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

Percentage Accurate: 61.1% → 93.0%

Time: 16.8s

Alternatives: 13

Speedup: 37.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.4 \cdot 10^{+60}:\\ \;\;\;\;\frac{x\_m}{\frac{\sqrt{z\_m \cdot z\_m - t \cdot a}}{z\_m \cdot y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\frac{z\_m + t \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.4e+60)
      (/ x_m (/ (sqrt (- (* z_m z_m) (* t a))) (* z_m y_m)))
      (* y_m (/ x_m (/ (+ z_m (* t (* a (/ -0.5 z_m)))) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.4e+60) {
		tmp = x_m / (sqrt(((z_m * z_m) - (t * a))) / (z_m * y_m));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 1.4d+60) then
        tmp = x_m / (sqrt(((z_m * z_m) - (t * a))) / (z_m * y_m))
    else
        tmp = y_m * (x_m / ((z_m + (t * (a * ((-0.5d0) / z_m)))) / 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 <= 1.4e+60) {
		tmp = x_m / (Math.sqrt(((z_m * z_m) - (t * a))) / (z_m * y_m));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 1.4e+60:
		tmp = x_m / (math.sqrt(((z_m * z_m) - (t * a))) / (z_m * y_m))
	else:
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.4e+60)
		tmp = Float64(x_m / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))) / Float64(z_m * y_m)));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z_m + Float64(t * Float64(a * Float64(-0.5 / z_m)))) / 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 <= 1.4e+60)
		tmp = x_m / (sqrt(((z_m * z_m) - (t * a))) / (z_m * y_m));
	else
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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, 1.4e+60], N[(x$95$m / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(N[(z$95$m + N[(t * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.4e60

   1. Initial program 72.0%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 73.4%

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

   3. Simplified 73.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 76.0%

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

   6. Applied egg-rr 76.0%

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

   if 1.4e60 < z

   1. Initial program 36.9%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 75.5%

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

   5. Simplified 75.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. /-lowering-/.f64 90.9%

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

   7. Applied egg-rr 90.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-inv N/A

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

      5. clear-num N/A

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

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

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

      7. /-lowering-/.f64 93.6%

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

   9. Applied egg-rr 93.6%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.4 \cdot 10^{+60}:\\ \;\;\;\;\frac{x}{\frac{\sqrt{z \cdot z - t \cdot a}}{z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + t \cdot \left(a \cdot \frac{-0.5}{z}\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.2 \cdot 10^{-133}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{elif}\;z\_m \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;z\_m \cdot \left(x\_m \cdot \frac{y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\frac{z\_m + t \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 2.2e-133)
      (* x_m (/ (* z_m y_m) (sqrt (- 0.0 (* t a)))))
      (if (<= z_m 1.6e+50)
        (* z_m (* x_m (/ y_m (sqrt (- (* z_m z_m) (* t a))))))
        (* y_m (/ x_m (/ (+ z_m (* t (* a (/ -0.5 z_m)))) z_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.2e-133) {
		tmp = x_m * ((z_m * y_m) / sqrt((0.0 - (t * a))));
	} else if (z_m <= 1.6e+50) {
		tmp = z_m * (x_m * (y_m / sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 2.2d-133) then
        tmp = x_m * ((z_m * y_m) / sqrt((0.0d0 - (t * a))))
    else if (z_m <= 1.6d+50) then
        tmp = z_m * (x_m * (y_m / sqrt(((z_m * z_m) - (t * a)))))
    else
        tmp = y_m * (x_m / ((z_m + (t * (a * ((-0.5d0) / z_m)))) / 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 <= 2.2e-133) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt((0.0 - (t * a))));
	} else if (z_m <= 1.6e+50) {
		tmp = z_m * (x_m * (y_m / Math.sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 2.2e-133:
		tmp = x_m * ((z_m * y_m) / math.sqrt((0.0 - (t * a))))
	elif z_m <= 1.6e+50:
		tmp = z_m * (x_m * (y_m / math.sqrt(((z_m * z_m) - (t * a)))))
	else:
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.2e-133)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(0.0 - Float64(t * a)))));
	elseif (z_m <= 1.6e+50)
		tmp = Float64(z_m * Float64(x_m * Float64(y_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z_m + Float64(t * Float64(a * Float64(-0.5 / z_m)))) / 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 <= 2.2e-133)
		tmp = x_m * ((z_m * y_m) / sqrt((0.0 - (t * a))));
	elseif (z_m <= 1.6e+50)
		tmp = z_m * (x_m * (y_m / sqrt(((z_m * z_m) - (t * a)))));
	else
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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, 2.2e-133], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.6e+50], N[(z$95$m * N[(x$95$m * N[(y$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(N[(z$95$m + N[(t * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.2000000000000001e-133

