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

Percentage Accurate: 60.7% → 91.0%

Time: 13.7s

Alternatives: 13

Speedup: 37.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.7% 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: 91.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \cdot \left(z\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z\_m \cdot \left(z\_m \cdot z\_m\right)}{t}} + \frac{-0.5}{z\_m}\right)\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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+20)
      (* (/ x_m (sqrt (- (* z_m z_m) (* t a)))) (* z_m y_m))
      (*
       y_m
       (*
        x_m
        (/
         z_m
         (+
          z_m
          (*
           t
           (*
            a
            (+
             (/ (* a -0.125) (/ (* z_m (* z_m z_m)) t))
             (/ -0.5 z_m)))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.5e+20) {
		tmp = (x_m / sqrt(((z_m * z_m) - (t * a)))) * (z_m * y_m);
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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 <= 2.5d+20) then
        tmp = (x_m / sqrt(((z_m * z_m) - (t * a)))) * (z_m * y_m)
    else
        tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * (-0.125d0)) / ((z_m * (z_m * z_m)) / t)) + ((-0.5d0) / z_m)))))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 <= 2.5e+20) {
		tmp = (x_m / Math.sqrt(((z_m * z_m) - (t * a)))) * (z_m * y_m);
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[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 <= 2.5e+20:
		tmp = (x_m / math.sqrt(((z_m * z_m) - (t * a)))) * (z_m * y_m)
	else:
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.5e+20)
		tmp = Float64(Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) * Float64(z_m * y_m));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / Float64(z_m + Float64(t * Float64(a * Float64(Float64(Float64(a * -0.125) / Float64(Float64(z_m * Float64(z_m * z_m)) / t)) + Float64(-0.5 / z_m))))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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 <= 2.5e+20)
		tmp = (x_m / sqrt(((z_m * z_m) - (t * a)))) * (z_m * y_m);
	else
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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+20], N[(N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(z$95$m + N[(t * N[(a * N[(N[(N[(a * -0.125), $MachinePrecision] / N[(N[(z$95$m * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.5e20

   1. Initial program 69.1%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 69.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 69.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\sqrt{z \cdot z - t \cdot a}\right)\right), \left(\color{blue}{y} \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

      10. *-lowering-*.f64 70.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   6. Applied egg-rr 70.4%

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

   if 2.5e20 < z

   1. Initial program 50.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{\color{blue}{{z}^{3}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{{\color{blue}{z}}^{3}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{z}\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{z}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), z\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{\color{blue}{{z}^{3}}}\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{{\color{blue}{z}}^{3}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)\right), \color{blue}{\left({z}^{3}\right)}\right)\right)\right)\right)\right) \]

   5. Simplified 62.2%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + t \cdot \left(\frac{a \cdot -0.5}{z} + \frac{-0.125 \cdot \left(a \cdot \left(a \cdot t\right)\right)}{z \cdot \left(z \cdot z\right)}\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{z}\right)\right)\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{z}\right)\right)\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{a \cdot t}{{z}^{3}}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{z}\right)\right)\right)\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot \frac{t}{{z}^{3}}\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \left(\frac{t}{{z}^{3}}\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left({z}^{3}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot {z}^{2}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 74.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 74.7%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z + t \cdot \color{blue}{\left(a \cdot \left(-0.125 \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{-0.5}{z}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right)\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 93.1%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.7 \cdot 10^{-115}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{elif}\;z\_m \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\frac{y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \cdot \left(z\_m \cdot x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z\_m \cdot \left(z\_m \cdot z\_m\right)}{t}} + \frac{-0.5}{z\_m}\right)\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 7.7e-115)
      (* (* z_m y_m) (/ x_m (sqrt (- 0.0 (* t a)))))
      (if (<= z_m 2e+134)
        (* (/ y_m (sqrt (- (* z_m z_m) (* t a)))) (* z_m x_m))
        (*
         y_m
         (*
          x_m
          (/
           z_m
           (+
            z_m
            (*
             t
             (*
              a
              (+
               (/ (* a -0.125) (/ (* z_m (* z_m z_m)) t))
               (/ -0.5 z_m))))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.7e-115) {
		tmp = (z_m * y_m) * (x_m / sqrt((0.0 - (t * a))));
	} else if (z_m <= 2e+134) {
		tmp = (y_m / sqrt(((z_m * z_m) - (t * a)))) * (z_m * x_m);
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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 <= 7.7d-115) then
        tmp = (z_m * y_m) * (x_m / sqrt((0.0d0 - (t * a))))
    else if (z_m <= 2d+134) then
        tmp = (y_m / sqrt(((z_m * z_m) - (t * a)))) * (z_m * x_m)
    else
        tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * (-0.125d0)) / ((z_m * (z_m * z_m)) / t)) + ((-0.5d0) / z_m)))))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 <= 7.7e-115) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt((0.0 - (t * a))));
	} else if (z_m <= 2e+134) {
		tmp = (y_m / Math.sqrt(((z_m * z_m) - (t * a)))) * (z_m * x_m);
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[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 <= 7.7e-115:
		tmp = (z_m * y_m) * (x_m / math.sqrt((0.0 - (t * a))))
	elif z_m <= 2e+134:
		tmp = (y_m / math.sqrt(((z_m * z_m) - (t * a)))) * (z_m * x_m)
	else:
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.7e-115)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(0.0 - Float64(t * a)))));
	elseif (z_m <= 2e+134)
		tmp = Float64(Float64(y_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) * Float64(z_m * x_m));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / Float64(z_m + Float64(t * Float64(a * Float64(Float64(Float64(a * -0.125) / Float64(Float64(z_m * Float64(z_m * z_m)) / t)) + Float64(-0.5 / z_m))))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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 <= 7.7e-115)
		tmp = (z_m * y_m) * (x_m / sqrt((0.0 - (t * a))));
	elseif (z_m <= 2e+134)
		tmp = (y_m / sqrt(((z_m * z_m) - (t * a)))) * (z_m * x_m);
	else
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 7.7e-115], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2e+134], N[(N[(y$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(z$95$m + N[(t * N[(a * N[(N[(N[(a * -0.125), $MachinePrecision] / N[(N[(z$95$m * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 7.7000000000000002e-115

