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

Percentage Accurate: 61.8% → 92.7%

Time: 14.0s

Alternatives: 12

Speedup: 37.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{y\_m}{\frac{\sqrt{z\_m \cdot z\_m - t \cdot a}}{z\_m}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{1 + \frac{a}{\frac{z\_m}{\frac{-0.5}{\frac{z\_m}{t}}}}}\\ \end{array}\right)\right) \end{array} \]

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 1.5e+152)
      (* (/ y_m (/ (sqrt (- (* z_m z_m) (* t a))) z_m)) x_m)
      (* y_m (/ x_m (+ 1.0 (/ a (/ z_m (/ -0.5 (/ z_m t))))))))))))

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 <= 1.5e+152) {
		tmp = (y_m / (sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m;
	} else {
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	}
	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 <= 1.5d+152) then
        tmp = (y_m / (sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m
    else
        tmp = y_m * (x_m / (1.0d0 + (a / (z_m / ((-0.5d0) / (z_m / t))))))
    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 <= 1.5e+152) {
		tmp = (y_m / (Math.sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m;
	} else {
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	}
	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 <= 1.5e+152:
		tmp = (y_m / (math.sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m
	else:
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))))
	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 <= 1.5e+152)
		tmp = Float64(Float64(y_m / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))) / z_m)) * x_m);
	else
		tmp = Float64(y_m * Float64(x_m / Float64(1.0 + Float64(a / Float64(z_m / Float64(-0.5 / Float64(z_m / t)))))));
	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 <= 1.5e+152)
		tmp = (y_m / (sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m;
	else
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	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, 1.5e+152], N[(N[(y$95$m / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(1.0 + N[(a / N[(z$95$m / N[(-0.5 / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.49999999999999995e152

   1. Initial program 69.5%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 72.7%

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

   4. Applied egg-rr 72.7%

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

   if 1.49999999999999995e152 < z

   1. Initial program 11.5%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 71.6%

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

   5. Simplified 71.6%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. div-inv N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. clear-num N/A

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

      14. associate-/r* N/A

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

      15. clear-num N/A

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

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

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

      17. /-lowering-/.f64 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. frac-times N/A

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

       2. associate-/l* N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-*l/ N/A

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

       11. associate-/r/ N/A

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

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

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

       13. clear-num N/A

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

       14. associate-/r* N/A

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

       15. associate-/r/ N/A

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

       16. associate-*l/ N/A

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

       17. clear-num N/A

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

       18. /-lowering-/.f64 N/A

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

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{y \cdot \frac{x}{1 + \frac{a}{\frac{z}{\frac{-0.5}{\frac{z}{t}}}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 90.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.6 \cdot 10^{-158}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{{\left(0 - t \cdot a\right)}^{0.5}}\\ \mathbf{elif}\;z\_m \leq 1.6 \cdot 10^{+152}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{1 + \frac{a}{\frac{z\_m}{\frac{-0.5}{\frac{z\_m}{t}}}}}\\ \end{array}\right)\right) \end{array} \]

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 1.6e-158)
      (* x_m (/ (* z_m y_m) (pow (- 0.0 (* t a)) 0.5)))
      (if (<= z_m 1.6e+152)
        (* (* z_m x_m) (/ y_m (sqrt (- (* z_m z_m) (* t a)))))
        (* y_m (/ x_m (+ 1.0 (/ a (/ z_m (/ -0.5 (/ z_m t)))))))))))))

