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

Percentage Accurate: 61.5% → 92.1%

Time: 19.9s

Alternatives: 15

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: 61.5% 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: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+15}:\\ \;\;\;\;\frac{x\_m}{\sqrt{{z\_m}^{2} - t \cdot a}} \cdot \left(z\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z\_m}, z\_m\right)}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(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+15)
      (* (/ x_m (sqrt (- (pow z_m 2.0) (* t a)))) (* z_m y_m))
      (* y_m (* x_m (/ z_m (fma -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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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+15) {
		tmp = (x_m / sqrt((pow(z_m, 2.0) - (t * a)))) * (z_m * y_m);
	} else {
		tmp = y_m * (x_m * (z_m / fma(-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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e+15)
		tmp = Float64(Float64(x_m / sqrt(Float64((z_m ^ 2.0) - Float64(t * a)))) * Float64(z_m * y_m));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / fma(-0.5, Float64(a * Float64(t / 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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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+15], N[(N[(x$95$m / N[Sqrt[N[(N[Power[z$95$m, 2.0], $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e15

   1. Initial program 62.1%

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

      1. associate-*l* 63.8%

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

      2. *-commutative 63.8%

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

      3. associate-*l* 62.0%

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

      4. *-commutative 62.0%

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

      5. associate-/l* 62.7%

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

   3. Simplified 62.7%

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

      1. associate-/l* 62.0%

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

      2. *-commutative 62.0%

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

      3. associate-*r* 63.8%

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

      4. *-commutative 63.8%

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

      5. associate-/l* 67.2%

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

      6. associate-/r/ 65.3%

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

      7. pow2 65.3%

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

      8. *-commutative 65.3%

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

   6. Applied egg-rr 65.3%

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

   if 1e15 < z

   1. Initial program 46.0%

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

      1. associate-/l* 50.4%

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

      2. associate-*l/ 50.5%

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

      3. *-commutative 50.5%

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

      4. associate-/l* 47.9%

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

   3. Simplified 47.9%

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

   5. Taylor expanded in z around inf 77.2%

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

      1. associate-/l* 81.5%

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

   7. Simplified 81.5%

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

      1. *-un-lft-identity 81.5%

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

      2. times-frac 96.7%

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

      3. +-commutative 96.7%

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

      4. *-commutative 96.7%

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

      5. fma-def 96.7%

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

      6. div-inv 96.7%

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

      7. clear-num 96.7%

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

   9. Applied egg-rr 96.7%

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

   10. Taylor expanded in y around 0 70.9%

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

       1. associate-*r* 75.4%

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

       2. remove-double-neg 75.4%

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

       3. distribute-rgt-neg-in 75.4%

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

       4. *-commutative 75.4%

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

       5. associate-*l* 67.8%

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

       6. distribute-rgt-neg-in 67.8%

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

       7. distribute-rgt-neg-in 67.8%

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

       8. remove-double-neg 67.8%

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

       9. +-commutative 67.8%

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

       10. associate-*r/ 72.0%

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

       11. fma-udef 72.0%

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

       12. associate-*r/ 81.5%

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

       13. associate-*r/ 96.7%

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

   12. Simplified 96.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+15}:\\ \;\;\;\;\frac{x}{\sqrt{{z}^{2} - t \cdot a}} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \sqrt{t \cdot \left(-a\right)}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{t\_1}\\ \mathbf{elif}\;z\_m \leq 1.3 \cdot 10^{-60} \lor \neg \left(z\_m \leq 4.9 \cdot 10^{-48}\right):\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z\_m}, z\_m\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;z\_m \cdot \frac{y\_m}{\frac{t\_1}{x\_m}}\\ \end{array}\right)\right) \end{array} \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (sqrt (* t (- a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 6.6e-103)
        (/ (* x_m (* z_m y_m)) t_1)
        (if (or (<= z_m 1.3e-60) (not (<= z_m 4.9e-48)))
          (* y_m (* x_m (/ z_m (fma -0.5 (* a (/ t z_m)) z_m))))
          (* z_m (/ y_m (/ t_1 x_m))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
y_m = fabs(y);
y_s = copysign(1.0, y);
z_m = fabs(z);
z_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 = sqrt((t * -a));
	double tmp;
	if (z_m <= 6.6e-103) {
		tmp = (x_m * (z_m * y_m)) / t_1;
	} else if ((z_m <= 1.3e-60) || !(z_m <= 4.9e-48)) {
		tmp = y_m * (x_m * (z_m / fma(-0.5, (a * (t / z_m)), z_m)));
	} else {
		tmp = z_m * (y_m / (t_1 / x_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
y_m = abs(y)
y_s = copysign(1.0, y)
z_m = abs(z)
z_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(Float64(t * Float64(-a)))
	tmp = 0.0
	if (z_m <= 6.6e-103)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / t_1);
	elseif ((z_m <= 1.3e-60) || !(z_m <= 4.9e-48))
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / fma(-0.5, Float64(a * Float64(t / z_m)), z_m))));
	else
		tmp = Float64(z_m * Float64(y_m / Float64(t_1 / x_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6.6e-103], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[Or[LessEqual[z$95$m, 1.3e-60], N[Not[LessEqual[z$95$m, 4.9e-48]], $MachinePrecision]], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z$95$m * N[(y$95$m / N[(t$95$1 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 6.59999999999999979e-103