   1. Initial program 65.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 64.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 64.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\sqrt{z \cdot z - t \cdot a}\right)\right), \left(\color{blue}{y} \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

      10. *-lowering-*.f64 66.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   6. Applied egg-rr 66.5%

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

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)}\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]

      4. *-lowering-*.f64 44.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]

   9. Simplified 44.1%

      \[\leadsto \frac{x}{\sqrt{\color{blue}{0 - a \cdot t}}} \cdot \left(y \cdot z\right) \]

   if 7.7000000000000002e-115 < z < 1.99999999999999984e134

   1. Initial program 92.2%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 92.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 92.0%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(z \cdot x\right), \color{blue}{\left(\frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot z\right), \left(\frac{\color{blue}{y}}{\sqrt{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), \left(\frac{\color{blue}{y}}{\sqrt{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 92.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, z\right), \mathsf{/.f64}\left(y, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 92.1%

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

   if 1.99999999999999984e134 < z

   1. Initial program 28.2%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{\color{blue}{{z}^{3}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{{\color{blue}{z}}^{3}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{z}\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{z}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), z\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{\color{blue}{{z}^{3}}}\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{{\color{blue}{z}}^{3}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)\right), \color{blue}{\left({z}^{3}\right)}\right)\right)\right)\right)\right) \]

   5. Simplified 53.4%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + t \cdot \left(\frac{a \cdot -0.5}{z} + \frac{-0.125 \cdot \left(a \cdot \left(a \cdot t\right)\right)}{z \cdot \left(z \cdot z\right)}\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{z}\right)\right)\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{z}\right)\right)\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{a \cdot t}{{z}^{3}}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{z}\right)\right)\right)\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot \frac{t}{{z}^{3}}\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \left(\frac{t}{{z}^{3}}\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left({z}^{3}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot {z}^{2}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 70.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 70.8%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z + t \cdot \color{blue}{\left(a \cdot \left(-0.125 \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{-0.5}{z}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right)\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 97.8%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.7 \cdot 10^{-115}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{0 - t \cdot a}}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\frac{y}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(z \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.9 \cdot 10^{-25}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z\_m \cdot \left(z\_m \cdot z\_m\right)}{t}} + \frac{-0.5}{z\_m}\right)\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 2.9e-25)
      (* (* z_m y_m) (/ x_m (sqrt (- 0.0 (* t a)))))
      (*
       y_m
       (*
        x_m
        (/
         z_m
         (+
          z_m
          (*
           t
           (*
            a
            (+
             (/ (* a -0.125) (/ (* z_m (* z_m z_m)) t))
             (/ -0.5 z_m)))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.9e-25) {
		tmp = (z_m * y_m) * (x_m / sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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 <= 2.9d-25) then
        tmp = (z_m * y_m) * (x_m / sqrt((0.0d0 - (t * a))))
    else
        tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * (-0.125d0)) / ((z_m * (z_m * z_m)) / t)) + ((-0.5d0) / z_m)))))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 <= 2.9e-25) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[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 <= 2.9e-25:
		tmp = (z_m * y_m) * (x_m / math.sqrt((0.0 - (t * a))))
	else:
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.9e-25)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(0.0 - Float64(t * a)))));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / Float64(z_m + Float64(t * Float64(a * Float64(Float64(Float64(a * -0.125) / Float64(Float64(z_m * Float64(z_m * z_m)) / t)) + Float64(-0.5 / z_m))))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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 <= 2.9e-25)
		tmp = (z_m * y_m) * (x_m / sqrt((0.0 - (t * a))));
	else
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 2.9e-25], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(z$95$m + N[(t * N[(a * N[(N[(N[(a * -0.125), $MachinePrecision] / N[(N[(z$95$m * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.9000000000000001e-25