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 <= 1.6e-158) {
		tmp = x_m * ((z_m * y_m) / pow((0.0 - (t * a)), 0.5));
	} else if (z_m <= 1.6e+152) {
		tmp = (z_m * x_m) * (y_m / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	}
	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 <= 1.6d-158) then
        tmp = x_m * ((z_m * y_m) / ((0.0d0 - (t * a)) ** 0.5d0))
    else if (z_m <= 1.6d+152) then
        tmp = (z_m * x_m) * (y_m / sqrt(((z_m * z_m) - (t * a))))
    else
        tmp = y_m * (x_m / (1.0d0 + (a / (z_m / ((-0.5d0) / (z_m / t))))))
    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 <= 1.6e-158) {
		tmp = x_m * ((z_m * y_m) / Math.pow((0.0 - (t * a)), 0.5));
	} else if (z_m <= 1.6e+152) {
		tmp = (z_m * x_m) * (y_m / Math.sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	}
	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 <= 1.6e-158:
		tmp = x_m * ((z_m * y_m) / math.pow((0.0 - (t * a)), 0.5))
	elif z_m <= 1.6e+152:
		tmp = (z_m * x_m) * (y_m / math.sqrt(((z_m * z_m) - (t * a))))
	else:
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))))
	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 <= 1.6e-158)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / (Float64(0.0 - Float64(t * a)) ^ 0.5)));
	elseif (z_m <= 1.6e+152)
		tmp = Float64(Float64(z_m * x_m) * Float64(y_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(1.0 + Float64(a / Float64(z_m / Float64(-0.5 / Float64(z_m / t)))))));
	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 <= 1.6e-158)
		tmp = x_m * ((z_m * y_m) / ((0.0 - (t * a)) ^ 0.5));
	elseif (z_m <= 1.6e+152)
		tmp = (z_m * x_m) * (y_m / sqrt(((z_m * z_m) - (t * a))));
	else
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	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, 1.6e-158], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Power[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.6e+152], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(1.0 + N[(a / N[(z$95$m / N[(-0.5 / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 1.59999999999999998e-158

   1. Initial program 61.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 35.4%

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

   5. Simplified 35.4%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{0 - a \cdot t}}} \]
   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. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left({\left(0 - a \cdot t\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{pow.f64}\left(\left(0 - a \cdot t\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right), \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 35.8%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right), \frac{1}{2}\right)\right)\right) \]

   7. Applied egg-rr 35.8%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot z}{{\left(0 - a \cdot t\right)}^{0.5}}} \]

   if 1.59999999999999998e-158 < z < 1.60000000000000003e152

   1. Initial program 89.3%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-/l* N/A

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      11. *-lowering-*.f64 80.6%

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

   4. Applied egg-rr 80.6%

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

   if 1.60000000000000003e152 < z

   1. Initial program 11.5%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 71.6%

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

   5. Simplified 71.6%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. div-inv N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. clear-num N/A

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

      14. associate-/r* N/A

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

      15. clear-num N/A

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

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

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

      17. /-lowering-/.f64 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. frac-times N/A

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

       2. associate-/l* N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-*l/ N/A

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

       11. associate-/r/ N/A

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

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

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

       13. clear-num N/A

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

       14. associate-/r* N/A

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

       15. associate-/r/ N/A

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

       16. associate-*l/ N/A

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

       17. clear-num N/A

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

       18. /-lowering-/.f64 N/A

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

   11. Applied egg-rr 100.0%

       \[\leadsto \color{blue}{y \cdot \frac{x}{1 + \frac{a}{\frac{z}{\frac{-0.5}{\frac{z}{t}}}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.6 \cdot 10^{-158}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{{\left(0 - t \cdot a\right)}^{0.5}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+152}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{1 + \frac{a}{\frac{z}{\frac{-0.5}{\frac{z}{t}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.1 \cdot 10^{+36}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{1 + \frac{a}{\frac{z\_m}{\frac{-0.5}{\frac{z\_m}{t}}}}}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 4.1e+36)
      (* (* z_m y_m) (/ x_m (sqrt (- (* z_m z_m) (* t a)))))
      (* y_m (/ x_m (+ 1.0 (/ a (/ z_m (/ -0.5 (/ z_m t))))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.1e+36) {
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	}
	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 <= 4.1d+36) then
        tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))))
    else
        tmp = y_m * (x_m / (1.0d0 + (a / (z_m / ((-0.5d0) / (z_m / t))))))
    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 <= 4.1e+36) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	}
	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 <= 4.1e+36:
		tmp = (z_m * y_m) * (x_m / math.sqrt(((z_m * z_m) - (t * a))))
	else:
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.1e+36)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(1.0 + Float64(a / Float64(z_m / Float64(-0.5 / Float64(z_m / t)))))));
	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 <= 4.1e+36)
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	else
		tmp = y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t))))));
	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, 4.1e+36], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(1.0 + N[(a / N[(z$95$m / N[(-0.5 / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.10000000000000013e36

   1. Initial program 66.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 64.0%

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

   4. Applied egg-rr 64.0%

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

   if 4.10000000000000013e36 < z

   1. Initial program 42.9%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 76.4%

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

   5. Simplified 76.4%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. div-inv N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. clear-num N/A