   1. Initial program 57.8%

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

      1. associate-*l* 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in z around 0 39.8%

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

      1. mul-1-neg 40.3%

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

      2. *-commutative 40.3%

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

      3. distribute-rgt-neg-in 40.3%

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

   7. Simplified 39.8%

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

   if 6.59999999999999979e-103 < z < 1.2999999999999999e-60 or 4.9000000000000002e-48 < z

   1. Initial program 52.7%

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

      1. associate-/l* 57.3%

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

      2. associate-*l/ 57.4%

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

      3. *-commutative 57.4%

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

      4. associate-/l* 54.3%

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

   3. Simplified 54.3%

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

   5. Taylor expanded in z around inf 78.5%

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

      1. associate-/l* 82.2%

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

   7. Simplified 82.2%

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

      1. *-un-lft-identity 82.2%

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

      2. times-frac 96.2%

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

      3. +-commutative 96.2%

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

      4. *-commutative 96.2%

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

      5. fma-def 96.2%

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

      6. div-inv 96.2%

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

      7. clear-num 96.2%

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

   9. Applied egg-rr 96.2%

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

   10. Taylor expanded in y around 0 73.1%

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

       1. associate-*r* 77.0%

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

       2. remove-double-neg 77.0%

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

       3. distribute-rgt-neg-in 77.0%

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

       4. *-commutative 77.0%

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

       5. associate-*l* 69.6%

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

       6. distribute-rgt-neg-in 69.6%

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

       7. distribute-rgt-neg-in 69.6%

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

       8. remove-double-neg 69.6%

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

       9. +-commutative 69.6%

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

       10. associate-*r/ 73.2%

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

       11. fma-udef 73.2%

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

       12. associate-*r/ 82.2%

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

       13. associate-*r/ 96.2%

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

   12. Simplified 96.2%

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

   if 1.2999999999999999e-60 < z < 4.9000000000000002e-48

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 84.4%

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

      4. *-commutative 84.4%

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

      5. associate-/l* 60.2%

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

   3. Simplified 60.2%

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

      1. associate-/r* 75.8%

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

      2. associate-/r/ 99.0%

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

      3. pow2 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in z around 0 76.2%

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

      1. neg-mul-1 76.2%

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

      2. distribute-rgt-neg-in 76.2%

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

   9. Simplified 76.2%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.6 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-60} \lor \neg \left(z \leq 4.9 \cdot 10^{-48}\right):\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{\frac{\sqrt{t \cdot \left(-a\right)}}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := y\_m \cdot \frac{z\_m \cdot x\_m}{\sqrt{t \cdot \left(-a\right)}}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.1 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z\_m \leq 4.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{z\_m \cdot \left(x\_m \cdot y\_m\right)}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\\ \mathbf{elif}\;z\_m \leq 2.2 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (* y_m (/ (* z_m x_m) (sqrt (* t (- a)))))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 3.1e-104)
        t_1
        (if (<= z_m 4.3e-60)
          (/ (* z_m (* x_m y_m)) (+ z_m (* -0.5 (/ (* t a) z_m))))
          (if (<= z_m 2.2e-47) 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 = y_m * ((z_m * x_m) / sqrt((t * -a)));
	double tmp;
	if (z_m <= 3.1e-104) {
		tmp = t_1;
	} else if (z_m <= 4.3e-60) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	} else if (z_m <= 2.2e-47) {
		tmp = t_1;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y_m * ((z_m * x_m) / sqrt((t * -a)))
    if (z_m <= 3.1d-104) then
        tmp = t_1
    else if (z_m <= 4.3d-60) then
        tmp = (z_m * (x_m * y_m)) / (z_m + ((-0.5d0) * ((t * a) / z_m)))
    else if (z_m <= 2.2d-47) then
        tmp = t_1
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = y_m * ((z_m * x_m) / Math.sqrt((t * -a)));
	double tmp;
	if (z_m <= 3.1e-104) {
		tmp = t_1;
	} else if (z_m <= 4.3e-60) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	} else if (z_m <= 2.2e-47) {
		tmp = t_1;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = y_m * ((z_m * x_m) / math.sqrt((t * -a)))
	tmp = 0
	if z_m <= 3.1e-104:
		tmp = t_1
	elif z_m <= 4.3e-60:
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / z_m)))
	elif z_m <= 2.2e-47:
		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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(y_m * Float64(Float64(z_m * x_m) / sqrt(Float64(t * Float64(-a)))))
	tmp = 0.0
	if (z_m <= 3.1e-104)
		tmp = t_1;
	elseif (z_m <= 4.3e-60)
		tmp = Float64(Float64(z_m * Float64(x_m * y_m)) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / z_m))));
	elseif (z_m <= 2.2e-47)
		tmp = t_1;
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = y_m * ((z_m * x_m) / sqrt((t * -a)));
	tmp = 0.0;
	if (z_m <= 3.1e-104)
		tmp = t_1;
	elseif (z_m <= 4.3e-60)
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / z_m)));
	elseif (z_m <= 2.2e-47)
		tmp = t_1;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.1e-104], t$95$1, If[LessEqual[z$95$m, 4.3e-60], N[(N[(z$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.2e-47], t$95$1, N[(x$95$m * y$95$m), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 3.09999999999999976e-104 or 4.3000000000000001e-60 < z < 2.20000000000000019e-47