   1. Initial program 67.0%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 67.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 67.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{x}{\sqrt{z \cdot z - t \cdot a}}\right), \color{blue}{\left(y \cdot z\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \left(\sqrt{z \cdot z - t \cdot a}\right)\right), \left(\color{blue}{y} \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right), \left(y \cdot z\right)\right) \]

      10. *-lowering-*.f64 69.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right), \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   6. Applied egg-rr 69.0%

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

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)}\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]

      4. *-lowering-*.f64 47.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(x, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right), \mathsf{*.f64}\left(y, z\right)\right) \]

   9. Simplified 47.0%

      \[\leadsto \frac{x}{\sqrt{\color{blue}{0 - a \cdot t}}} \cdot \left(y \cdot z\right) \]

   if 2.9000000000000001e-25 < z

   1. Initial program 58.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{\color{blue}{{z}^{3}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{{\color{blue}{z}}^{3}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{z}\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{z}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), z\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{\color{blue}{{z}^{3}}}\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{{\color{blue}{z}}^{3}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)\right), \color{blue}{\left({z}^{3}\right)}\right)\right)\right)\right)\right) \]

   5. Simplified 62.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + t \cdot \left(\frac{a \cdot -0.5}{z} + \frac{-0.125 \cdot \left(a \cdot \left(a \cdot t\right)\right)}{z \cdot \left(z \cdot z\right)}\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{z}\right)\right)\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{z}\right)\right)\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{a \cdot t}{{z}^{3}}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{z}\right)\right)\right)\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot \frac{t}{{z}^{3}}\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \left(\frac{t}{{z}^{3}}\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left({z}^{3}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot {z}^{2}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 73.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 73.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z + t \cdot \color{blue}{\left(a \cdot \left(-0.125 \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{-0.5}{z}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right)\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 88.5%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-25}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3 \cdot 10^{-25}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z\_m \cdot \left(z\_m \cdot z\_m\right)}{t}} + \frac{-0.5}{z\_m}\right)\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 3e-25)
      (* x_m (/ (* z_m y_m) (sqrt (- 0.0 (* t a)))))
      (*
       y_m
       (*
        x_m
        (/
         z_m
         (+
          z_m
          (*
           t
           (*
            a
            (+
             (/ (* a -0.125) (/ (* z_m (* z_m z_m)) t))
             (/ -0.5 z_m)))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3e-25) {
		tmp = x_m * ((z_m * y_m) / sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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 <= 3d-25) then
        tmp = x_m * ((z_m * y_m) / sqrt((0.0d0 - (t * a))))
    else
        tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * (-0.125d0)) / ((z_m * (z_m * z_m)) / t)) + ((-0.5d0) / z_m)))))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 <= 3e-25) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[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 <= 3e-25:
		tmp = x_m * ((z_m * y_m) / math.sqrt((0.0 - (t * a))))
	else:
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3e-25)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(0.0 - Float64(t * a)))));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / Float64(z_m + Float64(t * Float64(a * Float64(Float64(Float64(a * -0.125) / Float64(Float64(z_m * Float64(z_m * z_m)) / t)) + Float64(-0.5 / z_m))))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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 <= 3e-25)
		tmp = x_m * ((z_m * y_m) / sqrt((0.0 - (t * a))));
	else
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 3e-25], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(z$95$m + N[(t * N[(a * N[(N[(N[(a * -0.125), $MachinePrecision] / N[(N[(z$95$m * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.9999999999999998e-25

   1. Initial program 67.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot t\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right) \]

      4. *-lowering-*.f64 47.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right) \]

   5. Simplified 47.2%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{0 - a \cdot t}}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{\color{blue}{0 - a \cdot t}}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{y \cdot z}{\sqrt{0 - a \cdot t}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y \cdot z}{\sqrt{0 - a \cdot t}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\sqrt{0 - a \cdot t}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\sqrt{\color{blue}{0 - a \cdot t}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 46.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 46.0%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot z}{\sqrt{0 - a \cdot t}}} \]

   if 2.9999999999999998e-25 < z

   1. Initial program 58.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{\color{blue}{{z}^{3}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{{\color{blue}{z}}^{3}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{z}\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{z}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), z\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{\color{blue}{{z}^{3}}}\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{{\color{blue}{z}}^{3}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)\right), \color{blue}{\left({z}^{3}\right)}\right)\right)\right)\right)\right) \]