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

      14. associate-/r* N/A

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

      15. clear-num N/A

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

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

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

      17. /-lowering-/.f64 96.4%

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

   7. Applied egg-rr 96.4%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 96.4%

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

   9. Applied egg-rr 96.4%

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

       1. frac-times N/A

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

       2. associate-/l* N/A

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

       3. associate-*r/ N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

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

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

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

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

       8. associate-*r/ N/A

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

       9. *-commutative N/A

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

       10. associate-*l/ N/A

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

       11. associate-/r/ N/A

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

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

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

       13. clear-num N/A

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

       14. associate-/r* N/A

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

       15. associate-/r/ N/A

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

       16. associate-*l/ N/A

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

       17. clear-num N/A

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

       18. /-lowering-/.f64 N/A

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

   11. Applied egg-rr 96.4%

       \[\leadsto \color{blue}{y \cdot \frac{x}{1 + \frac{a}{\frac{z}{\frac{-0.5}{\frac{z}{t}}}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.1 \cdot 10^{+36}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{1 + \frac{a}{\frac{z}{\frac{-0.5}{\frac{z}{t}}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.9 \cdot 10^{-83}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{{\left(0 - t \cdot a\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{1 + \frac{-0.5}{\frac{z\_m}{t}} \cdot \frac{a}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 2.9e-83)
      (* x_m (/ (* z_m y_m) (pow (- 0.0 (* t a)) 0.5)))
      (/ (* y_m x_m) (+ 1.0 (* (/ -0.5 (/ z_m t)) (/ a 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-83) {
		tmp = x_m * ((z_m * y_m) / pow((0.0 - (t * a)), 0.5));
	} else {
		tmp = (y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / 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-83) then
        tmp = x_m * ((z_m * y_m) / ((0.0d0 - (t * a)) ** 0.5d0))
    else
        tmp = (y_m * x_m) / (1.0d0 + (((-0.5d0) / (z_m / t)) * (a / 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-83) {
		tmp = x_m * ((z_m * y_m) / Math.pow((0.0 - (t * a)), 0.5));
	} else {
		tmp = (y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / 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-83:
		tmp = x_m * ((z_m * y_m) / math.pow((0.0 - (t * a)), 0.5))
	else:
		tmp = (y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / 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-83)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / (Float64(0.0 - Float64(t * a)) ^ 0.5)));
	else
		tmp = Float64(Float64(y_m * x_m) / Float64(1.0 + Float64(Float64(-0.5 / Float64(z_m / t)) * Float64(a / 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-83)
		tmp = x_m * ((z_m * y_m) / ((0.0 - (t * a)) ^ 0.5));
	else
		tmp = (y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / 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-83], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Power[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(1.0 + N[(N[(-0.5 / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision] * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.8999999999999999e-83

   1. Initial program 63.8%

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

   3. Taylor expanded in z around 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 37.2%

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

   5. Simplified 37.2%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{0 - a \cdot t}}} \]
   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. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left({\left(0 - a \cdot t\right)}^{\color{blue}{\frac{1}{2}}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{pow.f64}\left(\left(0 - a \cdot t\right), \color{blue}{\frac{1}{2}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(a \cdot t\right)\right), \frac{1}{2}\right)\right)\right) \]

      9. *-lowering-*.f64 37.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(a, t\right)\right), \frac{1}{2}\right)\right)\right) \]

   7. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{x \cdot \frac{y \cdot z}{{\left(0 - a \cdot t\right)}^{0.5}}} \]

   if 2.8999999999999999e-83 < z

   1. Initial program 52.4%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 78.2%

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

   5. Simplified 78.2%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. div-inv N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. clear-num N/A

