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

Percentage Accurate: 60.3% → 93.2%

Time: 12.9s

Alternatives: 11

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: 60.3% 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: 93.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(t, \frac{a}{z\_m \cdot -2}, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 4.8e+55)
      (* (/ (* z_m y_m) (sqrt (- (* z_m z_m) (* t a)))) x_m)
      (* (/ z_m (fma t (/ a (* z_m -2.0)) z_m)) (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.8e+55) {
		tmp = ((z_m * y_m) / sqrt(((z_m * z_m) - (t * a)))) * x_m;
	} else {
		tmp = (z_m / fma(t, (a / (z_m * -2.0)), z_m)) * (y_m * x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.8e+55)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) * x_m);
	else
		tmp = Float64(Float64(z_m / fma(t, Float64(a / Float64(z_m * -2.0)), z_m)) * Float64(y_m * x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4.8e+55], N[(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] * x$95$m), $MachinePrecision], N[(N[(z$95$m / N[(t * N[(a / N[(z$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.7999999999999998e55

   1. Initial program 65.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 65.9

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

   4. Applied egg-rr 65.9%

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

   if 4.7999999999999998e55 < z

   1. Initial program 40.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 75.5

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

   5. Simplified 75.5%

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

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

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

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

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

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lowering-*.f64 96.4

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

   7. Applied egg-rr 96.4%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.8 \cdot 10^{+55}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(t, \frac{a}{z \cdot -2}, z\right)} \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{-128}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{-t \cdot a}}\\ \mathbf{elif}\;z\_m \leq 8.2 \cdot 10^{+108}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(t, \frac{a}{z\_m \cdot -2}, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.5e-128)
      (* x_m (/ (* z_m y_m) (sqrt (- (* t a)))))
      (if (<= z_m 8.2e+108)
        (* (* z_m x_m) (/ y_m (sqrt (- (* z_m z_m) (* t a)))))
        (* (/ z_m (fma t (/ a (* z_m -2.0)) z_m)) (* y_m x_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.5e-128) {
		tmp = x_m * ((z_m * y_m) / sqrt(-(t * a)));
	} else if (z_m <= 8.2e+108) {
		tmp = (z_m * x_m) * (y_m / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = (z_m / fma(t, (a / (z_m * -2.0)), z_m)) * (y_m * x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.5e-128)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(-Float64(t * a)))));
	elseif (z_m <= 8.2e+108)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(Float64(z_m / fma(t, Float64(a / Float64(z_m * -2.0)), z_m)) * Float64(y_m * x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.5e-128], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[(-N[(t * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 8.2e+108], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(t * N[(a / N[(z$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 3.5e-128

   1. Initial program 60.7%

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

      1. 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 59.5%

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

   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 36.5

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

   7. Simplified 36.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. un-div-inv N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 37.2

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

   9. Applied egg-rr 37.2%

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

   if 3.5e-128 < z < 8.1999999999999998e108

   1. Initial program 85.1%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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-*r* N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 83.5

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

   4. Applied egg-rr 83.5%

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

   if 8.1999999999999998e108 < z

   1. Initial program 30.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 75.5

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

   5. Simplified 75.5%

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

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

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

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

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

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lowering-*.f64 98.5

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

   7. Applied egg-rr 98.5%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{-t \cdot a}}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+108}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(t, \frac{a}{z \cdot -2}, z\right)} \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+134}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1e+134)
      (* (* y_m x_m) (/ z_m (sqrt (- (* z_m z_m) (* t a)))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e+134) {
		tmp = (y_m * x_m) * (z_m / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1d+134) then
        tmp = (y_m * x_m) * (z_m / sqrt(((z_m * z_m) - (t * a))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e+134) {
		tmp = (y_m * x_m) * (z_m / Math.sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1e+134:
		tmp = (y_m * x_m) * (z_m / math.sqrt(((z_m * z_m) - (t * a))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e+134)
		tmp = Float64(Float64(y_m * x_m) * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1e+134)
		tmp = (y_m * x_m) * (z_m / sqrt(((z_m * z_m) - (t * a))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1e+134], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - 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 < 9.99999999999999921e133

   1. Initial program 66.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 71.3

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

   4. Applied egg-rr 71.3%

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

   if 9.99999999999999921e133 < z

   1. Initial program 24.9%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 100.0

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

   5. Simplified 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+134}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.5 \cdot 10^{+108}:\\ \;\;\;\;z\_m \cdot \left(x\_m \cdot \frac{y\_m}{\sqrt{\mathsf{fma}\left(a, -t, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(t, \frac{a}{z\_m \cdot -2}, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.5e+108)
      (* z_m (* x_m (/ y_m (sqrt (fma a (- t) (* z_m z_m))))))
      (* (/ z_m (fma t (/ a (* z_m -2.0)) z_m)) (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.5e+108) {
		tmp = z_m * (x_m * (y_m / sqrt(fma(a, -t, (z_m * z_m)))));
	} else {
		tmp = (z_m / fma(t, (a / (z_m * -2.0)), z_m)) * (y_m * x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.5e+108)
		tmp = Float64(z_m * Float64(x_m * Float64(y_m / sqrt(fma(a, Float64(-t), Float64(z_m * z_m))))));
	else
		tmp = Float64(Float64(z_m / fma(t, Float64(a / Float64(z_m * -2.0)), z_m)) * Float64(y_m * x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.5e+108], N[(z$95$m * N[(x$95$m * N[(y$95$m / N[Sqrt[N[(a * (-t) + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(t * N[(a / N[(z$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.49999999999999995e108