   5. Simplified 62.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + t \cdot \left(\frac{a \cdot -0.5}{z} + \frac{-0.125 \cdot \left(a \cdot \left(a \cdot t\right)\right)}{z \cdot \left(z \cdot z\right)}\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{z}\right)\right)\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{z}\right)\right)\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{a \cdot t}{{z}^{3}}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{z}\right)\right)\right)\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot \frac{t}{{z}^{3}}\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \left(\frac{t}{{z}^{3}}\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left({z}^{3}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot {z}^{2}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 73.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 73.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z + t \cdot \color{blue}{\left(a \cdot \left(-0.125 \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{-0.5}{z}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right)\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 88.5%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.2 \cdot 10^{-25}:\\ \;\;\;\;x\_m \cdot \left(z\_m \cdot \frac{y\_m}{\sqrt{0 - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z\_m \cdot \left(z\_m \cdot z\_m\right)}{t}} + \frac{-0.5}{z\_m}\right)\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 3.2e-25)
      (* x_m (* z_m (/ y_m (sqrt (- 0.0 (* t a))))))
      (*
       y_m
       (*
        x_m
        (/
         z_m
         (+
          z_m
          (*
           t
           (*
            a
            (+
             (/ (* a -0.125) (/ (* z_m (* z_m z_m)) t))
             (/ -0.5 z_m)))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.2e-25) {
		tmp = x_m * (z_m * (y_m / sqrt((0.0 - (t * a)))));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.2d-25) then
        tmp = x_m * (z_m * (y_m / sqrt((0.0d0 - (t * a)))))
    else
        tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * (-0.125d0)) / ((z_m * (z_m * z_m)) / t)) + ((-0.5d0) / z_m)))))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.2e-25) {
		tmp = x_m * (z_m * (y_m / Math.sqrt((0.0 - (t * a)))));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.2e-25:
		tmp = x_m * (z_m * (y_m / math.sqrt((0.0 - (t * a)))))
	else:
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.2e-25)
		tmp = Float64(x_m * Float64(z_m * Float64(y_m / sqrt(Float64(0.0 - Float64(t * a))))));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / Float64(z_m + Float64(t * Float64(a * Float64(Float64(Float64(a * -0.125) / Float64(Float64(z_m * Float64(z_m * z_m)) / t)) + Float64(-0.5 / z_m))))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.2e-25)
		tmp = x_m * (z_m * (y_m / sqrt((0.0 - (t * a)))));
	else
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.2e-25], N[(x$95$m * N[(z$95$m * N[(y$95$m / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(z$95$m + N[(t * N[(a * N[(N[(N[(a * -0.125), $MachinePrecision] / N[(N[(z$95$m * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.2000000000000001e-25

   1. Initial program 67.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot t\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right) \]

      4. *-lowering-*.f64 47.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right) \]

   5. Simplified 47.2%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{0 - a \cdot t}}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\sqrt{\color{blue}{0 - a \cdot t}}} \]

      2. associate-/l* N/A

         \[\leadsto x \cdot \color{blue}{\frac{y \cdot z}{\sqrt{0 - a \cdot t}}} \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y \cdot z}{\sqrt{0 - a \cdot t}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\sqrt{0 - a \cdot t}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\sqrt{\color{blue}{0 - a \cdot t}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 46.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 46.0%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot z}{\sqrt{0 - a \cdot t}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{z \cdot y}{\sqrt{\color{blue}{0 - a \cdot t}}}\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(z \cdot \color{blue}{\frac{y}{\sqrt{0 - a \cdot t}}}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{y}{\sqrt{0 - a \cdot t}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(y, \color{blue}{\left(\sqrt{0 - a \cdot t}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(y, \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(y, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(y, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(t \cdot a\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 45.4%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(y, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(t, a\right)\right)\right)\right)\right)\right) \]

   9. Applied egg-rr 45.4%

      \[\leadsto x \cdot \color{blue}{\left(z \cdot \frac{y}{\sqrt{0 - t \cdot a}}\right)} \]

   if 3.2000000000000001e-25 < z

   1. Initial program 58.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{\color{blue}{{z}^{3}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{{\color{blue}{z}}^{3}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{z}\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{z}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), z\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{\color{blue}{{z}^{3}}}\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{{\color{blue}{z}}^{3}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)\right), \color{blue}{\left({z}^{3}\right)}\right)\right)\right)\right)\right) \]