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

      14. associate-/r* N/A

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

      15. clear-num N/A

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

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

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

      17. /-lowering-/.f64 95.0%

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

   7. Applied egg-rr 95.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 95.0%

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

   9. Applied egg-rr 95.0%

      \[\leadsto \frac{x \cdot y}{1 + \color{blue}{\frac{a}{z} \cdot \frac{-0.5}{\frac{z}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{-83}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{{\left(0 - t \cdot a\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{-0.5}{\frac{z}{t}} \cdot \frac{a}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{1 + \frac{-0.5}{\frac{z\_m}{t}} \cdot \frac{a}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 4.5e-82)
      (/ (* x_m (* z_m y_m)) (sqrt (- 0.0 (* t a))))
      (/ (* y_m x_m) (+ 1.0 (* (/ -0.5 (/ z_m t)) (/ a 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 <= 4.5e-82) {
		tmp = (x_m * (z_m * y_m)) / sqrt((0.0 - (t * a)));
	} else {
		tmp = (y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / 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 <= 4.5d-82) then
        tmp = (x_m * (z_m * y_m)) / sqrt((0.0d0 - (t * a)))
    else
        tmp = (y_m * x_m) / (1.0d0 + (((-0.5d0) / (z_m / t)) * (a / 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 <= 4.5e-82) {
		tmp = (x_m * (z_m * y_m)) / Math.sqrt((0.0 - (t * a)));
	} else {
		tmp = (y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / 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 <= 4.5e-82:
		tmp = (x_m * (z_m * y_m)) / math.sqrt((0.0 - (t * a)))
	else:
		tmp = (y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / 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 <= 4.5e-82)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(0.0 - Float64(t * a))));
	else
		tmp = Float64(Float64(y_m * x_m) / Float64(1.0 + Float64(Float64(-0.5 / Float64(z_m / t)) * Float64(a / 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 <= 4.5e-82)
		tmp = (x_m * (z_m * y_m)) / sqrt((0.0 - (t * a)));
	else
		tmp = (y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / 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, 4.5e-82], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(1.0 + N[(N[(-0.5 / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision] * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.4999999999999998e-82

   1. Initial program 63.8%

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

   3. Taylor expanded in z around 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 37.2%

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

   5. Simplified 37.2%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 36.5%

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

   7. Applied egg-rr 36.5%

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

   if 4.4999999999999998e-82 < z

   1. Initial program 52.4%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 78.2%

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

   5. Simplified 78.2%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. div-inv N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. clear-num N/A

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

      14. associate-/r* N/A

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

      15. clear-num N/A

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

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

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

      17. /-lowering-/.f64 95.0%

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

   7. Applied egg-rr 95.0%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 95.0%

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

   9. Applied egg-rr 95.0%

      \[\leadsto \frac{x \cdot y}{1 + \color{blue}{\frac{a}{z} \cdot \frac{-0.5}{\frac{z}{t}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-82}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{-0.5}{\frac{z}{t}} \cdot \frac{a}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.75 \cdot 10^{-117}:\\ \;\;\;\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{1 + \frac{-0.5}{\frac{\frac{z\_m}{\frac{t}{z\_m}}}{a}}}\\ \end{array}\right)\right) \end{array} \]

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 2.75e-117)
      (/ (* z_m (* y_m x_m)) (sqrt (- 0.0 (* t a))))
      (/ (* y_m x_m) (+ 1.0 (/ -0.5 (/ (/ z_m (/ t z_m)) a)))))))))

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 <= 2.75e-117) {
		tmp = (z_m * (y_m * x_m)) / sqrt((0.0 - (t * a)));
	} else {
		tmp = (y_m * x_m) / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a)));
	}
	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 <= 2.75d-117) then
        tmp = (z_m * (y_m * x_m)) / sqrt((0.0d0 - (t * a)))
    else
        tmp = (y_m * x_m) / (1.0d0 + ((-0.5d0) / ((z_m / (t / z_m)) / a)))
    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 <= 2.75e-117) {
		tmp = (z_m * (y_m * x_m)) / Math.sqrt((0.0 - (t * a)));
	} else {
		tmp = (y_m * x_m) / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a)));
	}
	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 <= 2.75e-117:
		tmp = (z_m * (y_m * x_m)) / math.sqrt((0.0 - (t * a)))
	else:
		tmp = (y_m * x_m) / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a)))
	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 <= 2.75e-117)
		tmp = Float64(Float64(z_m * Float64(y_m * x_m)) / sqrt(Float64(0.0 - Float64(t * a))));
	else
		tmp = Float64(Float64(y_m * x_m) / Float64(1.0 + Float64(-0.5 / Float64(Float64(z_m / Float64(t / z_m)) / a))));
	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 <= 2.75e-117)
		tmp = (z_m * (y_m * x_m)) / sqrt((0.0 - (t * a)));
	else
		tmp = (y_m * x_m) / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a)));
	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, 2.75e-117], N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(1.0 + N[(-0.5 / N[(N[(z$95$m / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.75000000000000013e-117

