Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2

Percentage Accurate: 61.9% → 92.0%

Time: 17.3s

Alternatives: 14

Speedup: 37.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 61.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))

C

double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / sqrt(((z * z) - (t * a)));
}

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((x * y) * z) / Math.sqrt(((z * z) - (t * a)));
}

Python

def code(x, y, z, t, a):
	return ((x * y) * z) / math.sqrt(((z * z) - (t * a)))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(x * y) * z) / sqrt(Float64(Float64(z * z) - Float64(t * a))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((x * y) * z) / sqrt(((z * z) - (t * a)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(x * y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{z_m \cdot \left(x_m \cdot y_m\right)}{\sqrt{z_m \cdot z_m - a \cdot t}}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;\left(x_m \cdot y_m\right) \cdot \frac{z_m}{z_m + -0.5 \cdot \frac{a}{\frac{z_m}{t}}}\\ \mathbf{elif}\;t_1 \leq 10^{+251}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (/ (* z_m (* x_m y_m)) (sqrt (- (* z_m z_m) (* a t))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= t_1 0.0)
        (* (* x_m y_m) (/ z_m (+ z_m (* -0.5 (/ a (/ z_m t))))))
        (if (<= t_1 1e+251) t_1 (* 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);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (z_m * (x_m * y_m)) / sqrt(((z_m * z_m) - (a * t)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t)))));
	} else if (t_1 <= 1e+251) {
		tmp = t_1;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z_m * (x_m * y_m)) / sqrt(((z_m * z_m) - (a * t)))
    if (t_1 <= 0.0d0) then
        tmp = (x_m * y_m) * (z_m / (z_m + ((-0.5d0) * (a / (z_m / t)))))
    else if (t_1 <= 1d+251) then
        tmp = t_1
    else
        tmp = x_m * y_m
    end if
    code = z_s * (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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (z_m * (x_m * y_m)) / Math.sqrt(((z_m * z_m) - (a * t)));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t)))));
	} else if (t_1 <= 1e+251) {
		tmp = t_1;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = (z_m * (x_m * y_m)) / math.sqrt(((z_m * z_m) - (a * t)))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t)))))
	elif t_1 <= 1e+251:
		tmp = t_1
	else:
		tmp = x_m * y_m
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(z_m * Float64(x_m * y_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(a / Float64(z_m / t))))));
	elseif (t_1 <= 1e+251)
		tmp = t_1;
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * 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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = (z_m * (x_m * y_m)) / sqrt(((z_m * z_m) - (a * t)));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = (x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t)))));
	elseif (t_1 <= 1e+251)
		tmp = t_1;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(z$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, 0.0], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(a / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+251], t$95$1, N[(x$95$m * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 0.0

   1. Initial program 66.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 56.0%

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

      1. associate-*r/ 56.0%

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

   5. Simplified 56.0%

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

   6. Taylor expanded in x around 0 52.9%

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

      1. associate-*r* 56.0%

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

      2. associate-*r/ 56.0%

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

      3. associate-*l* 56.0%

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

      4. associate-*r/ 57.5%

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

      5. associate-*l* 57.5%

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

      6. associate-*r/ 57.5%

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

      7. associate-/l* 57.5%

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

   8. Simplified 57.5%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 1e251

   1. Initial program 99.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   if 1e251 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 7.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 17.8%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 17.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 18.1%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 18.2%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 11.6%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 11.6%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 43.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}} \leq 0:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{elif}\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}} \leq 10^{+251}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{x_m}{\frac{\sqrt{{z_m}^{2} - a \cdot t}}{z_m \cdot y_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y_m}{\mathsf{fma}\left(-0.5, \frac{a}{\frac{x_m \cdot {z_m}^{2}}{t}}, \frac{1}{x_m}\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.8e+81)
      (/ x_m (/ (sqrt (- (pow z_m 2.0) (* a t))) (* z_m y_m)))
      (/ y_m (fma -0.5 (/ a (/ (* x_m (pow z_m 2.0)) t)) (/ 1.0 x_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.8e+81) {
		tmp = x_m / (sqrt((pow(z_m, 2.0) - (a * t))) / (z_m * y_m));
	} else {
		tmp = y_m / fma(-0.5, (a / ((x_m * pow(z_m, 2.0)) / t)), (1.0 / x_m));
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.8e+81)
		tmp = Float64(x_m / Float64(sqrt(Float64((z_m ^ 2.0) - Float64(a * t))) / Float64(z_m * y_m)));
	else
		tmp = Float64(y_m / fma(-0.5, Float64(a / Float64(Float64(x_m * (z_m ^ 2.0)) / t)), Float64(1.0 / x_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.8e+81], N[(x$95$m / N[(N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(-0.5 * N[(a / N[(N[(x$95$m * N[Power[z$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.79999999999999995e81

   1. Initial program 68.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 69.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 69.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 66.2%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 67.6%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 69.1%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 69.1%

