Taylor expanded in x around inf 21.8%

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

      1. associate-*r/ 21.8%

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

      2. associate-/l* 21.8%

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

      3. *-commutative 21.8%

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

   5. Simplified 21.8%

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

      1. associate-/r/ 21.8%

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

      2. *-commutative 21.8%

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

      3. associate-*r* 37.1%

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

   7. Applied egg-rr 37.1%

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

4. Final simplification 89.0%

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

Alternative 3: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 9.6 \cdot 10^{+221}:\\ \;\;\;\;\frac{y_m}{x_m} \cdot \frac{\cosh x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m \cdot \left(x_m \cdot 0.5 + \frac{1}{x_m}\right)}{z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= x_m 9.6e+221)
     (* (/ y_m x_m) (/ (cosh x_m) z))
     (/ (* y_m (+ (* x_m 0.5) (/ 1.0 x_m))) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 9.6e+221) {
		tmp = (y_m / x_m) * (cosh(x_m) / z);
	} else {
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 9.6d+221) then
        tmp = (y_m / x_m) * (cosh(x_m) / z)
    else
        tmp = (y_m * ((x_m * 0.5d0) + (1.0d0 / x_m))) / z
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 9.6e+221) {
		tmp = (y_m / x_m) * (Math.cosh(x_m) / z);
	} else {
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z;
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if x_m <= 9.6e+221:
		tmp = (y_m / x_m) * (math.cosh(x_m) / z)
	else:
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 9.6e+221)
		tmp = Float64(Float64(y_m / x_m) * Float64(cosh(x_m) / z));
	else
		tmp = Float64(Float64(y_m * Float64(Float64(x_m * 0.5) + Float64(1.0 / x_m))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (x_m <= 9.6e+221)
		tmp = (y_m / x_m) * (cosh(x_m) / z);
	else
		tmp = (y_m * ((x_m * 0.5) + (1.0 / x_m))) / z;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 9.6e+221], N[(N[(y$95$m / x$95$m), $MachinePrecision] * N[(N[Cosh[x$95$m], $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(N[(x$95$m * 0.5), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.60000000000000077e221

   1. Initial program 85.4%

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

      1. associate-*l/ 85.3%

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

   3. Simplified 85.3%

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

   if 9.60000000000000077e221 < x

   1. Initial program 66.7%

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

   2. Taylor expanded in x around 0 67.7%

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

   3. Taylor expanded in y around 0 67.7%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9.6 \cdot 10^{+221}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\cosh x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(x \cdot 0.5 + \frac{1}{x}\right)}{z}\\ \end{array} \]

Alternative 4: 69.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{y_m}{x_m} + 0.5 \cdot \left(x_m \cdot y_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m \cdot y_m}{z} + \frac{y_m}{x_m \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= y_m 4e+42)
     (/ (+ (/ y_m x_m) (* 0.5 (* x_m y_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 (y_m <= 4e+42) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_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 (y_m <= 4d+42) then
        tmp = ((y_m / x_m) + (0.5d0 * (x_m * y_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 (y_m <= 4e+42) {
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_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 y_m <= 4e+42:
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_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 (y_m <= 4e+42)
		tmp = Float64(Float64(Float64(y_m / x_m) + Float64(0.5 * Float64(x_m * y_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 (y_m <= 4e+42)
		tmp = ((y_m / x_m) + (0.5 * (x_m * y_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[y$95$m, 4e+42], 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[(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 2 regimes

2. if y < 4.00000000000000018e42

   1. Initial program 82.5%

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

   2. Taylor expanded in x around 0 66.6%

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

   if 4.00000000000000018e42 < y

   1. Initial program 91.7%

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

      1. associate-*l/ 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around 0 86.4%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{y}{x} + 0.5 \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot y}{z} + \frac{y}{x \cdot z}\\ \end{array} \]

Alternative 5: 65.8% accurate, 9.7× speedup

Math

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

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) (/ 1.0 x_m))) z))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

2. Taylor expanded in x around 0 69.1%

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

3. Taylor expanded in y around 0 69.0%

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

4. Final simplification 69.0%

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

Alternative 6: 65.9% 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 84.5%

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

2. Taylor expanded in x around 0 69.1%

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

3. Final simplification 69.1%

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

Alternative 7: 61.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.39:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x_m}{\frac{z}{y_m}}\\ \end{array}\right) \end{array} \]

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 (<= x_m 0.39) (/ (/ y_m x_m) z) (* 0.5 (/ x_m (/ z y_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 (x_m <= 0.39) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = 0.5 * (x_m / (z / y_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 (x_m <= 0.39d0) then
        tmp = (y_m / x_m) / z
    else
        tmp = 0.5d0 * (x_m / (z / y_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 (x_m <= 0.39) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = 0.5 * (x_m / (z / y_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 x_m <= 0.39:
		tmp = (y_m / x_m) / z
	else:
		tmp = 0.5 * (x_m / (z / y_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 (x_m <= 0.39)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(0.5 * Float64(x_m / Float64(z / y_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 (x_m <= 0.39)
		tmp = (y_m / x_m) / z;
	else
		tmp = 0.5 * (x_m / (z / y_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[x$95$m, 0.39], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(0.5 * N[(x$95$m / N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.39000000000000001