   1. Initial program 62.5%

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

   3. Taylor expanded in z around 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 36.4%

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

   5. Simplified 36.4%

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

   if 2.75000000000000013e-117 < z

   1. Initial program 55.4%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 75.3%

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

   5. Simplified 75.3%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. *-inverses N/A

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

      4. div-inv N/A

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

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

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

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

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

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

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

      8. associate-*r* N/A

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

      9. clear-num N/A

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

      10. un-div-inv N/A

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

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

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

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

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

      13. clear-num N/A

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

      14. associate-/r* N/A

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

      15. clear-num N/A

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

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

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

      17. /-lowering-/.f64 90.7%

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

   7. Applied egg-rr 90.7%

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

      1. *-commutative N/A

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

      2. div-inv N/A

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

      3. times-frac N/A

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

      4. clear-num N/A

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

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

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

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

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

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

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

      8. /-lowering-/.f64 90.7%

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

   9. Applied egg-rr 90.7%

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

       1. *-commutative N/A

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

       2. frac-times N/A

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

       3. associate-/r/ N/A

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

       4. associate-/l* N/A

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

       5. clear-num N/A

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

       6. un-div-inv N/A

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

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

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

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

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

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

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

       10. /-lowering-/.f64 90.7%

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

   11. Applied egg-rr 90.7%

       \[\leadsto \frac{x \cdot y}{1 + \color{blue}{\frac{-0.5}{\frac{\frac{z}{\frac{t}{z}}}{a}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.75 \cdot 10^{-117}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{1 + \frac{-0.5}{\frac{\frac{z}{\frac{t}{z}}}{a}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \frac{y\_m \cdot x\_m}{1 + \frac{-0.5}{\frac{\frac{z\_m}{\frac{t}{z\_m}}}{a}}}\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * ((y_m * x_m) / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a))))))

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(Float64(y_m * x_m) / Float64(1.0 + Float64(-0.5 / Float64(Float64(z_m / Float64(t / z_m)) / a)))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * ((y_m * x_m) / (1.0 + (-0.5 / ((z_m / (t / z_m)) / a))))));
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[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(1.0 + N[(-0.5 / N[(N[(z$95$m / N[(t / z$95$m), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in z around inf

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

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

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

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

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 42.8%

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

5. Simplified 42.8%

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

   1. associate-/l* N/A

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

   2. associate-/r* N/A

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

   3. *-inverses N/A

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

   4. div-inv N/A

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

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

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

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

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

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

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

   8. associate-*r* N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

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

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

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

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

   13. clear-num N/A

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

   14. associate-/r* N/A

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

   15. clear-num N/A

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

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

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

   17. /-lowering-/.f64 49.5%

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

7. Applied egg-rr 49.5%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. times-frac N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

   8. /-lowering-/.f64 49.5%

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

9. Applied egg-rr 49.5%

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

    1. *-commutative N/A

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

    2. frac-times N/A

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

    3. associate-/r/ N/A

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

    4. associate-/l* N/A

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

    5. clear-num N/A

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

    6. un-div-inv N/A

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

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

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

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

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

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

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

    10. /-lowering-/.f64 49.5%

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

11. Applied egg-rr 49.5%

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

12. Final simplification 49.5%

    \[\leadsto \frac{y \cdot x}{1 + \frac{-0.5}{\frac{\frac{z}{\frac{t}{z}}}{a}}} \]
13. Add Preprocessing

Alternative 8: 78.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \frac{y\_m \cdot x\_m}{1 + \frac{-0.5}{\frac{z\_m}{t}} \cdot \frac{a}{z\_m}}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (/ (* y_m x_m) (+ 1.0 (* (/ -0.5 (/ z_m t)) (/ a 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) {
	return z_s * (y_s * (x_s * ((y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / z_m))))));
}

Fortran

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

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * ((y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / z_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(Float64(y_m * x_m) / Float64(1.0 + Float64(Float64(-0.5 / Float64(z_m / t)) * Float64(a / z_m)))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * ((y_m * x_m) / (1.0 + ((-0.5 / (z_m / t)) * (a / z_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[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(1.0 + N[(N[(-0.5 / N[(z$95$m / t), $MachinePrecision]), $MachinePrecision] * N[(a / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in z around inf

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

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

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

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

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 42.8%

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

5. Simplified 42.8%

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

   1. associate-/l* N/A

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

   2. associate-/r* N/A

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

   3. *-inverses N/A

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

   4. div-inv N/A

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

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

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

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

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

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

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

   8. associate-*r* N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

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

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

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

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

   13. clear-num N/A

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

   14. associate-/r* N/A

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

   15. clear-num N/A

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

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

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

   17. /-lowering-/.f64 49.5%

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

7. Applied egg-rr 49.5%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. times-frac N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