   1. Initial program 65.2%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 46.2%

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

   7. Simplified 46.2%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. *-lowering-*.f64 44.9%

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

   9. Applied egg-rr 44.9%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{{\left(0 - a \cdot t\right)}^{0.5}}} \]
   10. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

       7. unpow1/2 N/A

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

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

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

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

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

       10. *-lowering-*.f64 45.5%

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

   11. Applied egg-rr 45.5%

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

   if 2.2000000000000001e-133 < z < 1.59999999999999991e50

   1. Initial program 89.3%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 91.2%

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

   3. Simplified 91.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 93.2%

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

   6. Applied egg-rr 93.2%

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

   if 1.59999999999999991e50 < z

   1. Initial program 40.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 76.7%

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

   5. Simplified 76.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. /-lowering-/.f64 91.3%

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

   7. Applied egg-rr 91.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-inv N/A

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

      5. clear-num N/A

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

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

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

      7. /-lowering-/.f64 93.9%

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

   9. Applied egg-rr 93.9%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.2 \cdot 10^{-133}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{0 - t \cdot a}}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;z \cdot \left(x \cdot \frac{y}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + t \cdot \left(a \cdot \frac{-0.5}{z}\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.2 \cdot 10^{+50}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\frac{z\_m + t \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.2e+50)
      (* (* z_m y_m) (/ x_m (sqrt (- (* z_m z_m) (* t a)))))
      (* y_m (/ x_m (/ (+ z_m (* t (* a (/ -0.5 z_m)))) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.2e+50) {
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 1.2d+50) then
        tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))))
    else
        tmp = y_m * (x_m / ((z_m + (t * (a * ((-0.5d0) / z_m)))) / 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 <= 1.2e+50) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 1.2e+50:
		tmp = (z_m * y_m) * (x_m / math.sqrt(((z_m * z_m) - (t * a))))
	else:
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.2e+50)
		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(Float64(z_m + Float64(t * Float64(a * Float64(-0.5 / z_m)))) / 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 <= 1.2e+50)
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	else
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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, 1.2e+50], 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[(N[(z$95$m + N[(t * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.2000000000000001e50

