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

Percentage Accurate: 61.1% → 92.3%

Time: 21.0s

Alternatives: 13

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: 61.1% 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: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_1 := \frac{z_m \cdot \left(x_m \cdot y_m\right)}{\sqrt{z_m \cdot z_m - a \cdot t}}\\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t_1 \leq 0:\\ \;\;\;\;y_m \cdot \frac{x_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \mathbf{elif}\;t_1 \leq 10^{+305}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \end{array} \]

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)
        (* y_m (/ x_m (/ (fma -0.5 (* a (/ t z_m)) z_m) z_m)))
        (if (<= t_1 1e+305) 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 = y_m * (x_m / (fma(-0.5, (a * (t / z_m)), z_m) / z_m));
	} else if (t_1 <= 1e+305) {
		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(y_m * Float64(x_m / Float64(fma(-0.5, Float64(a * Float64(t / z_m)), z_m) / z_m)));
	elseif (t_1 <= 1e+305)
		tmp = t_1;
	else
		tmp = Float64(x_m * y_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[(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[(y$95$m * N[(x$95$m / N[(N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+305], 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 63.7%

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

      1. associate-/l* 65.5%

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

      2. associate-*l/ 66.6%

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

      3. *-commutative 66.6%

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

      4. associate-/l* 66.9%

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

   3. Simplified 66.9%

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

      1. associate-/l* 66.6%

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

      2. div-inv 66.6%

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

      3. pow2 66.6%

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

   6. Applied egg-rr 66.6%

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

      1. associate-*r/ 66.6%

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

      2. *-rgt-identity 66.6%

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

      3. *-commutative 66.6%

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

   8. Simplified 66.6%

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

   9. Taylor expanded in z around inf 53.4%

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

       1. +-commutative 53.4%

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

       2. associate-/l* 55.5%

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

       3. fma-udef 55.5%

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

       4. associate-/l* 53.4%

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

       5. associate-*r/ 55.5%

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

   11. Simplified 55.5%

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

   if 0.0 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 9.9999999999999994e304

   1. Initial program 99.6%

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

   if 9.9999999999999994e304 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 11.9%

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

      1. associate-/l* 15.1%

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

      2. associate-*l/ 17.7%

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

      3. *-commutative 17.7%

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

      4. associate-/l* 16.7%

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

   3. Simplified 16.7%

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

   5. Taylor expanded in z around inf 50.2%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}} \leq 0:\\ \;\;\;\;y \cdot \frac{x}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \mathbf{elif}\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}} \leq 10^{+305}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.2 \cdot 10^{+66}:\\ \;\;\;\;x_m \cdot \left({\left({z_m}^{2} - a \cdot t\right)}^{-0.5} \cdot \left(z_m \cdot y_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.2e+66)
      (* x_m (* (pow (- (pow z_m 2.0) (* a t)) -0.5) (* z_m y_m)))
      (* y_m (/ x_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 <= 1.2e+66) {
		tmp = x_m * (pow((pow(z_m, 2.0) - (a * t)), -0.5) * (z_m * y_m));
	} else {
		tmp = y_m * (x_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 <= 1.2e+66)
		tmp = Float64(x_m * Float64((Float64((z_m ^ 2.0) - Float64(a * t)) ^ -0.5) * Float64(z_m * y_m)));
	else
		tmp = Float64(y_m * Float64(x_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, 1.2e+66], N[(x$95$m * N[(N[Power[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$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 < 1.2000000000000001e66

