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

Percentage Accurate: 62.0% → 94.3%

Time: 14.6s

Alternatives: 16

Speedup: 7.5×

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: 62.0% 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: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.3 \cdot 10^{-169}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{-t}} \cdot \frac{x\_m}{\sqrt{a}}\\ \mathbf{elif}\;z\_m \leq 5.5 \cdot 10^{+128}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(z\_m, z\_m, t \cdot \left(-a\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 4.3e-169)
      (* (/ (* z_m y_m) (sqrt (- t))) (/ x_m (sqrt a)))
      (if (<= z_m 5.5e+128)
        (* y_m (* x_m (/ z_m (sqrt (fma z_m z_m (* t (- a)))))))
        (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.3e-169) {
		tmp = ((z_m * y_m) / sqrt(-t)) * (x_m / sqrt(a));
	} else if (z_m <= 5.5e+128) {
		tmp = y_m * (x_m * (z_m / sqrt(fma(z_m, z_m, (t * -a)))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.3e-169)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(Float64(-t))) * Float64(x_m / sqrt(a)));
	elseif (z_m <= 5.5e+128)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(fma(z_m, z_m, Float64(t * Float64(-a)))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 4.3e-169], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[(-t)], $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 5.5e+128], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[Sqrt[N[(z$95$m * z$95$m + N[(t * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 4.29999999999999984e-169

   1. Initial program 62.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 64.4

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

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 37.7

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

   7. Simplified 37.7%

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

      1. associate-*l/ N/A

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

      2. pow1/2 N/A

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

      3. *-commutative N/A

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

      4. unpow-prod-down N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. times-frac N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. pow1/2 N/A

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

      14. sqrt-lowering-sqrt.f64 N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. /-lowering-/.f64 N/A

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

      17. pow1/2 N/A

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

      18. sqrt-lowering-sqrt.f64 16.3

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

   9. Applied egg-rr 16.3%

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

   if 4.29999999999999984e-169 < z < 5.4999999999999998e128

   1. Initial program 79.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 87.9

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

   4. Applied egg-rr 87.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sub-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-lowering-neg.f64 92.6

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

   6. Applied egg-rr 92.6%

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

   if 5.4999999999999998e128 < z

   1. Initial program 25.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.3 \cdot 10^{-169}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{-t}} \cdot \frac{x}{\sqrt{a}}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+128}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{\mathsf{fma}\left(z, z, t \cdot \left(-a\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 10^{-205}:\\ \;\;\;\;\frac{y\_m \cdot \left(z\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (/ (* z_m (* y_m x_m)) (sqrt (- (* z_m z_m) (* t a)))) 1e-205)
      (/ (* y_m (* z_m x_m)) z_m)
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (((z_m * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 1e-205) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 1d-205) then
        tmp = (y_m * (z_m * x_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (((z_m * (y_m * x_m)) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 1e-205) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if ((z_m * (y_m * x_m)) / math.sqrt(((z_m * z_m) - (t * a)))) <= 1e-205:
		tmp = (y_m * (z_m * x_m)) / z_m
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(Float64(z_m * Float64(y_m * x_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 1e-205)
		tmp = Float64(Float64(y_m * Float64(z_m * x_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (((z_m * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 1e-205)
		tmp = (y_m * (z_m * x_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-205], N[(N[(y$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 1e-205

   1. Initial program 66.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 51.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 47.2

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

   7. Applied egg-rr 47.2%

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

   if 1e-205 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 46.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 40.6

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

   5. Simplified 40.6%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}} \leq 10^{-205}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.35 \cdot 10^{-214}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{a}} \cdot \frac{x\_m}{\sqrt{-t}}\\ \mathbf{elif}\;z\_m \leq 3.2 \cdot 10^{+128}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(z\_m, z\_m, t \cdot \left(-a\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.35e-214)
      (* (/ (* z_m y_m) (sqrt a)) (/ x_m (sqrt (- t))))
      (if (<= z_m 3.2e+128)
        (* y_m (* x_m (/ z_m (sqrt (fma z_m z_m (* t (- a)))))))
        (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.35e-214) {
		tmp = ((z_m * y_m) / sqrt(a)) * (x_m / sqrt(-t));
	} else if (z_m <= 3.2e+128) {
		tmp = y_m * (x_m * (z_m / sqrt(fma(z_m, z_m, (t * -a)))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.35e-214)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(a)) * Float64(x_m / sqrt(Float64(-t))));
	elseif (z_m <= 3.2e+128)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(fma(z_m, z_m, Float64(t * Float64(-a)))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.35e-214], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[a], $MachinePrecision]), $MachinePrecision] * N[(x$95$m / N[Sqrt[(-t)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.2e+128], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[Sqrt[N[(z$95$m * z$95$m + N[(t * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.3500000000000002e-214