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

      1. associate-/r/ 67.9%

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

      2. associate-*l/ 67.9%

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

      3. *-commutative 67.9%

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

      4. associate-*l* 65.3%

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

      5. *-commutative 65.3%

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

      6. associate-/l* 65.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]

      7. div-inv 65.3%

         \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]

      8. pow2 65.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}}{y \cdot z}} \]

      9. *-commutative 65.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{\color{blue}{z \cdot y}}} \]

   6. Applied egg-rr 65.3%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 65.3%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}}} \]

      2. *-rgt-identity 65.3%

         \[\leadsto \frac{\color{blue}{x}}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}} \]

      3. *-commutative 65.3%

         \[\leadsto \frac{x}{\frac{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}}{z \cdot y}} \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{{z}^{2} - a \cdot t}}{z \cdot y}}} \]

   if 2.79999999999999995e81 < z

   1. Initial program 37.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 40.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 40.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 39.2%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 42.2%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 36.9%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 36.9%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 87.3%

      \[\leadsto \frac{y}{\color{blue}{-0.5 \cdot \frac{a \cdot t}{x \cdot {z}^{2}} + \frac{1}{x}}} \]
   6. Step-by-step derivation

      1. fma-def 87.3%

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a \cdot t}{x \cdot {z}^{2}}, \frac{1}{x}\right)}} \]

      2. associate-/l* 93.4%

         \[\leadsto \frac{y}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{a}{\frac{x \cdot {z}^{2}}{t}}}, \frac{1}{x}\right)} \]

      3. *-commutative 93.4%

         \[\leadsto \frac{y}{\mathsf{fma}\left(-0.5, \frac{a}{\frac{\color{blue}{{z}^{2} \cdot x}}{t}}, \frac{1}{x}\right)} \]

   7. Simplified 93.4%

      \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{a}{\frac{{z}^{2} \cdot x}{t}}, \frac{1}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.8 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{{z}^{2} - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(-0.5, \frac{a}{\frac{x \cdot {z}^{2}}{t}}, \frac{1}{x}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 4.5 \cdot 10^{+103}:\\ \;\;\;\;x_m \cdot \left(z_m \cdot \frac{y_m}{\sqrt{{z_m}^{2} - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 4.5e+103)
      (* x_m (* z_m (/ y_m (sqrt (- (pow z_m 2.0) (* a t))))))
      (* 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);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.5e+103) {
		tmp = x_m * (z_m * (y_m / sqrt((pow(z_m, 2.0) - (a * t)))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 4.5d+103) then
        tmp = x_m * (z_m * (y_m / sqrt(((z_m ** 2.0d0) - (a * t)))))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.5e+103) {
		tmp = x_m * (z_m * (y_m / Math.sqrt((Math.pow(z_m, 2.0) - (a * t)))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 4.5e+103:
		tmp = x_m * (z_m * (y_m / math.sqrt((math.pow(z_m, 2.0) - (a * t)))))
	else:
		tmp = x_m * y_m
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.5e+103)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(Float64((z_m ^ 2.0) - Float64(a * t))))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * 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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4.5e+103)
		tmp = x_m * (z_m * (y_m / sqrt(((z_m ^ 2.0) - (a * t)))));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4.5e+103], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.50000000000000001e103

   1. Initial program 68.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 69.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 69.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 66.7%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 68.6%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 69.5%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 69.5%

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

      1. associate-/r/ 68.4%

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

      2. *-commutative 68.4%

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

      3. associate-*r* 70.1%

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

      4. pow2 70.1%

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

   6. Applied egg-rr 70.1%

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

   if 4.50000000000000001e103 < z

   1. Initial program 32.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 36.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 36.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 34.5%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 36.1%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 32.0%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 32.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 94.6%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{{z}^{2} - a \cdot t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3 \cdot 10^{+79}:\\ \;\;\;\;\frac{x_m}{\frac{\sqrt{{z_m}^{2} - a \cdot t}}{z_m \cdot y_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{y_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3e+79)
      (/ x_m (/ (sqrt (- (pow z_m 2.0) (* a t))) (* z_m y_m)))
      (* x_m (/ y_m (/ (fma -0.5 (* a (/ t z_m)) z_m) z_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3e+79) {
		tmp = x_m / (sqrt((pow(z_m, 2.0) - (a * t))) / (z_m * y_m));
	} else {
		tmp = x_m * (y_m / (fma(-0.5, (a * (t / z_m)), z_m) / z_m));
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3e+79)
		tmp = Float64(x_m / Float64(sqrt(Float64((z_m ^ 2.0) - Float64(a * t))) / Float64(z_m * y_m)));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(fma(-0.5, Float64(a * Float64(t / z_m)), z_m) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3e+79], N[(x$95$m / N[(N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.99999999999999974e79

   1. Initial program 68.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 69.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 69.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 66.2%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 67.6%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 69.1%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 69.1%

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

      1. associate-/r/ 67.9%

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

      2. associate-*l/ 67.9%

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

      3. *-commutative 67.9%

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

      4. associate-*l* 65.3%

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

      5. *-commutative 65.3%

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

      6. associate-/l* 65.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]