   5. Simplified 62.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + t \cdot \left(\frac{a \cdot -0.5}{z} + \frac{-0.125 \cdot \left(a \cdot \left(a \cdot t\right)\right)}{z \cdot \left(z \cdot z\right)}\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{z}\right)\right)\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{z}\right)\right)\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{a \cdot t}{{z}^{3}}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{z}\right)\right)\right)\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot \frac{t}{{z}^{3}}\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \left(\frac{t}{{z}^{3}}\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left({z}^{3}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot {z}^{2}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 73.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 73.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z + t \cdot \color{blue}{\left(a \cdot \left(-0.125 \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{-0.5}{z}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right)\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 88.5%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 83.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.9 \cdot 10^{-25}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\sqrt{0 - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z\_m \cdot \left(z\_m \cdot z\_m\right)}{t}} + \frac{-0.5}{z\_m}\right)\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 2.9e-25)
      (* x_m (* y_m (/ z_m (sqrt (- 0.0 (* t a))))))
      (*
       y_m
       (*
        x_m
        (/
         z_m
         (+
          z_m
          (*
           t
           (*
            a
            (+
             (/ (* a -0.125) (/ (* z_m (* z_m z_m)) t))
             (/ -0.5 z_m)))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.9e-25) {
		tmp = x_m * (y_m * (z_m / sqrt((0.0 - (t * a)))));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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 <= 2.9d-25) then
        tmp = x_m * (y_m * (z_m / sqrt((0.0d0 - (t * a)))))
    else
        tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * (-0.125d0)) / ((z_m * (z_m * z_m)) / t)) + ((-0.5d0) / z_m)))))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 <= 2.9e-25) {
		tmp = x_m * (y_m * (z_m / Math.sqrt((0.0 - (t * a)))));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[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 <= 2.9e-25:
		tmp = x_m * (y_m * (z_m / math.sqrt((0.0 - (t * a)))))
	else:
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.9e-25)
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / sqrt(Float64(0.0 - Float64(t * a))))));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / Float64(z_m + Float64(t * Float64(a * Float64(Float64(Float64(a * -0.125) / Float64(Float64(z_m * Float64(z_m * z_m)) / t)) + Float64(-0.5 / z_m))))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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 <= 2.9e-25)
		tmp = x_m * (y_m * (z_m / sqrt((0.0 - (t * a)))));
	else
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 2.9e-25], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(z$95$m + N[(t * N[(a * N[(N[(N[(a * -0.125), $MachinePrecision] / N[(N[(z$95$m * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.9000000000000001e-25

   1. Initial program 67.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\color{blue}{\left(-1 \cdot \left(a \cdot t\right)\right)}\right)\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\left(\mathsf{neg}\left(a \cdot t\right)\right)\right)\right) \]

      2. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right) \]

      4. *-lowering-*.f64 47.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right) \]

   5. Simplified 47.2%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{0 - a \cdot t}}} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{\sqrt{0 - a \cdot t}}} \]

      2. associate-*l* N/A

         \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{z}{\sqrt{0 - a \cdot t}}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(y \cdot \frac{z}{\sqrt{0 - a \cdot t}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{z}{\sqrt{0 - a \cdot t}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{\left(\sqrt{0 - a \cdot t}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{sqrt.f64}\left(\left(0 - a \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 44.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 44.9%

      \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{z}{\sqrt{0 - a \cdot t}}\right)} \]

   if 2.9000000000000001e-25 < z

   1. Initial program 58.1%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{\color{blue}{{z}^{3}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{{\color{blue}{z}}^{3}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{z}\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{z}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), z\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{\color{blue}{{z}^{3}}}\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{{\color{blue}{z}}^{3}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)\right), \color{blue}{\left({z}^{3}\right)}\right)\right)\right)\right)\right) \]

   5. Simplified 62.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + t \cdot \left(\frac{a \cdot -0.5}{z} + \frac{-0.125 \cdot \left(a \cdot \left(a \cdot t\right)\right)}{z \cdot \left(z \cdot z\right)}\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{z}\right)\right)\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{z}\right)\right)\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{a \cdot t}{{z}^{3}}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{z}\right)\right)\right)\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot \frac{t}{{z}^{3}}\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \left(\frac{t}{{z}^{3}}\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left({z}^{3}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot {z}^{2}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 73.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 73.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z + t \cdot \color{blue}{\left(a \cdot \left(-0.125 \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{-0.5}{z}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right)\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 88.5%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-25}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\sqrt{0 - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{z\_m + \frac{\left(t \cdot a\right) \cdot -0.5}{z\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z\_m \cdot \left(z\_m \cdot z\_m\right)}{t}} + \frac{-0.5}{z\_m}\right)\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 2.1e+20)
      (* x_m (/ (* z_m y_m) (+ z_m (/ (* (* t a) -0.5) z_m))))
      (*
       y_m
       (*
        x_m
        (/
         z_m
         (+
          z_m
          (*
           t
           (*
            a
            (+
             (/ (* a -0.125) (/ (* z_m (* z_m z_m)) t))
             (/ -0.5 z_m)))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.1e+20) {
		tmp = x_m * ((z_m * y_m) / (z_m + (((t * a) * -0.5) / z_m)));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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 <= 2.1d+20) then
        tmp = x_m * ((z_m * y_m) / (z_m + (((t * a) * (-0.5d0)) / z_m)))
    else
        tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * (-0.125d0)) / ((z_m * (z_m * z_m)) / t)) + ((-0.5d0) / z_m)))))))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 <= 2.1e+20) {
		tmp = x_m * ((z_m * y_m) / (z_m + (((t * a) * -0.5) / z_m)));
	} else {
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[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 <= 2.1e+20:
		tmp = x_m * ((z_m * y_m) / (z_m + (((t * a) * -0.5) / z_m)))
	else:
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.1e+20)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(Float64(Float64(t * a) * -0.5) / z_m))));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / Float64(z_m + Float64(t * Float64(a * Float64(Float64(Float64(a * -0.125) / Float64(Float64(z_m * Float64(z_m * z_m)) / t)) + Float64(-0.5 / z_m))))))));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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 <= 2.1e+20)
		tmp = x_m * ((z_m * y_m) / (z_m + (((t * a) * -0.5) / z_m)));
	else
		tmp = y_m * (x_m * (z_m / (z_m + (t * (a * (((a * -0.125) / ((z_m * (z_m * z_m)) / t)) + (-0.5 / z_m)))))));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 2.1e+20], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(N[(N[(t * a), $MachinePrecision] * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(z$95$m + N[(t * N[(a * N[(N[(N[(a * -0.125), $MachinePrecision] / N[(N[(z$95$m * N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] + N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.1e20