   1. Initial program 58.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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/ 63.7%

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

      3. *-commutative 63.7%

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

      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 0 41.1%

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

      1. mul-1-neg 41.1%

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

      2. *-commutative 41.1%

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

      3. distribute-rgt-neg-in 41.1%

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

   7. Simplified 41.1%

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

   if 3.09999999999999976e-104 < z < 4.3000000000000001e-60

   1. Initial program 87.9%

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

   3. Taylor expanded in z around inf 64.5%

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

   if 2.20000000000000019e-47 < z

   1. Initial program 50.8%

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

      1. associate-/l* 54.8%

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

      2. associate-*l/ 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-/l* 52.5%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in z around inf 95.3%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.1 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-47}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := z\_m \cdot \frac{y\_m}{\frac{\sqrt{t \cdot \left(-a\right)}}{x\_m}}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.5 \cdot 10^{-103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z\_m \leq 1.7 \cdot 10^{-60}:\\ \;\;\;\;\frac{z\_m \cdot \left(x\_m \cdot y\_m\right)}{z\_m + -0.5 \cdot \frac{a}{\frac{z\_m}{t}}}\\ \mathbf{elif}\;z\_m \leq 7.2 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (* z_m (/ y_m (/ (sqrt (* t (- a))) x_m)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 2.5e-103)
        t_1
        (if (<= z_m 1.7e-60)
          (/ (* z_m (* x_m y_m)) (+ z_m (* -0.5 (/ a (/ z_m t)))))
          (if (<= z_m 7.2e-48) 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 * (y_m / (sqrt((t * -a)) / x_m));
	double tmp;
	if (z_m <= 2.5e-103) {
		tmp = t_1;
	} else if (z_m <= 1.7e-60) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * (a / (z_m / t))));
	} else if (z_m <= 7.2e-48) {
		tmp = t_1;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
y_m = abs(y)
y_s = copysign(1.0d0, y)
z_m = abs(z)
z_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z_m * (y_m / (sqrt((t * -a)) / x_m))
    if (z_m <= 2.5d-103) then
        tmp = t_1
    else if (z_m <= 1.7d-60) then
        tmp = (z_m * (x_m * y_m)) / (z_m + ((-0.5d0) * (a / (z_m / t))))
    else if (z_m <= 7.2d-48) then
        tmp = t_1
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
y_m = Math.abs(y);
y_s = Math.copySign(1.0, y);
z_m = Math.abs(z);
z_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = z_m * (y_m / (Math.sqrt((t * -a)) / x_m));
	double tmp;
	if (z_m <= 2.5e-103) {
		tmp = t_1;
	} else if (z_m <= 1.7e-60) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * (a / (z_m / t))));
	} else if (z_m <= 7.2e-48) {
		tmp = t_1;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
y_m = math.fabs(y)
y_s = math.copysign(1.0, y)
z_m = math.fabs(z)
z_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = z_m * (y_m / (math.sqrt((t * -a)) / x_m))
	tmp = 0
	if z_m <= 2.5e-103:
		tmp = t_1
	elif z_m <= 1.7e-60:
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * (a / (z_m / t))))
	elif z_m <= 7.2e-48:
		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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = Float64(z_m * Float64(y_m / Float64(sqrt(Float64(t * Float64(-a))) / x_m)))
	tmp = 0.0
	if (z_m <= 2.5e-103)
		tmp = t_1;
	elseif (z_m <= 1.7e-60)
		tmp = Float64(Float64(z_m * Float64(x_m * y_m)) / Float64(z_m + Float64(-0.5 * Float64(a / Float64(z_m / t)))));
	elseif (z_m <= 7.2e-48)
		tmp = t_1;
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
y_m = abs(y);
y_s = sign(y) * abs(1.0);
z_m = abs(z);
z_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = z_m * (y_m / (sqrt((t * -a)) / x_m));
	tmp = 0.0;
	if (z_m <= 2.5e-103)
		tmp = t_1;
	elseif (z_m <= 1.7e-60)
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * (a / (z_m / t))));
	elseif (z_m <= 7.2e-48)
		tmp = t_1;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y_m = N[Abs[y], $MachinePrecision]
y_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z_m = N[Abs[z], $MachinePrecision]
z_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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[(z$95$m * N[(y$95$m / N[(N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.5e-103], t$95$1, If[LessEqual[z$95$m, 1.7e-60], N[(N[(z$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(a / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 7.2e-48], t$95$1, N[(x$95$m * y$95$m), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 2.49999999999999983e-103 or 1.70000000000000003e-60 < z < 7.2000000000000003e-48