   1. Initial program 62.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 63.5

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 35.0

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

   7. Simplified 35.0%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. pow1/2 N/A

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

      7. unpow-prod-down N/A

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

      8. times-frac N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. pow1/2 N/A

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

      13. sqrt-lowering-sqrt.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. pow1/2 N/A

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

      16. sqrt-lowering-sqrt.f64 N/A

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

      17. neg-lowering-neg.f64 16.3

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

   9. Applied egg-rr 16.3%

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

   if 2.3500000000000002e-214 < z < 3.19999999999999986e128

   1. Initial program 78.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 86.7

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

   4. Applied egg-rr 86.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sub-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-lowering-neg.f64 90.9

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

   6. Applied egg-rr 90.9%

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

   if 3.19999999999999986e128 < z

   1. Initial program 25.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.35 \cdot 10^{-214}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{a}} \cdot \frac{x}{\sqrt{-t}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+128}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{\mathsf{fma}\left(z, z, t \cdot \left(-a\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.9 \cdot 10^{-107}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{elif}\;z\_m \leq 2.5 \cdot 10^{+128}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(z\_m, z\_m, t \cdot \left(-a\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 3.9e-107)
      (* x_m (/ (* z_m y_m) (sqrt (- (* z_m z_m) (* t a)))))
      (if (<= z_m 2.5e+128)
        (* y_m (* x_m (/ z_m (sqrt (fma z_m z_m (* t (- a)))))))
        (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.9e-107) {
		tmp = x_m * ((z_m * y_m) / sqrt(((z_m * z_m) - (t * a))));
	} else if (z_m <= 2.5e+128) {
		tmp = y_m * (x_m * (z_m / sqrt(fma(z_m, z_m, (t * -a)))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.9e-107)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	elseif (z_m <= 2.5e+128)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(fma(z_m, z_m, Float64(t * Float64(-a)))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 3.9e-107], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.5e+128], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[Sqrt[N[(z$95$m * z$95$m + N[(t * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.9000000000000001e-107

   1. Initial program 62.9%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 67.4

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

   4. Applied egg-rr 67.4%

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

   if 3.9000000000000001e-107 < z < 2.5e128

   1. Initial program 83.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 87.4

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

   4. Applied egg-rr 87.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sub-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-lowering-neg.f64 90.9

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

   6. Applied egg-rr 90.9%

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

   if 2.5e128 < z

   1. Initial program 25.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.9 \cdot 10^{-107}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+128}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{\mathsf{fma}\left(z, z, t \cdot \left(-a\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.5 \cdot 10^{-107}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{elif}\;z\_m \leq 3.2 \cdot 10^{+128}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(z\_m, z\_m, t \cdot \left(-a\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.5e-107)
      (* (* z_m y_m) (/ x_m (sqrt (- (* z_m z_m) (* t a)))))
      (if (<= z_m 3.2e+128)
        (* y_m (* x_m (/ z_m (sqrt (fma z_m z_m (* t (- a)))))))
        (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.5e-107) {
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	} else if (z_m <= 3.2e+128) {
		tmp = y_m * (x_m * (z_m / sqrt(fma(z_m, z_m, (t * -a)))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.5e-107)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	elseif (z_m <= 3.2e+128)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(fma(z_m, z_m, Float64(t * Float64(-a)))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.5e-107], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.2e+128], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[Sqrt[N[(z$95$m * z$95$m + N[(t * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.49999999999999985e-107

   1. Initial program 62.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 65.7

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

   4. Applied egg-rr 65.7%

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

   if 2.49999999999999985e-107 < z < 3.19999999999999986e128

   1. Initial program 83.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 87.4

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

   4. Applied egg-rr 87.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. sqrt-lowering-sqrt.f64 N/A

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

      6. sub-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. *-lowering-*.f64 N/A