   1. Initial program 66.3%

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

      1. *-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 68.8

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

   4. Applied egg-rr 68.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-lowering-*.f64 66.2

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

   6. Applied egg-rr 66.2%

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

   if 2.49999999999999995e108 < z

   1. Initial program 30.6%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 75.5

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

   5. Simplified 75.5%

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

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

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

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

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

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lowering-*.f64 98.5

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

   7. Applied egg-rr 98.5%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+108}:\\ \;\;\;\;z \cdot \left(x \cdot \frac{y}{\sqrt{\mathsf{fma}\left(a, -t, z \cdot z\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(t, \frac{a}{z \cdot -2}, z\right)} \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.75 \cdot 10^{-124}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{-t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z\_m}{\mathsf{fma}\left(t, \frac{a}{z\_m \cdot -2}, z\_m\right)} \cdot \left(y\_m \cdot x\_m\right)\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.75e-124)
      (* x_m (/ (* z_m y_m) (sqrt (- (* t a)))))
      (* (/ z_m (fma t (/ a (* z_m -2.0)) z_m)) (* y_m x_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.75e-124) {
		tmp = x_m * ((z_m * y_m) / sqrt(-(t * a)));
	} else {
		tmp = (z_m / fma(t, (a / (z_m * -2.0)), z_m)) * (y_m * x_m);
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.75e-124)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(-Float64(t * a)))));
	else
		tmp = Float64(Float64(z_m / fma(t, Float64(a / Float64(z_m * -2.0)), z_m)) * Float64(y_m * x_m));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.75e-124], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[(-N[(t * a), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z$95$m / N[(t * N[(a / N[(z$95$m * -2.0), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision] * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.7499999999999999e-124

   1. Initial program 60.7%

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

      1. 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 59.5%

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

   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 36.5

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

   7. Simplified 36.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. un-div-inv N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 37.2

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

   9. Applied egg-rr 37.2%

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

   if 1.7499999999999999e-124 < z

   1. Initial program 53.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 71.4

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

   5. Simplified 71.4%

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

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

      6. associate-/l* N/A

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

      7. associate-*l* N/A

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

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

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

      9. clear-num N/A

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

      10. associate-*l/ N/A

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

      11. *-lft-identity N/A

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

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

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

      13. div-inv N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

      17. metadata-eval N/A

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

      18. *-lowering-*.f64 87.3

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

   7. Applied egg-rr 87.3%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.75 \cdot 10^{-124}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{-t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\mathsf{fma}\left(t, \frac{a}{z \cdot -2}, z\right)} \cdot \left(y \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.45 \cdot 10^{-126}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{-t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2.45e-126)
      (* x_m (/ (* z_m y_m) (sqrt (- (* t a)))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.45e-126) {
		tmp = x_m * ((z_m * y_m) / sqrt(-(t * a)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.45d-126) then
        tmp = x_m * ((z_m * y_m) / sqrt(-(t * a)))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.45e-126) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt(-(t * a)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.45e-126:
		tmp = x_m * ((z_m * y_m) / math.sqrt(-(t * a)))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.45e-126)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(-Float64(t * a)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.45e-126)
		tmp = x_m * ((z_m * y_m) / sqrt(-(t * a)));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.45e-126], N[(x$95$m * N[(N[(z$95$m * y$95$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 < 2.45000000000000005e-126

   1. Initial program 60.7%

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

      1. 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 59.5%

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

   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 36.5

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

   7. Simplified 36.5%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. un-div-inv N/A

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 37.2

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

   9. Applied egg-rr 37.2%

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

   if 2.45000000000000005e-126 < z

   1. Initial program 53.5%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 86.9

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

   5. Simplified 86.9%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.45 \cdot 10^{-126}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{-t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.9 \cdot 10^{-124}:\\ \;\;\;\;z\_m \cdot \frac{y\_m \cdot x\_m}{\sqrt{-t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.9e-124)
      (* z_m (/ (* y_m x_m) (sqrt (- (* t a)))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-124) {
		tmp = z_m * ((y_m * x_m) / sqrt(-(t * a)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.9d-124) then
        tmp = z_m * ((y_m * x_m) / sqrt(-(t * a)))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-124) {
		tmp = z_m * ((y_m * x_m) / Math.sqrt(-(t * a)));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.9e-124:
		tmp = z_m * ((y_m * x_m) / math.sqrt(-(t * a)))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.9e-124)
		tmp = Float64(z_m * Float64(Float64(y_m * x_m) / sqrt(Float64(-Float64(t * a)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.9e-124)
		tmp = z_m * ((y_m * x_m) / sqrt(-(t * a)));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.9e-124], 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 < 1.90000000000000006e-124