   1. Initial program 69.1%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 69.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 69.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{\left(z \cdot \left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)\right)}\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \color{blue}{\left(1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \left(\frac{\frac{-1}{2} \cdot \left(a \cdot t\right)}{\color{blue}{{z}^{2}}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(a \cdot t\right)\right), \color{blue}{\left({z}^{2}\right)}\right)\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(a \cdot t\right) \cdot \frac{-1}{2}\right), \left({\color{blue}{z}}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a \cdot t\right), \frac{-1}{2}\right), \left({\color{blue}{z}}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, t\right), \frac{-1}{2}\right), \left({z}^{2}\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, t\right), \frac{-1}{2}\right), \left(z \cdot \color{blue}{z}\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 29.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, t\right), \frac{-1}{2}\right), \mathsf{*.f64}\left(z, \color{blue}{z}\right)\right)\right)\right)\right) \]

   7. Simplified 29.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y \cdot z}{z \cdot \left(1 + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z \cdot z}\right)}\right), \color{blue}{x}\right) \]

   9. Applied egg-rr 31.3%

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

   if 2.1e20 < z

   1. Initial program 50.6%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{\color{blue}{{z}^{3}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{{\color{blue}{z}}^{3}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{z}\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{z}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), z\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{\color{blue}{{z}^{3}}}\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{{\color{blue}{z}}^{3}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)\right), \color{blue}{\left({z}^{3}\right)}\right)\right)\right)\right)\right) \]

   5. Simplified 62.2%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + t \cdot \left(\frac{a \cdot -0.5}{z} + \frac{-0.125 \cdot \left(a \cdot \left(a \cdot t\right)\right)}{z \cdot \left(z \cdot z\right)}\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{z}\right)\right)\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{z}\right)\right)\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{a \cdot t}{{z}^{3}}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{z}\right)\right)\right)\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot \frac{t}{{z}^{3}}\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \left(\frac{t}{{z}^{3}}\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left({z}^{3}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot {z}^{2}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 74.7%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 74.7%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z + t \cdot \color{blue}{\left(a \cdot \left(-0.125 \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{-0.5}{z}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right)\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 93.1%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{+20}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + \frac{\left(t \cdot a\right) \cdot -0.5}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6 \cdot 10^{+156}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + \frac{\left(t \cdot a\right) \cdot -0.5}{z\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 6e+156)
      (* y_m (* x_m (/ z_m (+ z_m (/ (* (* t a) -0.5) z_m)))))
      (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
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 <= 6e+156) {
		tmp = y_m * (x_m * (z_m / (z_m + (((t * a) * -0.5) / z_m))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
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 <= 6d+156) then
        tmp = y_m * (x_m * (z_m / (z_m + (((t * a) * (-0.5d0)) / z_m))))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
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 <= 6e+156) {
		tmp = y_m * (x_m * (z_m / (z_m + (((t * a) * -0.5) / z_m))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[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 <= 6e+156:
		tmp = y_m * (x_m * (z_m / (z_m + (((t * a) * -0.5) / z_m))))
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
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 <= 6e+156)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / Float64(z_m + Float64(Float64(Float64(t * a) * -0.5) / z_m)))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
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 <= 6e+156)
		tmp = y_m * (x_m * (z_m / (z_m + (((t * a) * -0.5) / z_m))));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
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, 6e+156], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[(z$95$m + N[(N[(N[(t * a), $MachinePrecision] * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.9999999999999999e156