   1. Initial program 70.7%

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

      1. associate-*l* 68.6%

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

      2. *-commutative 68.6%

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

      3. associate-*l* 70.8%

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

      4. *-commutative 70.8%

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

      5. associate-/l* 71.2%

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

   3. Simplified 71.2%

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

      1. associate-/l* 70.8%

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

      2. div-inv 70.8%

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

      3. *-commutative 70.8%

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

      4. associate-*r* 68.7%

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

      5. *-commutative 68.7%

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

      6. associate-*r* 70.6%

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

      7. *-commutative 70.6%

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

      8. associate-*l* 69.6%

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

      9. pow1/2 69.6%

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

      10. pow-flip 69.6%

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

      11. pow2 69.6%

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

      12. metadata-eval 69.6%

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

   6. Applied egg-rr 69.6%

      \[\leadsto \color{blue}{z \cdot \left(\left(x \cdot y\right) \cdot {\left({z}^{2} - t \cdot a\right)}^{-0.5}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 70.6%

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

      2. *-commutative 70.6%

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

      3. associate-*r* 68.6%

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

      4. *-commutative 68.6%

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

      5. associate-*r* 68.6%

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

      6. *-commutative 68.6%

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

      7. *-commutative 68.6%

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

   8. Simplified 68.6%

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

   if 1.2000000000000001e66 < z

   1. Initial program 35.2%

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

      1. associate-/l* 38.5%

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

      2. associate-*l/ 38.6%

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

      3. *-commutative 38.6%

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

      4. associate-/l* 36.4%

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

   3. Simplified 36.4%

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

      1. associate-/l* 38.6%

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

      2. div-inv 38.6%

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

      3. pow2 38.6%

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

   6. Applied egg-rr 38.6%

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

      1. associate-*r/ 38.6%

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

      2. *-rgt-identity 38.6%

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

      3. *-commutative 38.6%

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

   8. Simplified 38.6%

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

   9. Taylor expanded in z around inf 89.8%

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

       1. +-commutative 89.8%

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

       2. associate-/l* 96.5%

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

       3. fma-udef 96.5%

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

       4. associate-/l* 89.8%

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

       5. associate-*r/ 96.5%

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

   11. Simplified 96.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+66}:\\ \;\;\;\;x \cdot \left({\left({z}^{2} - a \cdot t\right)}^{-0.5} \cdot \left(z \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.7 \cdot 10^{-146}:\\ \;\;\;\;\frac{z_m \cdot y_m}{\frac{\sqrt{a \cdot \left(-t\right)}}{x_m}}\\ \mathbf{elif}\;z_m \leq 6 \cdot 10^{+105}:\\ \;\;\;\;y_m \cdot \frac{z_m \cdot x_m}{\sqrt{z_m \cdot z_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.7e-146)
      (/ (* z_m y_m) (/ (sqrt (* a (- t))) x_m))
      (if (<= z_m 6e+105)
        (* y_m (/ (* z_m x_m) (sqrt (- (* z_m z_m) (* a t)))))
        (* y_m (/ x_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 <= 1.7e-146) {
		tmp = (z_m * y_m) / (sqrt((a * -t)) / x_m);
	} else if (z_m <= 6e+105) {
		tmp = y_m * ((z_m * x_m) / sqrt(((z_m * z_m) - (a * t))));
	} else {
		tmp = y_m * (x_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 <= 1.7e-146)
		tmp = Float64(Float64(z_m * y_m) / Float64(sqrt(Float64(a * Float64(-t))) / x_m));
	elseif (z_m <= 6e+105)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t)))));
	else
		tmp = Float64(y_m * Float64(x_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, 1.7e-146], N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 6e+105], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$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 < 1.7e-146