      7. div-inv 65.3%

         \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{\sqrt{z \cdot z - t \cdot a}}{y \cdot z}}} \]

      8. pow2 65.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{\color{blue}{{z}^{2}} - t \cdot a}}{y \cdot z}} \]

      9. *-commutative 65.3%

         \[\leadsto x \cdot \frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{\color{blue}{z \cdot y}}} \]

   6. Applied egg-rr 65.3%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}}} \]
   7. Step-by-step derivation

      1. associate-*r/ 65.3%

         \[\leadsto \color{blue}{\frac{x \cdot 1}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}}} \]

      2. *-rgt-identity 65.3%

         \[\leadsto \frac{\color{blue}{x}}{\frac{\sqrt{{z}^{2} - t \cdot a}}{z \cdot y}} \]

      3. *-commutative 65.3%

         \[\leadsto \frac{x}{\frac{\sqrt{{z}^{2} - \color{blue}{a \cdot t}}}{z \cdot y}} \]

   8. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{{z}^{2} - a \cdot t}}{z \cdot y}}} \]

   if 2.99999999999999974e79 < z

   1. Initial program 37.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 66.8%

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

      1. associate-*r/ 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around 0 57.4%

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

      1. *-commutative 57.4%

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

      2. associate-*r/ 57.4%

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

      3. associate-*l* 57.4%

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

      4. associate-*r/ 72.5%

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

      5. +-commutative 72.5%

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

      6. associate-*l* 72.5%

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

      7. *-commutative 72.5%

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

      8. associate-*r/ 72.5%

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

      9. fma-def 72.5%

         \[\leadsto x \cdot \frac{z \cdot y}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t \cdot a}{z}, z\right)}} \]

      10. associate-/l* 74.4%

          \[\leadsto x \cdot \frac{z \cdot y}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{t}{\frac{z}{a}}}, z\right)} \]

   8. Simplified 74.4%

      \[\leadsto \color{blue}{x \cdot \frac{z \cdot y}{\mathsf{fma}\left(-0.5, \frac{t}{\frac{z}{a}}, z\right)}} \]

   9. Taylor expanded in y around 0 72.5%

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

       1. *-commutative 72.5%

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

       2. associate-*l/ 74.4%

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

       3. +-commutative 74.4%

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

       4. fma-def 74.4%

          \[\leadsto x \cdot \frac{y \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}} \]

       5. associate-/l* 92.6%

          \[\leadsto x \cdot \color{blue}{\frac{y}{\frac{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}{z}}} \]

       6. *-commutative 92.6%

          \[\leadsto x \cdot \frac{y}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{a \cdot \frac{t}{z}}, z\right)}{z}} \]

   11. Simplified 92.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{{z}^{2} - a \cdot t}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{a \cdot \left(-t\right)}}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.63 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z_m \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;x_m \cdot \frac{z_m \cdot y_m}{z_m + \frac{t \cdot \left(a \cdot -0.5\right)}{z_m}}\\ \mathbf{elif}\;z_m \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{y_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (/ (* x_m (* z_m y_m)) (sqrt (* a (- t))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 1.63e-130)
        t_1
        (if (<= z_m 7.5e-116)
          (* x_m (/ (* z_m y_m) (+ z_m (/ (* t (* a -0.5)) z_m))))
          (if (<= z_m 1.15e-77)
            t_1
            (* x_m (/ y_m (/ (fma -0.5 (* a (/ t z_m)) z_m) z_m)))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (x_m * (z_m * y_m)) / sqrt((a * -t));
	double tmp;
	if (z_m <= 1.63e-130) {
		tmp = t_1;
	} else if (z_m <= 7.5e-116) {
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)));
	} else if (z_m <= 1.15e-77) {
		tmp = t_1;
	} else {
		tmp = x_m * (y_m / (fma(-0.5, (a * (t / z_m)), z_m) / z_m));
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(a * Float64(-t))))
	tmp = 0.0
	if (z_m <= 1.63e-130)
		tmp = t_1;
	elseif (z_m <= 7.5e-116)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(Float64(t * Float64(a * -0.5)) / z_m))));
	elseif (z_m <= 1.15e-77)
		tmp = t_1;
	else
		tmp = Float64(x_m * Float64(y_m / Float64(fma(-0.5, Float64(a * Float64(t / z_m)), z_m) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.63e-130], t$95$1, If[LessEqual[z$95$m, 7.5e-116], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(N[(t * N[(a * -0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.15e-77], t$95$1, N[(x$95$m * N[(y$95$m / N[(N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 1.6300000000000001e-130 or 7.5000000000000004e-116 < z < 1.14999999999999999e-77