   8. /-lowering-/.f64 49.5%

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

9. Applied egg-rr 49.5%

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

10. Final simplification 49.5%

    \[\leadsto \frac{y \cdot x}{1 + \frac{-0.5}{\frac{z}{t}} \cdot \frac{a}{z}} \]
11. Add Preprocessing

Alternative 9: 78.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \left(y\_m \cdot \frac{x\_m}{1 + \frac{a}{\frac{z\_m}{\frac{-0.5}{\frac{z\_m}{t}}}}}\right)\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * (y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t)))))))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(y_m * Float64(x_m / Float64(1.0 + Float64(a / Float64(z_m / Float64(-0.5 / Float64(z_m / t))))))))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (y_m * (x_m / (1.0 + (a / (z_m / (-0.5 / (z_m / t)))))))));
end

Wolfram

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

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in z around inf

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

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

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

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

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 42.8%

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

5. Simplified 42.8%

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

   1. associate-/l* N/A

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

   2. associate-/r* N/A

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

   3. *-inverses N/A

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

   4. div-inv N/A

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

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

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

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

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

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

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

   8. associate-*r* N/A

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

   9. clear-num N/A

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

   10. un-div-inv N/A

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

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

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

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

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

   13. clear-num N/A

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

   14. associate-/r* N/A

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

   15. clear-num N/A

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

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

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

   17. /-lowering-/.f64 49.5%

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

7. Applied egg-rr 49.5%

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

   1. *-commutative N/A

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

   2. div-inv N/A

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

   3. times-frac N/A

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

   4. clear-num N/A

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

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

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

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

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

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

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

   8. /-lowering-/.f64 49.5%

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

9. Applied egg-rr 49.5%

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

    1. frac-times N/A

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

    2. associate-/l* N/A

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

    3. associate-*r/ N/A

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

    4. *-commutative N/A

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

    5. associate-/l* N/A

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

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

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

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

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

    8. associate-*r/ N/A

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

    9. *-commutative N/A

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

    10. associate-*l/ N/A

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

    11. associate-/r/ N/A

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

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

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

    13. clear-num N/A

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

    14. associate-/r* N/A

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

    15. associate-/r/ N/A

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

    16. associate-*l/ N/A

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

    17. clear-num N/A

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

    18. /-lowering-/.f64 N/A

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

11. Applied egg-rr 49.2%

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

Alternative 10: 75.7% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{y\_m \cdot \left(z\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

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 2.4e-105) (/ (* y_m (* z_m x_m)) z_m) (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.4e-105) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.4d-105) then
        tmp = (y_m * (z_m * x_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.4e-105) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.4e-105:
		tmp = (y_m * (z_m * x_m)) / z_m
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.4e-105)
		tmp = Float64(Float64(y_m * Float64(z_m * x_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.4e-105)
		tmp = (y_m * (z_m * x_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.4e-105], N[(N[(y$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.40000000000000015e-105

   1. Initial program 62.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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 19.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(x \cdot y\right)\right), z\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(z \cdot x\right) \cdot y\right), z\right) \]

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

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

      4. *-lowering-*.f64 19.8%

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

   7. Applied egg-rr 19.8%

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

   if 2.40000000000000015e-105 < z

   1. Initial program 55.4%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 87.4%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 87.4%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (* y_s (* x_s (if (<= z_m 2e-64) (/ (* z_m (* y_m x_m)) z_m) (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e-64) {
		tmp = (z_m * (y_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2d-64) then
        tmp = (z_m * (y_m * x_m)) / z_m
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e-64) {
		tmp = (z_m * (y_m * x_m)) / z_m;
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2e-64:
		tmp = (z_m * (y_m * x_m)) / z_m
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e-64)
		tmp = Float64(Float64(z_m * Float64(y_m * x_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2e-64)
		tmp = (z_m * (y_m * x_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2e-64], N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.99999999999999993e-64

   1. Initial program 64.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 21.4%

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

   if 1.99999999999999993e-64 < z

   1. Initial program 51.3%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 91.6%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   5. Simplified 91.6%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{-64}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.2% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.8%

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

3. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 42.7%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

5. Simplified 42.7%

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

6. Final simplification 42.7%

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

Developer Target 1: 87.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024160 
(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)))))