   1. Initial program 65.1%

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

      1. associate-/l* 65.7%

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

      2. associate-*l/ 67.1%

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

      3. *-commutative 67.1%

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

      4. associate-/l* 66.1%

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

   3. Simplified 66.1%

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

      1. associate-*r/ 65.3%

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

      2. *-commutative 65.3%

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

      3. associate-*r* 63.7%

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

      4. associate-/l* 61.1%

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

      5. *-commutative 61.1%

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

      6. pow2 61.1%

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

   6. Applied egg-rr 61.1%

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

   7. Taylor expanded in z around 0 38.1%

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

      1. mul-1-neg 38.1%

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

      2. distribute-rgt-neg-in 38.1%

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

   9. Simplified 38.1%

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

   if 1.7e-146 < z < 6.0000000000000001e105

   1. Initial program 95.1%

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

      1. associate-/l* 97.7%

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

      2. associate-*l/ 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-/l* 94.9%

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

   3. Simplified 94.9%

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

   if 6.0000000000000001e105 < z

   1. Initial program 21.1%

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

      1. associate-/l* 23.4%

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

      2. associate-*l/ 23.5%

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

      3. *-commutative 23.5%

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

      4. associate-/l* 22.7%

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

   3. Simplified 22.7%

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

      1. associate-/l* 23.5%

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

      2. div-inv 23.5%

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

      3. pow2 23.5%

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

   6. Applied egg-rr 23.5%

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

      1. associate-*r/ 23.5%

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

      2. *-rgt-identity 23.5%

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

      3. *-commutative 23.5%

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

   8. Simplified 23.5%

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

   9. Taylor expanded in z around inf 87.3%

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

       1. +-commutative 87.3%

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

       2. associate-/l* 95.6%

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

       3. fma-udef 95.6%

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

       4. associate-/l* 87.3%

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

       5. associate-*r/ 95.6%

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

   11. Simplified 95.6%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-146}:\\ \;\;\;\;\frac{z \cdot y}{\frac{\sqrt{a \cdot \left(-t\right)}}{x}}\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{z_m \cdot y_m}{\frac{\sqrt{a \cdot \left(-t\right)}}{x_m}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.12e-60)
      (/ (* z_m y_m) (/ (sqrt (* a (- t))) x_m))
      (* y_m (/ x_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 <= 1.12e-60) {
		tmp = (z_m * y_m) / (sqrt((a * -t)) / x_m);
	} else {
		tmp = y_m * (x_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 <= 1.12e-60)
		tmp = Float64(Float64(z_m * y_m) / Float64(sqrt(Float64(a * Float64(-t))) / x_m));
	else
		tmp = Float64(y_m * Float64(x_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, 1.12e-60], N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$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 < 1.12e-60

   1. Initial program 67.4%

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

      1. associate-/l* 68.1%

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

      2. associate-*l/ 69.4%

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

      3. *-commutative 69.4%

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

      4. associate-/l* 68.4%

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

   3. Simplified 68.4%

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

      1. associate-*r/ 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-*r* 65.2%

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

      4. associate-/l* 62.8%

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

      5. *-commutative 62.8%

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

      6. pow2 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in z around 0 40.7%

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

      1. mul-1-neg 40.7%

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

      2. distribute-rgt-neg-in 40.7%

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

   9. Simplified 40.7%

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

   if 1.12e-60 < z

   1. Initial program 49.6%

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

      1. associate-/l* 52.2%

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

      2. associate-*l/ 51.9%

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

      3. *-commutative 51.9%

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

      4. associate-/l* 50.2%

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

   3. Simplified 50.2%

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

      1. associate-/l* 51.9%

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

      2. div-inv 51.9%

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

      3. pow2 51.9%

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

   6. Applied egg-rr 51.9%

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

      1. associate-*r/ 51.9%

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

      2. *-rgt-identity 51.9%

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

      3. *-commutative 51.9%

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

   8. Simplified 51.9%

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

   9. Taylor expanded in z around inf 86.9%

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

       1. +-commutative 86.9%

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

       2. associate-/l* 92.1%

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

       3. fma-udef 92.1%

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

       4. associate-/l* 86.9%

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

       5. associate-*r/ 92.1%

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

   11. Simplified 92.1%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{z \cdot y}{\frac{\sqrt{a \cdot \left(-t\right)}}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 1.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{z_m \cdot z_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z_m}, z_m\right)}{z_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.8e+66)
      (/ (* x_m (* z_m y_m)) (sqrt (- (* z_m z_m) (* a t))))
      (* y_m (/ x_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 <= 1.8e+66) {
		tmp = (x_m * (z_m * y_m)) / sqrt(((z_m * z_m) - (a * t)));
	} else {
		tmp = y_m * (x_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 <= 1.8e+66)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))));
	else
		tmp = Float64(y_m * Float64(x_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, 1.8e+66], 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[(y$95$m * N[(x$95$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 < 1.8e66