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

      10. neg-lowering-neg.f64 90.9

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

   6. Applied egg-rr 90.9%

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

   if 3.19999999999999986e128 < z

   1. Initial program 25.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-107}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{+128}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{\mathsf{fma}\left(z, z, t \cdot \left(-a\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.05 \cdot 10^{-42}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{elif}\;z\_m \leq 3.1 \cdot 10^{+128}:\\ \;\;\;\;x\_m \cdot \left(z\_m \cdot \frac{y\_m}{\sqrt{\mathsf{fma}\left(t, -a, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.05e-42)
      (* (* z_m y_m) (/ x_m (sqrt (- (* z_m z_m) (* t a)))))
      (if (<= z_m 3.1e+128)
        (* x_m (* z_m (/ y_m (sqrt (fma t (- a) (* z_m z_m))))))
        (* y_m x_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.05e-42) {
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	} else if (z_m <= 3.1e+128) {
		tmp = x_m * (z_m * (y_m / sqrt(fma(t, -a, (z_m * z_m)))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.05e-42)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	elseif (z_m <= 3.1e+128)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(fma(t, Float64(-a), Float64(z_m * z_m))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.05e-42], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 3.1e+128], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(t * (-a) + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.0500000000000001e-42

   1. Initial program 62.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 66.8

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

   4. Applied egg-rr 66.8%

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

   if 2.0500000000000001e-42 < z < 3.10000000000000004e128

   1. Initial program 88.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 88.6

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

   4. Applied egg-rr 88.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   6. Applied egg-rr 90.7%

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

   if 3.10000000000000004e128 < z

   1. Initial program 25.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.05 \cdot 10^{-42}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+128}:\\ \;\;\;\;x \cdot \left(z \cdot \frac{y}{\sqrt{\mathsf{fma}\left(t, -a, z \cdot z\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 0.9× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2e-15)
      (* (* z_m y_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

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e-15) {
		tmp = (z_m * y_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 x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e-15)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / fma(-0.5, Float64(a * Float64(t / z_m)), z_m)));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2e-15], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] * 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]

Derivation

1. Split input into 2 regimes

2. if z < 2.0000000000000002e-15

   1. Initial program 63.9%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. --lowering--.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 67.2

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

   4. Applied egg-rr 67.2%

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

   if 2.0000000000000002e-15 < z

   1. Initial program 52.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 80.6

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

   5. Simplified 80.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 89.5

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

   7. Applied egg-rr 89.5%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. /-lowering-/.f64 89.5

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

   9. Applied egg-rr 89.5%

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

4. Final simplification 74.6%

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

Alternative 8: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.7 \cdot 10^{-87}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \left(x\_m \cdot \sqrt{\frac{-1}{t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z\_m}, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.7e-87)
      (* (* z_m y_m) (* x_m (sqrt (/ -1.0 (* t a)))))
      (* (* y_m x_m) (/ z_m (fma -0.5 (* a (/ t z_m)) z_m))))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.7e-87) {
		tmp = (z_m * y_m) * (x_m * sqrt((-1.0 / (t * a))));
	} else {
		tmp = (y_m * x_m) * (z_m / fma(-0.5, (a * (t / z_m)), z_m));
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.7e-87)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m * sqrt(Float64(-1.0 / Float64(t * a)))));
	else
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / fma(-0.5, Float64(a * Float64(t / z_m)), z_m)));
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.7e-87], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m * N[Sqrt[N[(-1.0 / N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] * 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]

Derivation

1. Split input into 2 regimes

2. if z < 1.6999999999999999e-87

   1. Initial program 63.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 66.0%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 39.4

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

   7. Simplified 39.4%

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

   if 1.6999999999999999e-87 < z

   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 t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 78.1

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

   5. Simplified 78.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 85.9

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

   7. Applied egg-rr 85.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. div-inv N/A

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

      4. associate-*l* N/A

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. /-lowering-/.f64 85.9

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

   9. Applied egg-rr 85.9%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.7 \cdot 10^{-87}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(x \cdot \sqrt{\frac{-1}{t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z}, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.5 \cdot 10^{-65}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \left(x\_m \cdot \sqrt{\frac{-1}{t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.5e-65)
      (* (* z_m y_m) (* x_m (sqrt (/ -1.0 (* t a)))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.5e-65) {
		tmp = (z_m * y_m) * (x_m * sqrt((-1.0 / (t * a))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.5d-65) then
        tmp = (z_m * y_m) * (x_m * sqrt(((-1.0d0) / (t * a))))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.5e-65) {
		tmp = (z_m * y_m) * (x_m * Math.sqrt((-1.0 / (t * a))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.5e-65:
		tmp = (z_m * y_m) * (x_m * math.sqrt((-1.0 / (t * a))))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.5e-65)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m * sqrt(Float64(-1.0 / Float64(t * a)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.5e-65)
		tmp = (z_m * y_m) * (x_m * sqrt((-1.0 / (t * a))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.5e-65], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m * N[Sqrt[N[(-1.0 / N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.49999999999999991e-65