   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* 60.8%

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

      2. *-commutative 60.8%

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

      3. associate-*l* 59.5%

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

      4. *-commutative 59.5%

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

      5. associate-/l* 60.2%

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

   3. Simplified 60.2%

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

      1. associate-/r* 60.8%

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

      2. associate-/r/ 57.8%

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

      3. pow2 57.8%

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

   6. Applied egg-rr 57.8%

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

   7. Taylor expanded in z around 0 40.3%

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

      1. neg-mul-1 40.3%

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

      2. distribute-rgt-neg-in 40.3%

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

   9. Simplified 40.3%

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

   if 2.49999999999999983e-103 < z < 1.70000000000000003e-60

   1. Initial program 84.2%

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

   3. Taylor expanded in z around inf 84.2%

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

      1. associate-/l* 83.7%

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

   5. Simplified 84.2%

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

   if 7.2000000000000003e-48 < z

   1. Initial program 50.8%

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

      1. associate-/l* 54.8%

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

      2. associate-*l/ 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-/l* 52.5%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in z around inf 95.3%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-103}:\\ \;\;\;\;z \cdot \frac{y}{\frac{\sqrt{t \cdot \left(-a\right)}}{x}}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-60}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-48}:\\ \;\;\;\;z \cdot \frac{y}{\frac{\sqrt{t \cdot \left(-a\right)}}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \sqrt{t \cdot \left(-a\right)}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.15 \cdot 10^{-103}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{t\_1}\\ \mathbf{elif}\;z\_m \leq 7 \cdot 10^{-61}:\\ \;\;\;\;\frac{z\_m \cdot \left(x\_m \cdot y\_m\right)}{z\_m + -0.5 \cdot \frac{a}{\frac{z\_m}{t}}}\\ \mathbf{elif}\;z\_m \leq 4.9 \cdot 10^{-48}:\\ \;\;\;\;z\_m \cdot \frac{y\_m}{\frac{t\_1}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (sqrt (* t (- a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 2.15e-103)
        (/ (* x_m (* z_m y_m)) t_1)
        (if (<= z_m 7e-61)
          (/ (* z_m (* x_m y_m)) (+ z_m (* -0.5 (/ a (/ z_m t)))))
          (if (<= z_m 4.9e-48) (* z_m (/ y_m (/ t_1 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 = sqrt((t * -a));
	double tmp;
	if (z_m <= 2.15e-103) {
		tmp = (x_m * (z_m * y_m)) / t_1;
	} else if (z_m <= 7e-61) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * (a / (z_m / t))));
	} else if (z_m <= 4.9e-48) {
		tmp = z_m * (y_m / (t_1 / 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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sqrt((t * -a))
    if (z_m <= 2.15d-103) then
        tmp = (x_m * (z_m * y_m)) / t_1
    else if (z_m <= 7d-61) then
        tmp = (z_m * (x_m * y_m)) / (z_m + ((-0.5d0) * (a / (z_m / t))))
    else if (z_m <= 4.9d-48) then
        tmp = z_m * (y_m / (t_1 / 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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double t_1 = Math.sqrt((t * -a));
	double tmp;
	if (z_m <= 2.15e-103) {
		tmp = (x_m * (z_m * y_m)) / t_1;
	} else if (z_m <= 7e-61) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * (a / (z_m / t))));
	} else if (z_m <= 4.9e-48) {
		tmp = z_m * (y_m / (t_1 / 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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	t_1 = math.sqrt((t * -a))
	tmp = 0
	if z_m <= 2.15e-103:
		tmp = (x_m * (z_m * y_m)) / t_1
	elif z_m <= 7e-61:
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * (a / (z_m / t))))
	elif z_m <= 4.9e-48:
		tmp = z_m * (y_m / (t_1 / 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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(Float64(t * Float64(-a)))
	tmp = 0.0
	if (z_m <= 2.15e-103)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / t_1);
	elseif (z_m <= 7e-61)
		tmp = Float64(Float64(z_m * Float64(x_m * y_m)) / Float64(z_m + Float64(-0.5 * Float64(a / Float64(z_m / t)))));
	elseif (z_m <= 4.9e-48)
		tmp = Float64(z_m * Float64(y_m / Float64(t_1 / 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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt((t * -a));
	tmp = 0.0;
	if (z_m <= 2.15e-103)
		tmp = (x_m * (z_m * y_m)) / t_1;
	elseif (z_m <= 7e-61)
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * (a / (z_m / t))));
	elseif (z_m <= 4.9e-48)
		tmp = z_m * (y_m / (t_1 / 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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.15e-103], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z$95$m, 7e-61], N[(N[(z$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(a / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 4.9e-48], N[(z$95$m * N[(y$95$m / N[(t$95$1 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if z < 2.15000000000000011e-103