   1. Initial program 70.7%

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

      1. associate-*l* 68.6%

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

   3. Simplified 68.6%

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

   if 1.8e66 < z

   1. Initial program 35.2%

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

      1. associate-/l* 38.5%

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

      2. associate-*l/ 38.6%

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

      3. *-commutative 38.6%

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

      4. associate-/l* 36.4%

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

   3. Simplified 36.4%

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

      1. associate-/l* 38.6%

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

      2. div-inv 38.6%

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

      3. pow2 38.6%

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

   6. Applied egg-rr 38.6%

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

      1. associate-*r/ 38.6%

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

      2. *-rgt-identity 38.6%

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

      3. *-commutative 38.6%

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

   8. Simplified 38.6%

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

   9. Taylor expanded in z around inf 89.8%

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

       1. +-commutative 89.8%

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

       2. associate-/l* 96.5%

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

       3. fma-udef 96.5%

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

       4. associate-/l* 89.8%

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

       5. associate-*r/ 96.5%

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

   11. Simplified 96.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.4 \cdot 10^{-61}:\\ \;\;\;\;y_m \cdot \frac{z_m \cdot x_m}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

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 2.4e-61)
      (* y_m (/ (* z_m x_m) (sqrt (* 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 <= 2.4e-61) {
		tmp = y_m * ((z_m * x_m) / sqrt((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 <= 2.4d-61) then
        tmp = y_m * ((z_m * x_m) / sqrt((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 <= 2.4e-61) {
		tmp = y_m * ((z_m * x_m) / Math.sqrt((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 <= 2.4e-61:
		tmp = y_m * ((z_m * x_m) / math.sqrt((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 <= 2.4e-61)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / sqrt(Float64(a * Float64(-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 <= 2.4e-61)
		tmp = y_m * ((z_m * x_m) / sqrt((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, 2.4e-61], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $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 < 2.4000000000000001e-61

   1. Initial program 67.4%

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

      1. associate-/l* 68.1%

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

      2. associate-*l/ 69.4%

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

      3. *-commutative 69.4%

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

      4. associate-/l* 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in z around 0 44.5%

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

      1. mul-1-neg 44.5%

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

      2. *-commutative 44.5%

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

      3. distribute-rgt-neg-in 44.5%

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

   7. Simplified 44.5%

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

   if 2.4000000000000001e-61 < z

   1. Initial program 49.6%

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

      1. associate-/l* 52.2%

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

      2. associate-*l/ 51.9%

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

      3. *-commutative 51.9%

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

      4. associate-/l* 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in z around inf 90.2%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-61}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 2.35 \cdot 10^{-60}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

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 2.35e-60)
      (/ (* x_m (* z_m y_m)) (sqrt (* 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 <= 2.35e-60) {
		tmp = (x_m * (z_m * y_m)) / sqrt((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 <= 2.35d-60) then
        tmp = (x_m * (z_m * y_m)) / sqrt((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 <= 2.35e-60) {
		tmp = (x_m * (z_m * y_m)) / Math.sqrt((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 <= 2.35e-60:
		tmp = (x_m * (z_m * y_m)) / math.sqrt((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 <= 2.35e-60)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(a * Float64(-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 <= 2.35e-60)
		tmp = (x_m * (z_m * y_m)) / sqrt((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, 2.35e-60], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.35e-60

   1. Initial program 67.4%

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

      1. associate-*l* 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in z around 0 42.4%