   1. Initial program 62.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 66.0%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 39.4

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

   7. Simplified 39.4%

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

   if 2.49999999999999991e-65 < z

   1. Initial program 55.8%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 86.5

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

   5. Simplified 86.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{-65}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \left(x \cdot \sqrt{\frac{-1}{t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.4 \cdot 10^{-65}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 4.4e-65)
      (* (* z_m y_m) (/ x_m (sqrt (* t (- a)))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.4e-65) {
		tmp = (z_m * y_m) * (x_m / sqrt((t * -a)));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 4.4d-65) then
        tmp = (z_m * y_m) * (x_m / sqrt((t * -a)))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.4e-65) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt((t * -a)));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 4.4e-65:
		tmp = (z_m * y_m) * (x_m / math.sqrt((t * -a)))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.4e-65)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4.4e-65)
		tmp = (z_m * y_m) * (x_m / sqrt((t * -a)));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 4.4e-65], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.40000000000000042e-65

   1. Initial program 62.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 66.8

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 41.8

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

   7. Simplified 41.8%

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

      1. associate-*l/ N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-*l* N/A

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

      8. div-inv N/A

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

      9. *-lowering-*.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. sqrt-lowering-sqrt.f64 N/A

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

      13. distribute-rgt-neg-out N/A

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

      14. *-commutative N/A

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

      15. neg-lowering-neg.f64 N/A

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

      16. *-commutative N/A

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

      17. *-lowering-*.f64 39.4

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

   9. Applied egg-rr 39.4%

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

   if 4.40000000000000042e-65 < z

   1. Initial program 55.8%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 86.5

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

   5. Simplified 86.5%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{-65}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.9e-65)
      (* x_m (* y_m (/ z_m (sqrt (* t (- a))))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.9e-65) {
		tmp = x_m * (y_m * (z_m / sqrt((t * -a))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.9d-65) then
        tmp = x_m * (y_m * (z_m / sqrt((t * -a))))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.9e-65) {
		tmp = x_m * (y_m * (z_m / Math.sqrt((t * -a))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.9e-65:
		tmp = x_m * (y_m * (z_m / math.sqrt((t * -a))))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.9e-65)
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / sqrt(Float64(t * Float64(-a))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.9e-65)
		tmp = x_m * (y_m * (z_m / sqrt((t * -a))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.9e-65], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.8999999999999998e-65

   1. Initial program 62.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 66.8

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

   4. Applied egg-rr 66.8%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 41.8

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

   7. Simplified 41.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. sqrt-lowering-sqrt.f64 N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. *-commutative N/A

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

      11. neg-lowering-neg.f64 N/A

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 40.8

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

   9. Applied egg-rr 40.8%

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

   if 2.8999999999999998e-65 < z

   1. Initial program 55.8%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 86.5

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

   5. Simplified 86.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-65}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.2 \cdot 10^{-76}:\\ \;\;\;\;z\_m \cdot \frac{y\_m \cdot x\_m}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 3.2e-76)
      (* z_m (/ (* y_m x_m) (sqrt (* t (- a)))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.2e-76) {
		tmp = z_m * ((y_m * x_m) / sqrt((t * -a)));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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-76) then
        tmp = z_m * ((y_m * x_m) / sqrt((t * -a)))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.2e-76) {
		tmp = z_m * ((y_m * x_m) / Math.sqrt((t * -a)));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.2e-76:
		tmp = z_m * ((y_m * x_m) / math.sqrt((t * -a)))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.2e-76)
		tmp = Float64(z_m * Float64(Float64(y_m * x_m) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.2e-76)
		tmp = z_m * ((y_m * x_m) / sqrt((t * -a)));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 3.2e-76], N[(z$95$m * N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.1999999999999998e-76