   1. Initial program 84.8%

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

   2. Taylor expanded in x around 0 62.2%

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

   if 0.39000000000000001 < x

   1. Initial program 83.3%

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

   2. Taylor expanded in x around 0 46.3%

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

   3. Taylor expanded in x around inf 46.3%

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

      1. associate-/l* 37.5%

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

   5. Simplified 37.5%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.39:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternative 8: 65.5% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.39:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \left(x_m \cdot \frac{0.5}{z}\right)\\ \end{array}\right) \end{array} \]

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 0.39) (/ (/ y_m x_m) z) (* 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 <= 0.39) {
		tmp = (y_m / x_m) / z;
	} 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 <= 0.39d0) then
        tmp = (y_m / x_m) / z
    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 <= 0.39) {
		tmp = (y_m / x_m) / z;
	} 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 <= 0.39:
		tmp = (y_m / x_m) / z
	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 <= 0.39)
		tmp = Float64(Float64(y_m / x_m) / z);
	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 <= 0.39)
		tmp = (y_m / x_m) / z;
	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, 0.39], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(0.5 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.39000000000000001

   1. Initial program 84.8%

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

   2. Taylor expanded in x around 0 62.2%

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

   if 0.39000000000000001 < x

   1. Initial program 83.3%

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

   2. Taylor expanded in x around 0 46.3%

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

   3. Taylor expanded in x around inf 46.3%

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

      1. associate-*r/ 46.3%

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

      2. associate-/l* 46.3%

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

      3. *-commutative 46.3%

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

   5. Simplified 46.3%

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

      1. associate-/r/ 46.3%

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

      2. *-commutative 46.3%

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

      3. associate-*r* 41.0%

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

   7. Applied egg-rr 41.0%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.39:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{0.5}{z}\right)\\ \end{array} \]

Alternative 9: 65.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.39:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x_m \cdot y_m\right)\\ \end{array}\right) \end{array} \]

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 0.39) (/ (/ y_m x_m) z) (* (/ 0.5 z) (* x_m y_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 (x_m <= 0.39) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (0.5 / z) * (x_m * y_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 (x_m <= 0.39d0) then
        tmp = (y_m / x_m) / z
    else
        tmp = (0.5d0 / z) * (x_m * y_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 (x_m <= 0.39) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (0.5 / z) * (x_m * y_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 x_m <= 0.39:
		tmp = (y_m / x_m) / z
	else:
		tmp = (0.5 / z) * (x_m * y_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 (x_m <= 0.39)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(0.5 / z) * Float64(x_m * y_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 (x_m <= 0.39)
		tmp = (y_m / x_m) / z;
	else
		tmp = (0.5 / z) * (x_m * y_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[x$95$m, 0.39], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(0.5 / z), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.39000000000000001

   1. Initial program 84.8%

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

   2. Taylor expanded in x around 0 62.2%

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

   if 0.39000000000000001 < x

   1. Initial program 83.3%

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

   2. Taylor expanded in x around 0 46.3%

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

   3. Taylor expanded in x around inf 46.3%

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

      1. associate-*r/ 46.3%

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

      2. associate-/l* 46.3%

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

      3. *-commutative 46.3%

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

   5. Simplified 46.3%

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

      1. associate-/r/ 46.3%

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

   7. Applied egg-rr 46.3%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.39:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{z} \cdot \left(x \cdot y\right)\\ \end{array} \]

Alternative 10: 65.6% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.39:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{z}{x_m \cdot y_m}}\\ \end{array}\right) \end{array} \]

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 (<= x_m 0.39) (/ (/ y_m x_m) z) (/ 0.5 (/ z (* x_m y_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 (x_m <= 0.39) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = 0.5 / (z / (x_m * y_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 (x_m <= 0.39d0) then
        tmp = (y_m / x_m) / z
    else
        tmp = 0.5d0 / (z / (x_m * y_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 (x_m <= 0.39) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = 0.5 / (z / (x_m * y_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 x_m <= 0.39:
		tmp = (y_m / x_m) / z
	else:
		tmp = 0.5 / (z / (x_m * y_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 (x_m <= 0.39)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(0.5 / Float64(z / Float64(x_m * y_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 (x_m <= 0.39)
		tmp = (y_m / x_m) / z;
	else
		tmp = 0.5 / (z / (x_m * y_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[x$95$m, 0.39], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(0.5 / N[(z / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.39000000000000001

   1. Initial program 84.8%

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

   2. Taylor expanded in x around 0 62.2%

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

   if 0.39000000000000001 < x

   1. Initial program 83.3%

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

   2. Taylor expanded in x around 0 46.3%

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

   3. Taylor expanded in x around inf 46.3%

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

      1. associate-*r/ 46.3%

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

      2. associate-/l* 46.3%

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

      3. *-commutative 46.3%

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

   5. Simplified 46.3%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.39:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{z}{x \cdot y}}\\ \end{array} \]

Alternative 11: 65.8% accurate, 11.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;x_m \leq 0.39:\\ \;\;\;\;\frac{1}{z \cdot \frac{x_m}{y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{z}{x_m \cdot y_m}}\\ \end{array}\right) \end{array} \]