   1. Initial program 63.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l* 60.3%

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

   3. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 38.1%

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

      1. mul-1-neg 38.1%

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

      2. distribute-rgt-neg-out 38.1%

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

   7. Simplified 38.1%

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

   if 1.6300000000000001e-130 < z < 7.5000000000000004e-116

   1. Initial program 62.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 62.8%

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

      1. associate-*r/ 62.8%

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

   5. Simplified 62.8%

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

      1. associate-*r* 62.4%

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

      2. *-un-lft-identity 62.4%

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

      3. times-frac 62.8%

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

      4. /-rgt-identity 62.8%

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

      5. *-commutative 62.8%

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

      6. associate-*r* 62.8%

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

   7. Applied egg-rr 62.8%

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

   if 1.14999999999999999e-77 < z

   1. Initial program 55.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 72.0%

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

      1. associate-*r/ 72.0%

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

   5. Simplified 72.0%

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

   6. Taylor expanded in x around 0 65.5%

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

      1. *-commutative 65.5%

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

      2. associate-*r/ 65.5%

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

      3. associate-*l* 65.5%

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

      4. associate-*r/ 75.9%

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

      5. +-commutative 75.9%

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

      6. associate-*l* 75.9%

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

      7. *-commutative 75.9%

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

      8. associate-*r/ 75.9%

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

      9. fma-def 75.9%

         \[\leadsto x \cdot \frac{z \cdot y}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t \cdot a}{z}, z\right)}} \]

      10. associate-/l* 77.2%

          \[\leadsto x \cdot \frac{z \cdot y}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{t}{\frac{z}{a}}}, z\right)} \]

   8. Simplified 77.2%

      \[\leadsto \color{blue}{x \cdot \frac{z \cdot y}{\mathsf{fma}\left(-0.5, \frac{t}{\frac{z}{a}}, z\right)}} \]

   9. Taylor expanded in y around 0 75.9%

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

       1. *-commutative 75.9%

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

       2. associate-*l/ 77.2%

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

       3. +-commutative 77.2%

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

       4. fma-def 77.2%

          \[\leadsto x \cdot \frac{y \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}} \]

       5. associate-/l* 89.7%

          \[\leadsto x \cdot \color{blue}{\frac{y}{\frac{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}{z}}} \]

       6. *-commutative 89.7%

          \[\leadsto x \cdot \frac{y}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{a \cdot \frac{t}{z}}, z\right)}{z}} \]

   11. Simplified 89.7%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.63 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + \frac{t \cdot \left(a \cdot -0.5\right)}{z}}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-77}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z_m \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\frac{y_m}{\frac{\sqrt{z_m \cdot z_m - a \cdot t}}{z_m \cdot x_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 7.2e-146)
      (/ (* x_m (* z_m y_m)) (sqrt (* a (- t))))
      (if (<= z_m 8e+104)
        (/ y_m (/ (sqrt (- (* z_m z_m) (* a t))) (* z_m x_m)))
        (* 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);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.2e-146) {
		tmp = (x_m * (z_m * y_m)) / sqrt((a * -t));
	} else if (z_m <= 8e+104) {
		tmp = y_m / (sqrt(((z_m * z_m) - (a * t))) / (z_m * x_m));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 7.2d-146) then
        tmp = (x_m * (z_m * y_m)) / sqrt((a * -t))
    else if (z_m <= 8d+104) then
        tmp = y_m / (sqrt(((z_m * z_m) - (a * t))) / (z_m * x_m))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.2e-146) {
		tmp = (x_m * (z_m * y_m)) / Math.sqrt((a * -t));
	} else if (z_m <= 8e+104) {
		tmp = y_m / (Math.sqrt(((z_m * z_m) - (a * t))) / (z_m * x_m));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7.2e-146:
		tmp = (x_m * (z_m * y_m)) / math.sqrt((a * -t))
	elif z_m <= 8e+104:
		tmp = y_m / (math.sqrt(((z_m * z_m) - (a * t))) / (z_m * x_m))
	else:
		tmp = x_m * y_m
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.2e-146)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(a * Float64(-t))));
	elseif (z_m <= 8e+104)
		tmp = Float64(y_m / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))) / Float64(z_m * x_m)));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * 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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.2e-146)
		tmp = (x_m * (z_m * y_m)) / sqrt((a * -t));
	elseif (z_m <= 8e+104)
		tmp = y_m / (sqrt(((z_m * z_m) - (a * t))) / (z_m * x_m));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.2e-146], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 8e+104], N[(y$95$m / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 7.19999999999999957e-146

   1. Initial program 62.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l* 58.6%

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

   3. Simplified 58.6%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 35.8%

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

      1. mul-1-neg 35.8%

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

      2. distribute-rgt-neg-out 35.8%

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

   7. Simplified 35.8%

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

   if 7.19999999999999957e-146 < z < 8e104

   1. Initial program 88.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 88.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 88.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 85.9%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 91.5%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 91.6%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 91.6%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   if 8e104 < z