   1. Initial program 57.8%

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

      1. associate-*l* 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in z around 0 39.8%

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

      1. mul-1-neg 40.3%

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

      2. *-commutative 40.3%

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

      3. distribute-rgt-neg-in 40.3%

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

   7. Simplified 39.8%

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

   if 2.15000000000000011e-103 < z < 7.0000000000000006e-61

   1. Initial program 84.2%

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

   3. Taylor expanded in z around inf 84.2%

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

      1. associate-/l* 83.7%

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

   5. Simplified 84.2%

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

   if 7.0000000000000006e-61 < z < 4.9000000000000002e-48

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. *-commutative 99.6%

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

      3. associate-*l* 84.4%

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

      4. *-commutative 84.4%

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

      5. associate-/l* 60.2%

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

   3. Simplified 60.2%

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

      1. associate-/r* 75.8%

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

      2. associate-/r/ 99.0%

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

      3. pow2 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in z around 0 76.2%

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

      1. neg-mul-1 76.2%

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

      2. distribute-rgt-neg-in 76.2%

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

   9. Simplified 76.2%

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

   if 4.9000000000000002e-48 < z

   1. Initial program 50.8%

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

      1. associate-/l* 54.8%

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

      2. associate-*l/ 54.9%

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

      3. *-commutative 54.9%

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

      4. associate-/l* 52.5%

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

   3. Simplified 52.5%

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

   5. Taylor expanded in z around inf 95.3%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{-103}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-61}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-48}:\\ \;\;\;\;z \cdot \frac{y}{\frac{\sqrt{t \cdot \left(-a\right)}}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8 \cdot 10^{-104}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z\_m \leq 2 \cdot 10^{+127}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z\_m}, z\_m\right)}\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
y_m = (fabs.f64 y)
y_s = (copysign.f64 1 y)
z_m = (fabs.f64 z)
z_s = (copysign.f64 1 z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(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 8e-104)
      (/ (* x_m (* z_m y_m)) (sqrt (* t (- a))))
      (if (<= z_m 2e+127)
        (* y_m (/ (* z_m x_m) (sqrt (- (* z_m z_m) (* t a)))))
        (* y_m (* x_m (/ z_m (fma -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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 8e-104) {
		tmp = (x_m * (z_m * y_m)) / sqrt((t * -a));
	} else if (z_m <= 2e+127) {
		tmp = y_m * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * (x_m * (z_m / fma(-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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8e-104)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(t * Float64(-a))));
	elseif (z_m <= 2e+127)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / fma(-0.5, Float64(a * Float64(t / 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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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, 8e-104], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2e+127], 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[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 7.99999999999999941e-104

   1. Initial program 57.8%

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

      1. associate-*l* 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in z around 0 39.8%