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 39.9

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

   5. Simplified 39.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-out N/A

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

      9. neg-lowering-neg.f64 N/A

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

      10. *-lowering-*.f64 41.8

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

   7. Applied egg-rr 41.8%

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

   if 3.1999999999999998e-76 < z

   1. Initial program 55.8%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 85.8

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

   5. Simplified 85.8%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{-76}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{-74}:\\ \;\;\;\;z\_m \cdot \left(y\_m \cdot \frac{x\_m}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.5e-74)
      (* z_m (* y_m (/ x_m (sqrt (* t (- a))))))
      (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.5e-74) {
		tmp = z_m * (y_m * (x_m / sqrt((t * -a))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1.5d-74) then
        tmp = z_m * (y_m * (x_m / sqrt((t * -a))))
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.5e-74) {
		tmp = z_m * (y_m * (x_m / Math.sqrt((t * -a))));
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.5e-74:
		tmp = z_m * (y_m * (x_m / math.sqrt((t * -a))))
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.5e-74)
		tmp = Float64(z_m * Float64(y_m * Float64(x_m / sqrt(Float64(t * Float64(-a))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.5e-74)
		tmp = z_m * (y_m * (x_m / sqrt((t * -a))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.5e-74], N[(z$95$m * N[(y$95$m * N[(x$95$m / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.50000000000000003e-74

   1. Initial program 62.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 N/A

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

      11. sqrt-lowering-sqrt.f64 N/A

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 66.4

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

   4. Applied egg-rr 66.4%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

      4. *-lowering-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 41.6

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

   7. Simplified 41.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. sqrt-lowering-sqrt.f64 N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. *-commutative N/A

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

      10. neg-lowering-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 40.0

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

   9. Applied egg-rr 40.0%

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

   if 1.50000000000000003e-74 < z

   1. Initial program 55.8%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 85.8

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

   5. Simplified 85.8%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.5 \cdot 10^{-74}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{x}{\sqrt{t \cdot \left(-a\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq -1.2 \cdot 10^{-91}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= (* t a) -1.2e-91) (* (* z_m x_m) (/ y_m z_m)) (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -1.2e-91) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 * a) <= (-1.2d-91)) then
        tmp = (z_m * x_m) * (y_m / z_m)
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -1.2e-91) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (t * a) <= -1.2e-91:
		tmp = (z_m * x_m) * (y_m / z_m)
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(t * a) <= -1.2e-91)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((t * a) <= -1.2e-91)
		tmp = (z_m * x_m) * (y_m / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[N[(t * a), $MachinePrecision], -1.2e-91], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 t a) < -1.20000000000000005e-91

   1. Initial program 67.6%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 33.9

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

   5. Simplified 33.9%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. associate-/l* N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 30.0

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

   7. Applied egg-rr 30.0%

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

   if -1.20000000000000005e-91 < (*.f64 t a)

   1. Initial program 54.7%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 51.7

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

   5. Simplified 51.7%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot a \leq -1.2 \cdot 10^{-91}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (* z_s (if (<= z_m 5.4e-147) (/ (* z_m (* y_m x_m)) z_m) (* y_m x_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.4e-147) {
		tmp = (z_m * (y_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 5.4d-147) then
        tmp = (z_m * (y_m * x_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5.4e-147) {
		tmp = (z_m * (y_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 5.4e-147:
		tmp = (z_m * (y_m * x_m)) / z_m
	else:
		tmp = y_m * x_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5.4e-147)
		tmp = Float64(Float64(z_m * Float64(y_m * x_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5.4e-147)
		tmp = (z_m * (y_m * x_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 5.4e-147], N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.3999999999999999e-147

   1. Initial program 63.1%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 24.9%

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

   if 5.3999999999999999e-147 < z

   1. Initial program 56.0%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 78.7

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

   5. Simplified 78.7%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.2% accurate, 7.5× speedup

Math

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

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (* x_s (* y_s (* z_s (* y_m x_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	return x_s * (y_s * (z_s * (y_m * x_m)));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 = x_s * (y_s * (z_s * (y_m * x_m)))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	return x_s * (y_s * (z_s * (y_m * x_m)));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	return x_s * (y_s * (z_s * (y_m * x_m)))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	return Float64(x_s * Float64(y_s * Float64(z_s * Float64(y_m * x_m))))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = x_s * (y_s * (z_s * (y_m * x_m)));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, 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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.0%

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

3. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 44.3

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

5. Simplified 44.3%

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

6. Final simplification 44.3%

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

Developer Target 1: 88.6% accurate, 0.7× 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 2024204 
(FPCore (x y z t a)
  :name "Statistics.Math.RootFinding:ridders from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -31921305903852764000000000000000000000000000000) (- (* y x)) (if (< z 5976268120920894000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y)) (* y x))))

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