   1. Initial program 32.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 36.1%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 36.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 34.5%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 36.1%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 32.0%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 32.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 94.6%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-146}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+104}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{z \cdot z - a \cdot t}}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{a \cdot \left(-t\right)}}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.63 \cdot 10^{-130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z_m \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;x_m \cdot \frac{z_m \cdot y_m}{z_m + \frac{t \cdot \left(a \cdot -0.5\right)}{z_m}}\\ \mathbf{elif}\;z_m \leq 5.8 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(x_m \cdot y_m\right) \cdot \frac{z_m}{z_m + -0.5 \cdot \frac{a}{\frac{z_m}{t}}}\\ \end{array}\right)\right) \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (/ (* x_m (* z_m y_m)) (sqrt (* a (- t))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 1.63e-130)
        t_1
        (if (<= z_m 7.5e-116)
          (* x_m (/ (* z_m y_m) (+ z_m (/ (* t (* a -0.5)) z_m))))
          (if (<= z_m 5.8e-78)
            t_1
            (* (* x_m y_m) (/ z_m (+ z_m (* -0.5 (/ a (/ z_m t))))))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (x_m * (z_m * y_m)) / sqrt((a * -t));
	double tmp;
	if (z_m <= 1.63e-130) {
		tmp = t_1;
	} else if (z_m <= 7.5e-116) {
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)));
	} else if (z_m <= 5.8e-78) {
		tmp = t_1;
	} else {
		tmp = (x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t)))));
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x_m * (z_m * y_m)) / sqrt((a * -t))
    if (z_m <= 1.63d-130) then
        tmp = t_1
    else if (z_m <= 7.5d-116) then
        tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * (-0.5d0))) / z_m)))
    else if (z_m <= 5.8d-78) then
        tmp = t_1
    else
        tmp = (x_m * y_m) * (z_m / (z_m + ((-0.5d0) * (a / (z_m / t)))))
    end if
    code = z_s * (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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = (x_m * (z_m * y_m)) / Math.sqrt((a * -t));
	double tmp;
	if (z_m <= 1.63e-130) {
		tmp = t_1;
	} else if (z_m <= 7.5e-116) {
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)));
	} else if (z_m <= 5.8e-78) {
		tmp = t_1;
	} else {
		tmp = (x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t)))));
	}
	return z_s * (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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = (x_m * (z_m * y_m)) / math.sqrt((a * -t))
	tmp = 0
	if z_m <= 1.63e-130:
		tmp = t_1
	elif z_m <= 7.5e-116:
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)))
	elif z_m <= 5.8e-78:
		tmp = t_1
	else:
		tmp = (x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t)))))
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(a * Float64(-t))))
	tmp = 0.0
	if (z_m <= 1.63e-130)
		tmp = t_1;
	elseif (z_m <= 7.5e-116)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(Float64(t * Float64(a * -0.5)) / z_m))));
	elseif (z_m <= 5.8e-78)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / Float64(z_m + Float64(-0.5 * Float64(a / Float64(z_m / t))))));
	end
	return Float64(z_s * 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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = (x_m * (z_m * y_m)) / sqrt((a * -t));
	tmp = 0.0;
	if (z_m <= 1.63e-130)
		tmp = t_1;
	elseif (z_m <= 7.5e-116)
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)));
	elseif (z_m <= 5.8e-78)
		tmp = t_1;
	else
		tmp = (x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t)))));
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := Block[{t$95$1 = N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.63e-130], t$95$1, If[LessEqual[z$95$m, 7.5e-116], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(N[(t * N[(a * -0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 5.8e-78], t$95$1, N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(a / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 1.6300000000000001e-130 or 7.5000000000000004e-116 < z < 5.8000000000000001e-78

   1. Initial program 63.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l* 60.3%

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

   3. Simplified 60.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 38.1%

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

      1. mul-1-neg 38.1%

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

      2. distribute-rgt-neg-out 38.1%

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

   7. Simplified 38.1%

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

   if 1.6300000000000001e-130 < z < 7.5000000000000004e-116

   1. Initial program 62.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 62.8%

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

      1. associate-*r/ 62.8%

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

   5. Simplified 62.8%

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

      1. associate-*r* 62.4%

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

      2. *-un-lft-identity 62.4%

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

      3. times-frac 62.8%

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

      4. /-rgt-identity 62.8%

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

      5. *-commutative 62.8%

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

      6. associate-*r* 62.8%

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

   7. Applied egg-rr 62.8%

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

   if 5.8000000000000001e-78 < z

   1. Initial program 55.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 72.0%

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

      1. associate-*r/ 72.0%

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

   5. Simplified 72.0%

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

   6. Taylor expanded in x around 0 65.5%

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

      1. associate-*r* 72.0%

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

      2. associate-*r/ 72.0%

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

      3. associate-*l* 72.0%

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

      4. associate-*r/ 87.4%

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

      5. associate-*l* 87.4%

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

      6. associate-*r/ 87.4%

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

      7. associate-/l* 89.7%

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

   8. Simplified 89.7%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.63 \cdot 10^{-130}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-116}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + \frac{t \cdot \left(a \cdot -0.5\right)}{z}}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{z_m \cdot z_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot \frac{y_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.95e+79)
      (/ (* x_m (* z_m y_m)) (sqrt (- (* z_m z_m) (* a t))))
      (* x_m (/ y_m (/ (fma -0.5 (* a (/ t z_m)) z_m) z_m))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.95e+79) {
		tmp = (x_m * (z_m * y_m)) / sqrt(((z_m * z_m) - (a * t)));
	} else {
		tmp = x_m * (y_m / (fma(-0.5, (a * (t / z_m)), z_m) / z_m));
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.95e+79)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(fma(-0.5, Float64(a * Float64(t / z_m)), z_m) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.95e+79], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.95e79