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

      1. mul-1-neg 40.3%

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

      2. *-commutative 40.3%

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

      3. distribute-rgt-neg-in 40.3%

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

   7. Simplified 39.8%

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

   if 7.99999999999999941e-104 < z < 1.99999999999999991e127

   1. Initial program 92.8%

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

      1. associate-/l* 99.9%

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

      2. associate-*l/ 98.0%

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

      3. *-commutative 98.0%

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

      4. associate-/l* 93.2%

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

   3. Simplified 93.2%

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

   if 1.99999999999999991e127 < z

   1. Initial program 20.6%

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

      1. associate-/l* 22.6%

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

      2. associate-*l/ 22.8%

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

      3. *-commutative 22.8%

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

      4. associate-/l* 20.4%

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

   3. Simplified 20.4%

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

   5. Taylor expanded in z around inf 69.1%

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

      1. associate-/l* 75.7%

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

   7. Simplified 75.7%

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

      1. *-un-lft-identity 75.7%

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

      2. times-frac 97.9%

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

      3. +-commutative 97.9%

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

      4. *-commutative 97.9%

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

      5. fma-def 97.9%

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

      6. div-inv 97.9%

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

      7. clear-num 97.9%

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

   9. Applied egg-rr 97.9%

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

   10. Taylor expanded in y around 0 62.6%

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

       1. associate-*r* 69.4%

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

       2. remove-double-neg 69.4%

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

       3. distribute-rgt-neg-in 69.4%

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

       4. *-commutative 69.4%

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

       5. associate-*l* 57.7%

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

       6. distribute-rgt-neg-in 57.7%

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

       7. distribute-rgt-neg-in 57.7%

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

       8. remove-double-neg 57.7%

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

       9. +-commutative 57.7%

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

       10. associate-*r/ 64.3%

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

       11. fma-udef 64.3%

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

       12. associate-*r/ 75.7%

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

       13. associate-*r/ 97.9%

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

   12. Simplified 97.9%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8 \cdot 10^{-104}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+127}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \sqrt{z\_m \cdot z\_m - t \cdot a}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3 \cdot 10^{-60}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{t\_1}\\ \mathbf{elif}\;z\_m \leq 4.3 \cdot 10^{+126}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z\_m}, z\_m\right)}\right)\\ \end{array}\right)\right) \end{array} \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (let* ((t_1 (sqrt (- (* z_m z_m) (* t a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 3e-60)
        (/ (* x_m (* z_m y_m)) t_1)
        (if (<= z_m 4.3e+126)
          (* y_m (/ (* z_m x_m) t_1))
          (* y_m (* x_m (/ z_m (fma -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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 = sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (z_m <= 3e-60) {
		tmp = (x_m * (z_m * y_m)) / t_1;
	} else if (z_m <= 4.3e+126) {
		tmp = y_m * ((z_m * x_m) / t_1);
	} else {
		tmp = y_m * (x_m * (z_m / fma(-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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	t_1 = sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))
	tmp = 0.0
	if (z_m <= 3e-60)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / t_1);
	elseif (z_m <= 4.3e+126)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / t_1));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / fma(-0.5, Float64(a * Float64(t / 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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3e-60], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z$95$m, 4.3e+126], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(-0.5 * N[(a * N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 3.00000000000000019e-60

   1. Initial program 59.1%

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

      1. associate-*l* 60.9%

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

   3. Simplified 60.9%

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

   if 3.00000000000000019e-60 < z < 4.3000000000000002e126

   1. Initial program 93.8%

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

      1. associate-/l* 100.0%

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

      2. associate-*l/ 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-/l* 96.4%

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

   3. Simplified 96.4%

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

   if 4.3000000000000002e126 < z

   1. Initial program 20.6%

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

      1. associate-/l* 22.6%

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

      2. associate-*l/ 22.8%

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

      3. *-commutative 22.8%

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

      4. associate-/l* 20.4%

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

   3. Simplified 20.4%

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

   5. Taylor expanded in z around inf 69.1%

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

      1. associate-/l* 75.7%

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

   7. Simplified 75.7%

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

      1. *-un-lft-identity 75.7%

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

      2. times-frac 97.9%

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

      3. +-commutative 97.9%

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

      4. *-commutative 97.9%

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

      5. fma-def 97.9%

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

      6. div-inv 97.9%

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

      7. clear-num 97.9%

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

   9. Applied egg-rr 97.9%

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

   10. Taylor expanded in y around 0 62.6%

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

       1. associate-*r* 69.4%

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

       2. remove-double-neg 69.4%

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

       3. distribute-rgt-neg-in 69.4%

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

       4. *-commutative 69.4%

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

       5. associate-*l* 57.7%

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

       6. distribute-rgt-neg-in 57.7%

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

       7. distribute-rgt-neg-in 57.7%

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

       8. remove-double-neg 57.7%

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

       9. +-commutative 57.7%

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

       10. associate-*r/ 64.3%

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

       11. fma-udef 64.3%

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

       12. associate-*r/ 75.7%

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

       13. associate-*r/ 97.9%

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

   12. Simplified 97.9%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{-60}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{+126}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-60}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{z\_m + -0.5 \cdot \frac{a}{\frac{z\_m}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (* x_m y_m) 2e-60)
      (* y_m (/ (* z_m x_m) (+ z_m (* -0.5 (/ a (/ z_m 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 ((x_m * y_m) <= 2e-60) {
		tmp = y_m * ((z_m * x_m) / (z_m + (-0.5 * (a / (z_m / 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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 ((x_m * y_m) <= 2d-60) then
        tmp = y_m * ((z_m * x_m) / (z_m + ((-0.5d0) * (a / (z_m / 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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
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 ((x_m * y_m) <= 2e-60) {
		tmp = y_m * ((z_m * x_m) / (z_m + (-0.5 * (a / (z_m / 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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (x_m * y_m) <= 2e-60:
		tmp = y_m * ((z_m * x_m) / (z_m + (-0.5 * (a / (z_m / 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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 2e-60)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / Float64(z_m + Float64(-0.5 * Float64(a / Float64(z_m / 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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((x_m * y_m) <= 2e-60)
		tmp = y_m * ((z_m * x_m) / (z_m + (-0.5 * (a / (z_m / 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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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[N[(x$95$m * y$95$m), $MachinePrecision], 2e-60], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(a / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.9999999999999999e-60