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

      1. mul-1-neg 44.5%

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

      2. *-commutative 44.5%

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

      3. distribute-rgt-neg-in 44.5%

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

   7. Simplified 42.4%

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

   if 2.35e-60 < z

   1. Initial program 49.6%

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

      1. associate-/l* 52.2%

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

      2. associate-*l/ 51.9%

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

      3. *-commutative 51.9%

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

      4. associate-/l* 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in z around inf 90.2%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{-60}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{a \cdot \left(-t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3 \cdot 10^{-61}:\\ \;\;\;\;\frac{z_m \cdot y_m}{\frac{\sqrt{a \cdot \left(-t\right)}}{x_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

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 3e-61)
      (/ (* z_m y_m) (/ (sqrt (* a (- t))) 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 <= 3e-61) {
		tmp = (z_m * y_m) / (sqrt((a * -t)) / 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 <= 3d-61) then
        tmp = (z_m * y_m) / (sqrt((a * -t)) / 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 <= 3e-61) {
		tmp = (z_m * y_m) / (Math.sqrt((a * -t)) / 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 <= 3e-61:
		tmp = (z_m * y_m) / (math.sqrt((a * -t)) / 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 <= 3e-61)
		tmp = Float64(Float64(z_m * y_m) / Float64(sqrt(Float64(a * Float64(-t))) / 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 <= 3e-61)
		tmp = (z_m * y_m) / (sqrt((a * -t)) / 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, 3e-61], N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(N[Sqrt[N[(a * (-t)), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.00000000000000012e-61

   1. Initial program 67.4%

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

      1. associate-/l* 68.1%

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

      2. associate-*l/ 69.4%

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

      3. *-commutative 69.4%

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

      4. associate-/l* 68.4%

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

   3. Simplified 68.4%

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

      1. associate-*r/ 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-*r* 65.2%

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

      4. associate-/l* 62.8%

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

      5. *-commutative 62.8%

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

      6. pow2 62.8%

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

   6. Applied egg-rr 62.8%

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

   7. Taylor expanded in z around 0 40.7%

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

      1. mul-1-neg 40.7%

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

      2. distribute-rgt-neg-in 40.7%

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

   9. Simplified 40.7%

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

   if 3.00000000000000012e-61 < z

   1. Initial program 49.6%

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

      1. associate-/l* 52.2%

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

      2. associate-*l/ 51.9%

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

      3. *-commutative 51.9%

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

      4. associate-/l* 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in z around inf 90.2%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{-61}:\\ \;\;\;\;\frac{z \cdot y}{\frac{\sqrt{a \cdot \left(-t\right)}}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+48}:\\ \;\;\;\;y_m \cdot \frac{z_m \cdot x_m}{z_m + -0.5 \cdot \frac{1}{\frac{\frac{z_m}{a}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;y_m \cdot \frac{x_m}{\frac{z_m + -0.5 \cdot \frac{a \cdot t}{z_m}}{z_m}}\\ \end{array}\right)\right) \end{array} \]