   1. Initial program 68.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l* 65.3%

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

   3. Simplified 65.3%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing

   if 2.95e79 < z

   1. Initial program 37.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 66.8%

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

      1. associate-*r/ 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in x around 0 57.4%

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

      1. *-commutative 57.4%

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

      2. associate-*r/ 57.4%

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

      3. associate-*l* 57.4%

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

      4. associate-*r/ 72.5%

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

      5. +-commutative 72.5%

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

      6. associate-*l* 72.5%

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

      7. *-commutative 72.5%

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

      8. associate-*r/ 72.5%

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

      9. fma-def 72.5%

         \[\leadsto x \cdot \frac{z \cdot y}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t \cdot a}{z}, z\right)}} \]

      10. associate-/l* 74.4%

          \[\leadsto x \cdot \frac{z \cdot y}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{t}{\frac{z}{a}}}, z\right)} \]

   8. Simplified 74.4%

      \[\leadsto \color{blue}{x \cdot \frac{z \cdot y}{\mathsf{fma}\left(-0.5, \frac{t}{\frac{z}{a}}, z\right)}} \]

   9. Taylor expanded in y around 0 72.5%

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

       1. *-commutative 72.5%

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

       2. associate-*l/ 74.4%

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

       3. +-commutative 74.4%

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

       4. fma-def 74.4%

          \[\leadsto x \cdot \frac{y \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}} \]

       5. associate-/l* 92.6%

          \[\leadsto x \cdot \color{blue}{\frac{y}{\frac{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}{z}}} \]

       6. *-commutative 92.6%

          \[\leadsto x \cdot \frac{y}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{a \cdot \frac{t}{z}}, z\right)}{z}} \]

   11. Simplified 92.6%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.9% accurate, 6.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.7 \cdot 10^{+14}:\\ \;\;\;\;x_m \cdot \frac{z_m \cdot y_m}{z_m + \frac{t \cdot \left(a \cdot -0.5\right)}{z_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.7e+14)
      (* x_m (/ (* z_m y_m) (+ z_m (/ (* t (* a -0.5)) z_m))))
      (* 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);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e+14) {
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.7d+14) then
        tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * (-0.5d0))) / z_m)))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e+14) {
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.7e+14:
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)))
	else:
		tmp = x_m * y_m
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.7e+14)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(Float64(t * Float64(a * -0.5)) / z_m))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * 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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.7e+14)
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * (a * -0.5)) / z_m)));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.7e+14], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(N[(t * N[(a * -0.5), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.7e14

   1. Initial program 65.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 31.9%

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

      1. associate-*r/ 31.9%

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

   5. Simplified 31.9%

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

      1. associate-*r* 32.0%

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

      2. *-un-lft-identity 32.0%

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

      3. times-frac 32.0%

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

      4. /-rgt-identity 32.0%

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

      5. *-commutative 32.0%

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

      6. associate-*r* 32.0%

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

   7. Applied egg-rr 32.0%

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

   if 1.7e14 < z

   1. Initial program 48.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 50.2%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 50.2%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 47.8%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 51.4%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 48.3%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 48.3%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 93.6%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + \frac{t \cdot \left(a \cdot -0.5\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.0% accurate, 7.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.63 \cdot 10^{-130}:\\ \;\;\;\;x_m \cdot \frac{y_m}{\frac{a \cdot \left(-0.5 \cdot \frac{t}{z_m}\right)}{z_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.63e-130)
      (* x_m (/ y_m (/ (* a (* -0.5 (/ t z_m))) z_m)))
      (* 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);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.63e-130) {
		tmp = x_m * (y_m / ((a * (-0.5 * (t / z_m))) / z_m));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.63d-130) then
        tmp = x_m * (y_m / ((a * ((-0.5d0) * (t / z_m))) / z_m))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.63e-130) {
		tmp = x_m * (y_m / ((a * (-0.5 * (t / z_m))) / z_m));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.63e-130:
		tmp = x_m * (y_m / ((a * (-0.5 * (t / z_m))) / z_m))
	else:
		tmp = x_m * y_m
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.63e-130)
		tmp = Float64(x_m * Float64(y_m / Float64(Float64(a * Float64(-0.5 * Float64(t / z_m))) / z_m)));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * 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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.63e-130)
		tmp = x_m * (y_m / ((a * (-0.5 * (t / z_m))) / z_m));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.63e-130], N[(x$95$m * N[(y$95$m / N[(N[(a * N[(-0.5 * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.6300000000000001e-130