   1. Initial program 60.7%

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

      1. *-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 63.5

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

   4. Applied egg-rr 63.5%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 39.6

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

   7. Simplified 39.6%

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

   if 1.90000000000000006e-124 < z

   1. Initial program 53.5%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 86.9

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

   5. Simplified 86.9%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{-124}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{-t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.2 \cdot 10^{-126}:\\ \;\;\;\;z\_m \cdot \left(y\_m \cdot \frac{x\_m}{\sqrt{-t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 7.2e-126)
      (* z_m (* y_m (/ x_m (sqrt (- (* t a))))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.2e-126) {
		tmp = z_m * (y_m * (x_m / sqrt(-(t * a))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 7.2d-126) then
        tmp = z_m * (y_m * (x_m / sqrt(-(t * a))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.2e-126) {
		tmp = z_m * (y_m * (x_m / Math.sqrt(-(t * a))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7.2e-126:
		tmp = z_m * (y_m * (x_m / math.sqrt(-(t * a))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.2e-126)
		tmp = Float64(z_m * Float64(y_m * Float64(x_m / sqrt(Float64(-Float64(t * a))))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.2e-126)
		tmp = z_m * (y_m * (x_m / sqrt(-(t * a))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.2e-126], 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 < 7.1999999999999999e-126

   1. Initial program 60.7%

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

      1. 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 59.5%

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

   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 36.5

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

   7. Simplified 36.5%

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. frac-2neg N/A

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

      7. metadata-eval N/A

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

      8. sqrt-div N/A

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

      9. metadata-eval N/A

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

      10. un-div-inv N/A

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

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

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

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

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

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

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

      14. *-commutative N/A

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

      15. *-lowering-*.f64 38.4

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

   9. Applied egg-rr 38.4%

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

   if 7.1999999999999999e-126 < z

   1. Initial program 53.5%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 86.9

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

   5. Simplified 86.9%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.2 \cdot 10^{-126}:\\ \;\;\;\;z \cdot \left(y \cdot \frac{x}{\sqrt{-t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t \cdot a \leq -5.4 \cdot 10^{-148}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (* t a) -5.4e-148) (* y_m (/ (* z_m x_m) z_m)) (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -5.4e-148) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((t * a) <= (-5.4d-148)) then
        tmp = y_m * ((z_m * x_m) / z_m)
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if ((t * a) <= -5.4e-148) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if (t * a) <= -5.4e-148:
		tmp = y_m * ((z_m * x_m) / z_m)
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(t * a) <= -5.4e-148)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if ((t * a) <= -5.4e-148)
		tmp = y_m * ((z_m * x_m) / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(t * a), $MachinePrecision], -5.4e-148], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / 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) < -5.39999999999999976e-148

   1. Initial program 65.9%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 36.0

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

   5. Simplified 36.0%

      \[\leadsto \color{blue}{x \cdot y} \]
   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. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 41.5

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

   7. Applied egg-rr 41.5%

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

   if -5.39999999999999976e-148 < (*.f64 t a)

   1. Initial program 51.5%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 52.9

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

   5. Simplified 52.9%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \cdot a \leq -5.4 \cdot 10^{-148}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.6 \cdot 10^{-158}:\\ \;\;\;\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= z_m 2.6e-158) (/ (* z_m (* y_m x_m)) z_m) (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.6e-158) {
		tmp = (z_m * (y_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.6d-158) then
        tmp = (z_m * (y_m * x_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.6e-158) {
		tmp = (z_m * (y_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.6e-158:
		tmp = (z_m * (y_m * x_m)) / z_m
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.6e-158)
		tmp = Float64(Float64(z_m * Float64(y_m * x_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.6e-158)
		tmp = (z_m * (y_m * x_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.6e-158], 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 < 2.6e-158

   1. Initial program 60.5%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 22.0%

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

   if 2.6e-158 < z

   1. Initial program 54.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 84.2

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

   5. Simplified 84.2%

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

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.6 \cdot 10^{-158}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.1% accurate, 7.5× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (y_m * x_m)));
}

Fortran

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

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (y_m * x_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * (y_m * x_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(y_m * x_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (y_m * x_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 45.7

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

5. Simplified 45.7%

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

6. Final simplification 45.7%

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

Developer Target 1: 88.1% 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 2024198 
(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)))))