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 (<= t -5e+48)
      (* y_m (/ (* z_m x_m) (+ z_m (* -0.5 (/ 1.0 (/ (/ z_m a) t))))))
      (* y_m (/ x_m (/ (+ z_m (* -0.5 (/ (* a t) 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 (t <= -5e+48) {
		tmp = y_m * ((z_m * x_m) / (z_m + (-0.5 * (1.0 / ((z_m / a) / t)))));
	} else {
		tmp = y_m * (x_m / ((z_m + (-0.5 * ((a * t) / z_m))) / z_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 (t <= (-5d+48)) then
        tmp = y_m * ((z_m * x_m) / (z_m + ((-0.5d0) * (1.0d0 / ((z_m / a) / t)))))
    else
        tmp = y_m * (x_m / ((z_m + ((-0.5d0) * ((a * t) / z_m))) / z_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 (t <= -5e+48) {
		tmp = y_m * ((z_m * x_m) / (z_m + (-0.5 * (1.0 / ((z_m / a) / t)))));
	} else {
		tmp = y_m * (x_m / ((z_m + (-0.5 * ((a * t) / z_m))) / z_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 t <= -5e+48:
		tmp = y_m * ((z_m * x_m) / (z_m + (-0.5 * (1.0 / ((z_m / a) / t)))))
	else:
		tmp = y_m * (x_m / ((z_m + (-0.5 * ((a * t) / 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 (t <= -5e+48)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / Float64(z_m + Float64(-0.5 * Float64(1.0 / Float64(Float64(z_m / a) / t))))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z_m + Float64(-0.5 * Float64(Float64(a * t) / z_m))) / z_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 (t <= -5e+48)
		tmp = y_m * ((z_m * x_m) / (z_m + (-0.5 * (1.0 / ((z_m / a) / t)))));
	else
		tmp = y_m * (x_m / ((z_m + (-0.5 * ((a * t) / z_m))) / z_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[t, -5e+48], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(1.0 / N[(N[(z$95$m / a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(N[(z$95$m + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -4.99999999999999973e48

   1. Initial program 58.9%

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

      1. associate-/l* 61.1%

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

      2. associate-*l/ 61.3%

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

      3. *-commutative 61.3%

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

      4. associate-/l* 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in z around inf 47.0%

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

      1. clear-num 47.0%

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

      2. inv-pow 47.0%

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

   7. Applied egg-rr 47.0%

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

      1. unpow-1 47.0%

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

      2. associate-/r* 48.6%

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

   9. Simplified 48.6%

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

   if -4.99999999999999973e48 < t

   1. Initial program 62.3%

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

      1. associate-/l* 63.4%

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

      2. associate-*l/ 64.3%

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

      3. *-commutative 64.3%

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

      4. associate-/l* 62.8%

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

   3. Simplified 62.8%

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

      1. associate-/l* 64.3%

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

      2. div-inv 64.2%

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

      3. pow2 64.2%

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

   6. Applied egg-rr 64.2%

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

      1. associate-*r/ 64.3%

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

      2. *-rgt-identity 64.3%

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

      3. *-commutative 64.3%

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

   8. Simplified 64.3%

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

   9. Taylor expanded in z around inf 44.3%

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

4. Final simplification 45.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{1}{\frac{\frac{z}{a}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \frac{a \cdot t}{z}}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 10^{+67}:\\ \;\;\;\;y_m \cdot \frac{x_m}{\frac{z_m + -0.5 \cdot \frac{a \cdot t}{z_m}}{z_m}}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

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 1e+67)
      (* y_m (/ x_m (/ (+ z_m (* -0.5 (/ (* a 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 <= 1e+67) {
		tmp = y_m * (x_m / ((z_m + (-0.5 * ((a * 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 <= 1d+67) then
        tmp = y_m * (x_m / ((z_m + ((-0.5d0) * ((a * 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 <= 1e+67) {
		tmp = y_m * (x_m / ((z_m + (-0.5 * ((a * 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 <= 1e+67:
		tmp = y_m * (x_m / ((z_m + (-0.5 * ((a * 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 <= 1e+67)
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z_m + Float64(-0.5 * Float64(Float64(a * 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 <= 1e+67)
		tmp = y_m * (x_m / ((z_m + (-0.5 * ((a * 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, 1e+67], N[(y$95$m * N[(x$95$m / N[(N[(z$95$m + N[(-0.5 * N[(N[(a * t), $MachinePrecision] / 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 < 9.99999999999999983e66