   1. Initial program 63.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 28.1%

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

      1. associate-*r/ 28.1%

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

   5. Simplified 28.1%

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

   6. Taylor expanded in x around 0 28.1%

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

      1. *-commutative 28.1%

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

      2. associate-*r/ 28.1%

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

      3. associate-*l* 28.1%

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

      4. associate-*r/ 28.1%

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

      5. +-commutative 28.1%

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

      6. associate-*l* 28.1%

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

      7. *-commutative 28.1%

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

      8. associate-*r/ 28.1%

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

      9. fma-def 28.1%

         \[\leadsto x \cdot \frac{z \cdot y}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t \cdot a}{z}, z\right)}} \]

      10. associate-/l* 28.1%

          \[\leadsto x \cdot \frac{z \cdot y}{\mathsf{fma}\left(-0.5, \color{blue}{\frac{t}{\frac{z}{a}}}, z\right)} \]

   8. Simplified 28.1%

      \[\leadsto \color{blue}{x \cdot \frac{z \cdot y}{\mathsf{fma}\left(-0.5, \frac{t}{\frac{z}{a}}, z\right)}} \]

   9. Taylor expanded in y around 0 28.1%

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

       1. *-commutative 28.1%

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

       2. associate-*l/ 28.1%

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

       3. +-commutative 28.1%

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

       4. fma-def 28.1%

          \[\leadsto x \cdot \frac{y \cdot z}{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}} \]

       5. associate-/l* 28.1%

          \[\leadsto x \cdot \color{blue}{\frac{y}{\frac{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}{z}}} \]

       6. *-commutative 28.1%

          \[\leadsto x \cdot \frac{y}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{a \cdot \frac{t}{z}}, z\right)}{z}} \]

   11. Simplified 28.1%

       \[\leadsto x \cdot \color{blue}{\frac{y}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}} \]

   12. Taylor expanded in a around inf 23.9%

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

       1. associate-*r/ 25.2%

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

       2. *-commutative 25.2%

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

       3. associate-*l* 25.2%

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

   14. Simplified 25.2%

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

   if 1.6300000000000001e-130 < z

   1. Initial program 57.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 58.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 58.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 56.8%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 60.3%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 58.0%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 84.5%

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

4. Final simplification 48.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.63 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \frac{y}{\frac{a \cdot \left(-0.5 \cdot \frac{t}{z}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.63 \cdot 10^{-130}:\\ \;\;\;\;\frac{z_m}{-0.5 \cdot \left(\frac{a}{x_m} \cdot \frac{t}{z_m \cdot y_m}\right)}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.63e-130)
      (/ z_m (* -0.5 (* (/ a x_m) (/ t (* z_m y_m)))))
      (* 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);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.63e-130) {
		tmp = z_m / (-0.5 * ((a / x_m) * (t / (z_m * y_m))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.63d-130) then
        tmp = z_m / ((-0.5d0) * ((a / x_m) * (t / (z_m * y_m))))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.63e-130) {
		tmp = z_m / (-0.5 * ((a / x_m) * (t / (z_m * y_m))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.63e-130:
		tmp = z_m / (-0.5 * ((a / x_m) * (t / (z_m * y_m))))
	else:
		tmp = x_m * y_m
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.63e-130)
		tmp = Float64(z_m / Float64(-0.5 * Float64(Float64(a / x_m) * Float64(t / Float64(z_m * y_m)))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * 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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.63e-130)
		tmp = z_m / (-0.5 * ((a / x_m) * (t / (z_m * y_m))));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.63e-130], N[(z$95$m / N[(-0.5 * N[(N[(a / x$95$m), $MachinePrecision] * N[(t / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.6300000000000001e-130

   1. Initial program 63.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 28.1%

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

      1. associate-*r/ 28.1%

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

   5. Simplified 28.1%

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

   6. Taylor expanded in x around 0 28.1%

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

      1. associate-*r* 28.1%

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

      2. associate-*r/ 28.1%

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

      3. associate-*l* 28.1%

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

      4. associate-*r/ 28.1%

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

      5. associate-*l* 28.1%

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

      6. associate-*r/ 28.1%

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

      7. associate-/l* 28.1%

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

   8. Simplified 28.1%

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

   9. Taylor expanded in x around 0 28.1%

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

       1. associate-*r* 28.1%

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

       2. *-commutative 28.1%

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

       3. +-commutative 28.1%

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

       4. associate-/l* 28.0%

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

       5. *-commutative 28.0%

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

       6. fma-udef 28.0%

          \[\leadsto \frac{z \cdot \left(x \cdot y\right)}{\color{blue}{\mathsf{fma}\left(\frac{a}{\frac{z}{t}}, -0.5, z\right)}} \]

       7. associate-/l* 28.3%

          \[\leadsto \color{blue}{\frac{z}{\frac{\mathsf{fma}\left(\frac{a}{\frac{z}{t}}, -0.5, z\right)}{x \cdot y}}} \]