   1. Initial program 71.5%

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

   3. Taylor expanded in t around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)\right)}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{-1}{8} \cdot \frac{{a}^{2} \cdot t}{{z}^{3}}\right)}\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{\color{blue}{{z}^{3}}}\right)\right)\right)\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{{\color{blue}{z}}^{3}}\right)\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot \color{blue}{t}\right)\right)\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{-1}{2} \cdot \frac{a}{z} + \left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{-1}{2} \cdot \frac{a}{z}\right), \color{blue}{\left(\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)}\right)\right)\right)\right) \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\left(\frac{\frac{-1}{2} \cdot a}{z}\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot a\right), z\right), \left(\color{blue}{\left(\frac{-1}{8} \cdot \frac{{a}^{2}}{{z}^{3}}\right)} \cdot t\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(a \cdot \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\left(\color{blue}{\frac{-1}{8}} \cdot \frac{{a}^{2}}{{z}^{3}}\right) \cdot t\right)\right)\right)\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot {a}^{2}}{{z}^{3}} \cdot t\right)\right)\right)\right)\right) \]

      13. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\left(\frac{-1}{8} \cdot {a}^{2}\right) \cdot t}{\color{blue}{{z}^{3}}}\right)\right)\right)\right)\right) \]

      14. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \left(\frac{\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)}{{\color{blue}{z}}^{3}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(a, \frac{-1}{2}\right), z\right), \mathsf{/.f64}\left(\left(\frac{-1}{8} \cdot \left({a}^{2} \cdot t\right)\right), \color{blue}{\left({z}^{3}\right)}\right)\right)\right)\right)\right) \]

   5. Simplified 26.2%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + t \cdot \left(\frac{a \cdot -0.5}{z} + \frac{-0.125 \cdot \left(a \cdot \left(a \cdot t\right)\right)}{z \cdot \left(z \cdot z\right)}\right)}} \]

   6. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} - \frac{1}{2} \cdot \frac{1}{z}\right)}\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{1}{z}\right)\right)}\right)\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot 1}{z}\right)\right)\right)\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{z}\right)\right)\right)\right)\right)\right)\right) \]

      5. distribute-neg-frac N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\mathsf{neg}\left(\frac{1}{2}\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}} + \frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \frac{a \cdot t}{{z}^{3}}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(\frac{a \cdot t}{{z}^{3}}\right)\right), \left(\frac{\color{blue}{\frac{-1}{2}}}{z}\right)\right)\right)\right)\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \left(a \cdot \frac{t}{{z}^{3}}\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \left(\frac{t}{{z}^{3}}\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left({z}^{3}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      12. cube-mult N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \left(z \cdot {z}^{2}\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left({z}^{2}\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \left(z \cdot z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right)\right)\right) \]

      17. /-lowering-/.f64 37.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(z, \mathsf{*.f64}\left(z, z\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\frac{-1}{2}, \color{blue}{z}\right)\right)\right)\right)\right)\right) \]

   8. Simplified 37.1%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z + t \cdot \color{blue}{\left(a \cdot \left(-0.125 \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{-0.5}{z}\right)\right)}} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{z}{z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(t \cdot \left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \color{blue}{\left(a \cdot \left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)\right)}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \color{blue}{\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right) + \frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{+.f64}\left(\left(\frac{-1}{8} \cdot \left(a \cdot \frac{t}{z \cdot \left(z \cdot z\right)}\right)\right), \color{blue}{\left(\frac{\frac{-1}{2}}{z}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   10. Applied egg-rr 38.5%

       \[\leadsto \color{blue}{y \cdot \left(x \cdot \frac{z}{z + t \cdot \left(a \cdot \left(\frac{a \cdot -0.125}{\frac{z \cdot \left(z \cdot z\right)}{t}} + \frac{-0.5}{z}\right)\right)}\right)} \]

   11. Taylor expanded in t around 0

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{\left(z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}\right)}\right)\right)\right) \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \color{blue}{\left(\frac{-1}{2} \cdot \frac{a \cdot t}{z}\right)}\right)\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \left(\frac{\frac{-1}{2} \cdot \left(a \cdot t\right)}{\color{blue}{z}}\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(a \cdot t\right)\right), \color{blue}{z}\right)\right)\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(\left(a \cdot t\right) \cdot \frac{-1}{2}\right), z\right)\right)\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a \cdot t\right), \frac{-1}{2}\right), z\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 39.4%

          \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, t\right), \frac{-1}{2}\right), z\right)\right)\right)\right)\right) \]

   13. Simplified 39.4%

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

   if 5.9999999999999999e156 < z

   1. Initial program 16.9%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 16.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 16.7%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 100.0%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{+156}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{z + \frac{\left(t \cdot a\right) \cdot -0.5}{z}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.1% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{y\_m \cdot \left(z\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 1.45e-74) (/ (* y_m (* z_m x_m)) z_m) (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.45e-74) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.45d-74) then
        tmp = (y_m * (z_m * x_m)) / z_m
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.45e-74) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.45e-74:
		tmp = (y_m * (z_m * x_m)) / z_m
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.45e-74)
		tmp = Float64(Float64(y_m * Float64(z_m * x_m)) / z_m);
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.45e-74)
		tmp = (y_m * (z_m * x_m)) / z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.45e-74], N[(N[(y$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.45e-74