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 (<= x_m 0.39) (/ 1.0 (* z (/ x_m y_m))) (/ 0.5 (/ z (* x_m y_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 (x_m <= 0.39) {
		tmp = 1.0 / (z * (x_m / y_m));
	} else {
		tmp = 0.5 / (z / (x_m * y_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 (x_m <= 0.39d0) then
        tmp = 1.0d0 / (z * (x_m / y_m))
    else
        tmp = 0.5d0 / (z / (x_m * y_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 (x_m <= 0.39) {
		tmp = 1.0 / (z * (x_m / y_m));
	} else {
		tmp = 0.5 / (z / (x_m * y_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 x_m <= 0.39:
		tmp = 1.0 / (z * (x_m / y_m))
	else:
		tmp = 0.5 / (z / (x_m * y_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 (x_m <= 0.39)
		tmp = Float64(1.0 / Float64(z * Float64(x_m / y_m)));
	else
		tmp = Float64(0.5 / Float64(z / Float64(x_m * y_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 (x_m <= 0.39)
		tmp = 1.0 / (z * (x_m / y_m));
	else
		tmp = 0.5 / (z / (x_m * y_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[x$95$m, 0.39], N[(1.0 / N[(z * N[(x$95$m / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 / N[(z / N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.39000000000000001

   1. Initial program 84.8%

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

      1. associate-/l* 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in x around 0 62.2%

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

      1. clear-num 62.2%

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

      2. associate-/r/ 62.1%

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

      3. clear-num 62.4%

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

   6. Applied egg-rr 62.4%

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

   if 0.39000000000000001 < x

   1. Initial program 83.3%

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

   2. Taylor expanded in x around 0 46.3%

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

   3. Taylor expanded in x around inf 46.3%

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

      1. associate-*r/ 46.3%

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

      2. associate-/l* 46.3%

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

      3. *-commutative 46.3%

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

   5. Simplified 46.3%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.39:\\ \;\;\;\;\frac{1}{z \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{\frac{z}{x \cdot y}}\\ \end{array} \]

Alternative 12: 52.5% accurate, 15.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{x_m \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (* y_s (* x_s (if (<= y_m 2e+44) (/ (/ y_m x_m) z) (/ y_m (* x_m z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2e+44) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (x_m * z);
	}
	return y_s * (x_s * tmp);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y_m <= 2d+44) then
        tmp = (y_m / x_m) / z
    else
        tmp = y_m / (x_m * z)
    end if
    code = y_s * (x_s * tmp)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2e+44) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = y_m / (x_m * z);
	}
	return y_s * (x_s * tmp);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if y_m <= 2e+44:
		tmp = (y_m / x_m) / z
	else:
		tmp = y_m / (x_m * z)
	return y_s * (x_s * tmp)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2e+44)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(y_m / Float64(x_m * z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0;
	if (y_m <= 2e+44)
		tmp = (y_m / x_m) / z;
	else
		tmp = y_m / (x_m * z);
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[y$95$m, 2e+44], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m / N[(x$95$m * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.0000000000000002e44

   1. Initial program 82.8%

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

   2. Taylor expanded in x around 0 52.1%

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

   if 2.0000000000000002e44 < y

   1. Initial program 91.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   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. Taylor expanded in x around 0 54.5%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot z}\\ \end{array} \]

Alternative 13: 56.4% accurate, 15.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;y_m \leq 10^{-53}:\\ \;\;\;\;\frac{\frac{y_m}{x_m}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y_m}{z}}{x_m}\\ \end{array}\right) \end{array} \]

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 (<= y_m 1e-53) (/ (/ y_m x_m) z) (/ (/ 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) {
	double tmp;
	if (y_m <= 1e-53) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_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 (y_m <= 1d-53) then
        tmp = (y_m / x_m) / z
    else
        tmp = (y_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 (y_m <= 1e-53) {
		tmp = (y_m / x_m) / z;
	} else {
		tmp = (y_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 y_m <= 1e-53:
		tmp = (y_m / x_m) / z
	else:
		tmp = (y_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 (y_m <= 1e-53)
		tmp = Float64(Float64(y_m / x_m) / z);
	else
		tmp = Float64(Float64(y_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 (y_m <= 1e-53)
		tmp = (y_m / x_m) / z;
	else
		tmp = (y_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[y$95$m, 1e-53], N[(N[(y$95$m / x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(y$95$m / z), $MachinePrecision] / x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000003e-53

   1. Initial program 81.1%

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

   2. Taylor expanded in x around 0 51.7%

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

   if 1.00000000000000003e-53 < y

   1. Initial program 93.4%

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

      1. associate-*l/ 93.4%

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

   3. Simplified 93.4%

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

      1. associate-*r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 64.3%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-53}:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \end{array} \]

Alternative 14: 49.0% 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 84.5%

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

   1. associate-*l/ 84.5%

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

3. Simplified 84.5%

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

4. Taylor expanded in x around 0 50.1%

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

5. Final simplification 50.1%

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

Developer target: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023333 
(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))