       8. fma-udef 28.3%

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

       9. associate-/l* 28.2%

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

       10. *-commutative 28.2%

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

       11. associate-*l/ 28.3%

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

       12. *-commutative 28.3%

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

       13. fma-def 28.3%

           \[\leadsto \frac{z}{\frac{\color{blue}{\mathsf{fma}\left(-0.5, \frac{t}{z} \cdot a, z\right)}}{x \cdot y}} \]

       14. *-commutative 28.3%

           \[\leadsto \frac{z}{\frac{\mathsf{fma}\left(-0.5, \color{blue}{a \cdot \frac{t}{z}}, z\right)}{x \cdot y}} \]

   11. Simplified 28.3%

       \[\leadsto \color{blue}{\frac{z}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{x \cdot y}}} \]

   12. Taylor expanded in a around inf 26.1%

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

       1. times-frac 26.9%

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

       2. *-commutative 26.9%

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

   14. Simplified 26.9%

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

   if 1.6300000000000001e-130 < z

   1. Initial program 57.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 58.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 58.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 56.8%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 60.3%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 58.0%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 58.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 84.5%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.63 \cdot 10^{-130}:\\ \;\;\;\;\frac{z}{-0.5 \cdot \left(\frac{a}{x} \cdot \frac{t}{z \cdot y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 79.5% accurate, 7.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (* (* x_m y_m) (/ z_m (+ z_m (* -0.5 (/ a (/ z_m t))))))))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * ((x_m * y_m) * (z_m / (z_m + ((-0.5d0) * (a / (z_m / t))))))))
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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * ((x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t))))))));
}

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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * ((x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t))))))))

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * ((x_m * y_m) * (z_m / (z_m + (-0.5 * (a / (z_m / t))))))));
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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(z$95$m + N[(-0.5 * N[(a / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.7%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
2. Add Preprocessing

3. Taylor expanded in z around inf 43.9%

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

   1. associate-*r/ 43.9%

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

5. Simplified 43.9%

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

6. Taylor expanded in x around 0 41.6%

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

   1. associate-*r* 43.9%

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

   2. associate-*r/ 43.9%

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

   3. associate-*l* 43.9%

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

   4. associate-*r/ 50.0%

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

   5. associate-*l* 50.0%

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

   6. associate-*r/ 50.0%

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

   7. associate-/l* 50.8%

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

8. Simplified 50.8%

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

9. Final simplification 50.8%

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

Alternative 13: 74.9% accurate, 12.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{y_m}{\frac{z_m}{z_m \cdot x_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= z_m 4e-159) (/ y_m (/ z_m (* z_m x_m))) (* 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);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e-159) {
		tmp = y_m / (z_m / (z_m * x_m));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 4d-159) then
        tmp = y_m / (z_m / (z_m * x_m))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4e-159) {
		tmp = y_m / (z_m / (z_m * x_m));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 4e-159:
		tmp = y_m / (z_m / (z_m * x_m))
	else:
		tmp = x_m * y_m
	return z_s * (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)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4e-159)
		tmp = Float64(y_m / Float64(z_m / Float64(z_m * x_m)));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * 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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4e-159)
		tmp = y_m / (z_m / (z_m * x_m));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4e-159], N[(y$95$m / N[(z$95$m / N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.99999999999999995e-159

   1. Initial program 61.9%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 63.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 63.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 60.1%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 60.7%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 62.0%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 62.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 21.3%

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

   if 3.99999999999999995e-159 < z

   1. Initial program 59.0%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. associate-*l/ 60.6%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

      2. *-commutative 60.6%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

      3. associate-/l* 58.8%

         \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

      4. associate-/r/ 62.2%

         \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

      5. associate-/r* 60.0%

         \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

   3. Simplified 60.0%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 83.5%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-159}:\\ \;\;\;\;\frac{y}{\frac{z}{z \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.8% accurate, 37.7× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (* z_s (* y_s (* x_s (* 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);
z_m = fabs(z);
z_s = copysign(1.0, z);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (x_m * y_m)));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    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_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * (x_m * y_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);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (x_m * y_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)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * (x_m * y_m)))

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(x_m * y_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);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (x_m * y_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]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.7%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
2. Step-by-step derivation

   1. associate-*l/ 62.2%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot z} \]

   2. *-commutative 62.2%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\sqrt{z \cdot z - t \cdot a}} \cdot z \]

   3. associate-/l* 59.5%

      \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}} \cdot z \]

   4. associate-/r/ 61.3%

      \[\leadsto \color{blue}{\frac{y}{\frac{\frac{\sqrt{z \cdot z - t \cdot a}}{x}}{z}}} \]

   5. associate-/r* 61.2%

      \[\leadsto \frac{y}{\color{blue}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]

3. Simplified 61.2%

   \[\leadsto \color{blue}{\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{x \cdot z}}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf 44.1%

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

6. Final simplification 44.1%

   \[\leadsto x \cdot y \]
7. Add Preprocessing

Developer target: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024013 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z -3.1921305903852764e+46) (- (* y x)) (if (< z 5.976268120920894e+90) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x)))

  (/ (* (* x y) z) (sqrt (- (* z z) (* t a)))))