   1. Initial program 65.9%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 66.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 66.1%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{z}\right) \]
   6. Step-by-step derivation

   7. Simplified 25.5%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(x \cdot y\right)\right), z\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(z \cdot x\right) \cdot y\right), z\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(z \cdot x\right), y\right), z\right) \]

      5. *-lowering-*.f64 26.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, x\right), y\right), z\right) \]

   9. Applied egg-rr 26.0%

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

   if 1.45e-74 < z

   1. Initial program 61.1%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 58.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 58.6%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 83.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 83.6%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.45 \cdot 10^{-74}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{z\_m \cdot \left(x\_m \cdot y\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 7.5e-98) (/ (* z_m (* x_m y_m)) z_m) (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-98) {
		tmp = (z_m * (x_m * y_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 7.5d-98) then
        tmp = (z_m * (x_m * y_m)) / z_m
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7.5e-98) {
		tmp = (z_m * (x_m * y_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7.5e-98:
		tmp = (z_m * (x_m * y_m)) / z_m
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7.5e-98)
		tmp = Float64(Float64(z_m * Float64(x_m * y_m)) / z_m);
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7.5e-98)
		tmp = (z_m * (x_m * y_m)) / z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7.5e-98], N[(N[(z$95$m * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.5000000000000006e-98

   1. Initial program 65.6%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{z}\right) \]
   4. Step-by-step derivation

   5. Simplified 24.3%

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

   if 7.5000000000000006e-98 < z

   1. Initial program 61.7%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 60.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 60.4%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 81.1%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 81.1%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{z \cdot \left(x \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 7 \cdot 10^{-98}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 7e-98) (/ (* x_m (* z_m y_m)) z_m) (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7e-98) {
		tmp = (x_m * (z_m * y_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 7d-98) then
        tmp = (x_m * (z_m * y_m)) / z_m
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 7e-98) {
		tmp = (x_m * (z_m * y_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 7e-98:
		tmp = (x_m * (z_m * y_m)) / z_m
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 7e-98)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / z_m);
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 7e-98)
		tmp = (x_m * (z_m * y_m)) / z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 7e-98], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 7.0000000000000004e-98

   1. Initial program 65.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 65.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 65.3%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{z}\right) \]
   6. Step-by-step derivation

   7. Simplified 25.4%

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

   if 7.0000000000000004e-98 < z

   1. Initial program 61.7%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 60.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 60.4%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 81.1%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 81.1%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 7 \cdot 10^{-98}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.2 \cdot 10^{-183}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 2.2e-183) (* (* z_m x_m) (/ y_m z_m)) (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.2e-183) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.2d-183) then
        tmp = (z_m * x_m) * (y_m / z_m)
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.2e-183) {
		tmp = (z_m * x_m) * (y_m / z_m);
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.2e-183:
		tmp = (z_m * x_m) * (y_m / z_m)
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.2e-183)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.2e-183)
		tmp = (z_m * x_m) * (y_m / z_m);
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.2e-183], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.2e-183

   1. Initial program 63.4%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 63.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 63.6%

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

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{z}\right) \]
   6. Step-by-step derivation

   7. Simplified 24.1%

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

      1. div-inv N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(x \cdot y\right), \color{blue}{\left(z \cdot \frac{1}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), \left(\color{blue}{z} \cdot \frac{1}{z}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \color{blue}{\left(\frac{1}{z}\right)}\right)\right) \]

      7. /-lowering-/.f64 15.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), \mathsf{*.f64}\left(z, \mathsf{/.f64}\left(1, \color{blue}{z}\right)\right)\right) \]

   9. Applied egg-rr 15.5%

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

       1. associate-*r* N/A

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

       2. div-inv N/A

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

       3. *-commutative N/A

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

       4. associate-*r/ N/A

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

       5. associate-/l* N/A

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

       6. associate-*r* N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(\left(z \cdot x\right), \color{blue}{\left(\frac{y}{z}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, x\right), \left(\frac{\color{blue}{y}}{z}\right)\right) \]

       9. /-lowering-/.f64 20.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(z, x\right), \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   11. Applied egg-rr 20.3%

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

   if 2.2e-183 < z

   1. Initial program 65.6%

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

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 63.6%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 63.6%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 75.2%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 75.2%

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

Alternative 13: 72.7% accurate, 37.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 (* x_m y_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.3%

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

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

      \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

   8. *-lowering-*.f64 63.6%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

3. Simplified 63.6%

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

5. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 39.3%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

7. Simplified 39.3%

   \[\leadsto \color{blue}{x \cdot y} \]
8. Add Preprocessing

Developer Target 1: 87.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024138 
(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)))))