   1. Initial program 58.7%

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

      1. associate-/l* 62.1%

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

      2. associate-*l/ 61.6%

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

      3. *-commutative 61.6%

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

      4. associate-/l* 58.7%

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

   3. Simplified 58.7%

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

   5. Taylor expanded in z around inf 50.2%

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

      1. associate-/l* 51.9%

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

   7. Simplified 51.9%

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

   if 1.9999999999999999e-60 < (*.f64 x y)

   1. Initial program 51.7%

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

      1. associate-/l* 55.3%

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

      2. associate-*l/ 58.6%

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

      3. *-commutative 58.6%

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

      4. associate-/l* 56.0%

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

   3. Simplified 56.0%

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

   5. Taylor expanded in z around inf 38.0%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 2 \cdot 10^{-60}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + -0.5 \cdot \frac{a}{\frac{z}{t}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-169}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{z\_m + \frac{a}{z\_m} \cdot \left(t \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(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 -3e-169)
      (* y_m (/ (* z_m x_m) (+ z_m (* (/ a z_m) (* t -0.5)))))
      (* 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= -3e-169) {
		tmp = y_m * ((z_m * x_m) / (z_m + ((a / z_m) * (t * -0.5))));
	} 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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 <= (-3d-169)) then
        tmp = y_m * ((z_m * x_m) / (z_m + ((a / z_m) * (t * (-0.5d0)))))
    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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
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 <= -3e-169) {
		tmp = y_m * ((z_m * x_m) / (z_m + ((a / z_m) * (t * -0.5))));
	} 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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if t <= -3e-169:
		tmp = y_m * ((z_m * x_m) / (z_m + ((a / z_m) * (t * -0.5))))
	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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (t <= -3e-169)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / Float64(z_m + Float64(Float64(a / z_m) * Float64(t * -0.5)))));
	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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (t <= -3e-169)
		tmp = y_m * ((z_m * x_m) / (z_m + ((a / z_m) * (t * -0.5))));
	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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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, -3e-169], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[(z$95$m + N[(N[(a / z$95$m), $MachinePrecision] * N[(t * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -2.9999999999999999e-169

   1. Initial program 59.8%

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

      1. associate-/l* 61.6%

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

      2. associate-*l/ 61.7%

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

      3. *-commutative 61.7%

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

      4. associate-/l* 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in z around inf 50.5%

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

      1. associate-/l* 53.9%

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

   7. Simplified 53.9%

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

   8. Taylor expanded in a around 0 50.5%

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

      1. associate-*r/ 53.9%

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

      2. *-commutative 53.9%

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

      3. associate-*r/ 50.5%

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

      4. associate-*l/ 53.9%

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

      5. associate-*r* 53.9%

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

   10. Simplified 53.9%

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

   if -2.9999999999999999e-169 < t

   1. Initial program 53.5%

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

      1. associate-/l* 58.4%

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

      2. associate-*l/ 59.7%

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

      3. *-commutative 59.7%

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

      4. associate-/l* 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in z around inf 41.6%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-169}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z + \frac{a}{z} \cdot \left(t \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 7.2e-87)
      (/ (* x_m (* z_m y_m)) (+ z_m (* -0.5 (/ (* t a) 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.2e-87) {
		tmp = (x_m * (z_m * y_m)) / (z_m + (-0.5 * ((t * a) / 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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 7.2d-87) then
        tmp = (x_m * (z_m * y_m)) / (z_m + ((-0.5d0) * ((t * a) / 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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.2e-87) {
		tmp = (x_m * (z_m * y_m)) / (z_m + (-0.5 * ((t * a) / 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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7.2e-87:
		tmp = (x_m * (z_m * y_m)) / (z_m + (-0.5 * ((t * a) / 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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.2e-87)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / 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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.2e-87)
		tmp = (x_m * (z_m * y_m)) / (z_m + (-0.5 * ((t * a) / 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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.2e-87], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.19999999999999986e-87