   1. Initial program 70.7%

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

      1. associate-/l* 71.3%

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

      2. associate-*l/ 72.3%

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

      3. *-commutative 72.3%

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

      4. associate-/l* 71.4%

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

   3. Simplified 71.4%

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

      1. associate-/l* 72.3%

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

      2. div-inv 72.3%

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

      3. pow2 72.3%

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

   6. Applied egg-rr 72.3%

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

      1. associate-*r/ 72.3%

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

      2. *-rgt-identity 72.3%

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

      3. *-commutative 72.3%

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

   8. Simplified 72.3%

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

   9. Taylor expanded in z around inf 31.3%

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

   if 9.99999999999999983e66 < z

   1. Initial program 35.2%

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

      1. associate-/l* 38.5%

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

      2. associate-*l/ 38.6%

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

      3. *-commutative 38.6%

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

      4. associate-/l* 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in z around inf 96.2%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+67}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + -0.5 \cdot \frac{a \cdot t}{z}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 3.6 \cdot 10^{-125}:\\ \;\;\;\;y_m \cdot \frac{z_m \cdot x_m}{z_m}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

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 3.6e-125) (* y_m (/ (* z_m x_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 <= 3.6e-125) {
		tmp = y_m * ((z_m * x_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 <= 3.6d-125) then
        tmp = y_m * ((z_m * x_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 <= 3.6e-125) {
		tmp = y_m * ((z_m * x_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 <= 3.6e-125:
		tmp = y_m * ((z_m * x_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 <= 3.6e-125)
		tmp = Float64(y_m * Float64(Float64(z_m * x_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 <= 3.6e-125)
		tmp = y_m * ((z_m * x_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, 3.6e-125], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.6000000000000002e-125

   1. Initial program 66.2%

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

      1. associate-/l* 66.8%

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

      2. associate-*l/ 68.1%

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

      3. *-commutative 68.1%

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

      4. associate-/l* 67.1%

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

   3. Simplified 67.1%

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

   5. Taylor expanded in z around inf 16.6%

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

   if 3.6000000000000002e-125 < z

   1. Initial program 53.6%

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

      1. associate-/l* 56.2%

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

      2. associate-*l/ 55.9%

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

      3. *-commutative 55.9%

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

      4. associate-/l* 54.4%

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

   3. Simplified 54.4%

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

   5. Taylor expanded in z around inf 85.4%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-125}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.3% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ z_s \cdot \left(y_s \cdot \left(x_s \cdot \begin{array}{l} \mathbf{if}\;z_m \leq 6 \cdot 10^{-70}:\\ \;\;\;\;\frac{x_m \cdot \left(z_m \cdot y_m\right)}{z_m}\\ \mathbf{else}:\\ \;\;\;\;x_m \cdot y_m\\ \end{array}\right)\right) \end{array} \]

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 6e-70) (/ (* x_m (* z_m y_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 <= 6e-70) {
		tmp = (x_m * (z_m * y_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 <= 6d-70) then
        tmp = (x_m * (z_m * y_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 <= 6e-70) {
		tmp = (x_m * (z_m * y_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 <= 6e-70:
		tmp = (x_m * (z_m * y_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 <= 6e-70)
		tmp = Float64(Float64(x_m * Float64(z_m * y_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 <= 6e-70)
		tmp = (x_m * (z_m * y_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, 6e-70], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.0000000000000003e-70

   1. Initial program 67.4%

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

      1. associate-*l* 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in z around inf 23.7%

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

   if 6.0000000000000003e-70 < z

   1. Initial program 49.6%

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

      1. associate-/l* 52.2%

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

      2. associate-*l/ 51.9%

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

      3. *-commutative 51.9%

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

      4. associate-/l* 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in z around inf 90.2%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-70}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.2% 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 61.5%

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

   1. associate-/l* 62.8%

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

   2. associate-*l/ 63.6%

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

   3. *-commutative 63.6%

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

   4. associate-/l* 62.4%

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

3. Simplified 62.4%

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

5. Taylor expanded in z around inf 40.8%

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

6. Final simplification 40.8%

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

Developer target: 89.2% accurate, 0.9× 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 2024018 
(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)))))