   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* 60.8%

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

   3. Simplified 60.8%

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

   5. Taylor expanded in z around inf 19.5%

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

   if 7.19999999999999986e-87 < z

   1. Initial program 52.7%

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

      1. associate-/l* 57.3%

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

      2. associate-*l/ 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-/l* 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in z around inf 92.9%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{z\_m \cdot \left(x\_m \cdot y\_m\right)}{z\_m + -0.5 \cdot \frac{t \cdot a}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(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 3.2e-87)
      (/ (* z_m (* x_m y_m)) (+ z_m (* -0.5 (/ (* t a) 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 3.2e-87) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / 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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 <= 3.2d-87) then
        tmp = (z_m * (x_m * y_m)) / (z_m + ((-0.5d0) * ((t * a) / 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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
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 <= 3.2e-87) {
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / 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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.2e-87:
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / 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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.2e-87)
		tmp = Float64(Float64(z_m * Float64(x_m * y_m)) / Float64(z_m + Float64(-0.5 * Float64(Float64(t * a) / 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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.2e-87)
		tmp = (z_m * (x_m * y_m)) / (z_m + (-0.5 * ((t * a) / 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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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, 3.2e-87], N[(N[(z$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(z$95$m + N[(-0.5 * N[(N[(t * a), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.19999999999999979e-87

   1. Initial program 58.9%

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

   3. Taylor expanded in z around inf 19.5%

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

   if 3.19999999999999979e-87 < z

   1. Initial program 52.7%

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

      1. associate-/l* 57.3%

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

      2. associate-*l/ 56.5%

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

      3. *-commutative 56.5%

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

      4. associate-/l* 53.7%

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

   3. Simplified 53.7%

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

   5. Taylor expanded in z around inf 92.9%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{-87}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z + -0.5 \cdot \frac{t \cdot a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.35 \cdot 10^{-195}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(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 1.35e-195) (* 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 1.35e-195) {
		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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.35d-195) 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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.35e-195) {
		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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.35e-195:
		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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.35e-195)
		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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.35e-195)
		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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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, 1.35e-195], 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 < 1.35e-195

   1. Initial program 54.9%

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

      1. associate-/l* 58.0%

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

      2. associate-*l/ 63.2%

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

      3. *-commutative 63.2%

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

      4. associate-/l* 61.4%

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

   3. Simplified 61.4%

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

   5. Taylor expanded in z around inf 11.9%

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

   if 1.35e-195 < z

   1. Initial program 57.8%

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

      1. associate-/l* 61.6%

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

      2. associate-*l/ 58.0%

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

      3. *-commutative 58.0%

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

      4. associate-/l* 54.2%

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

   3. Simplified 54.2%

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

   5. Taylor expanded in z around inf 83.4%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.35 \cdot 10^{-195}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(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 8.5e-193) (/ (* 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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 8.5e-193) {
		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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 <= 8.5d-193) 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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
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 <= 8.5e-193) {
		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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 8.5e-193:
		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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8.5e-193)
		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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 8.5e-193)
		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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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, 8.5e-193], 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 < 8.50000000000000004e-193

   1. Initial program 55.2%

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

      1. associate-*l* 56.7%

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

   3. Simplified 56.7%

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

   5. Taylor expanded in z around inf 16.4%

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

   if 8.50000000000000004e-193 < z

   1. Initial program 57.4%

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

      1. associate-/l* 61.3%

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

      2. associate-*l/ 57.7%

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

      3. *-commutative 57.7%

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

      4. associate-/l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in z around inf 84.0%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.9% accurate, 9.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ y_m = \left|y\right| \\ y_s = \mathsf{copysign}\left(1, y\right) \\ z_m = \left|z\right| \\ z_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{z\_m \cdot \left(x\_m \cdot y\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(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 8.6e-193) (/ (* z_m (* x_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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 8.6e-193) {
		tmp = (z_m * (x_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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 <= 8.6d-193) then
        tmp = (z_m * (x_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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
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 <= 8.6e-193) {
		tmp = (z_m * (x_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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 8.6e-193:
		tmp = (z_m * (x_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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8.6e-193)
		tmp = Float64(Float64(z_m * Float64(x_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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 8.6e-193)
		tmp = (z_m * (x_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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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, 8.6e-193], N[(N[(z$95$m * N[(x$95$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 < 8.6000000000000004e-193

   1. Initial program 55.2%

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

   3. Taylor expanded in z around inf 16.3%

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

   if 8.6000000000000004e-193 < z

   1. Initial program 57.4%

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

      1. associate-/l* 61.3%

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

      2. associate-*l/ 57.7%

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

      3. *-commutative 57.7%

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

      4. associate-/l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in z around inf 84.0%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.6 \cdot 10^{-193}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.6% accurate, 37.7× speedup

Math

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

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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(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);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
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)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
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)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
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);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
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]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 56.3%

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

   1. associate-/l* 59.8%

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

   2. associate-*l/ 60.6%

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

   3. *-commutative 60.6%

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

   4. associate-/l* 57.8%

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

3. Simplified 57.8%

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

5. Taylor expanded in z around inf 47.0%

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

6. Final simplification 47.0%

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

Developer target: 87.6% 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 2024036 
(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)))))