   1. Initial program 71.3%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 72.9%

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

   3. Simplified 72.9%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 72.8%

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

   6. Applied egg-rr 72.8%

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

   if 1.2000000000000001e50 < z

   1. Initial program 40.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 76.7%

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

   5. Simplified 76.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. /-lowering-/.f64 91.3%

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

   7. Applied egg-rr 91.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-inv N/A

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

      5. clear-num N/A

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

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

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

      7. /-lowering-/.f64 93.9%

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

   9. Applied egg-rr 93.9%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+50}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + t \cdot \left(a \cdot \frac{-0.5}{z}\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.95 \cdot 10^{-111}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\frac{z\_m + t \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 1.95e-111)
      (* x_m (/ (* z_m y_m) (sqrt (- 0.0 (* t a)))))
      (* y_m (/ x_m (/ (+ z_m (* t (* a (/ -0.5 z_m)))) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e-111) {
		tmp = x_m * ((z_m * y_m) / sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 1.95d-111) then
        tmp = x_m * ((z_m * y_m) / sqrt((0.0d0 - (t * a))))
    else
        tmp = y_m * (x_m / ((z_m + (t * (a * ((-0.5d0) / z_m)))) / 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 <= 1.95e-111) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 1.95e-111:
		tmp = x_m * ((z_m * y_m) / math.sqrt((0.0 - (t * a))))
	else:
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.95e-111)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(0.0 - Float64(t * a)))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z_m + Float64(t * Float64(a * Float64(-0.5 / z_m)))) / 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 <= 1.95e-111)
		tmp = x_m * ((z_m * y_m) / sqrt((0.0 - (t * a))));
	else
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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, 1.95e-111], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(N[(z$95$m + N[(t * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.9500000000000001e-111

   1. Initial program 65.8%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 46.8%

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

   7. Simplified 46.8%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. *-lowering-*.f64 45.5%

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

   9. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{{\left(0 - a \cdot t\right)}^{0.5}}} \]
   10. Step-by-step derivation

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

       4. associate-*r/ N/A

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

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

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

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

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

       7. unpow1/2 N/A

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

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

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

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

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

       10. *-lowering-*.f64 46.1%

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

   11. Applied egg-rr 46.1%

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

   if 1.9500000000000001e-111 < z

   1. Initial program 57.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 74.3%

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

   5. Simplified 74.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. /-lowering-/.f64 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-inv N/A

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

      5. clear-num N/A

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

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

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

      7. /-lowering-/.f64 87.3%

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

   9. Applied egg-rr 87.3%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + t \cdot \left(a \cdot \frac{-0.5}{z}\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 8.5 \cdot 10^{-112}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\frac{z\_m + t \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 8.5e-112)
      (* (* z_m y_m) (/ x_m (sqrt (- 0.0 (* t a)))))
      (* y_m (/ x_m (/ (+ z_m (* t (* a (/ -0.5 z_m)))) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 8.5e-112) {
		tmp = (z_m * y_m) * (x_m / sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 8.5d-112) then
        tmp = (z_m * y_m) * (x_m / sqrt((0.0d0 - (t * a))))
    else
        tmp = y_m * (x_m / ((z_m + (t * (a * ((-0.5d0) / z_m)))) / 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 <= 8.5e-112) {
		tmp = (z_m * y_m) * (x_m / Math.sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 <= 8.5e-112:
		tmp = (z_m * y_m) * (x_m / math.sqrt((0.0 - (t * a))))
	else:
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 8.5e-112)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(0.0 - Float64(t * a)))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z_m + Float64(t * Float64(a * Float64(-0.5 / z_m)))) / 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 <= 8.5e-112)
		tmp = (z_m * y_m) * (x_m / sqrt((0.0 - (t * a))));
	else
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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, 8.5e-112], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(N[(z$95$m + N[(t * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 8.49999999999999992e-112

   1. Initial program 65.8%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.2%

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

   3. Simplified 67.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 67.2%

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

   6. Applied egg-rr 67.2%

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

   7. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 45.5%

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

   9. Simplified 45.5%

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

   if 8.49999999999999992e-112 < z

   1. Initial program 57.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 74.3%

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

   5. Simplified 74.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. /-lowering-/.f64 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-inv N/A

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

      5. clear-num N/A

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

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

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

      7. /-lowering-/.f64 87.3%

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

   9. Applied egg-rr 87.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 8.5 \cdot 10^{-112}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + t \cdot \left(a \cdot \frac{-0.5}{z}\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.1 \cdot 10^{-111}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{0 - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\frac{z\_m + t \cdot \left(a \cdot \frac{-0.5}{z\_m}\right)}{z\_m}}\\ \end{array}\right)\right) \end{array} \]

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 2.1e-111)
      (* y_m (* x_m (/ z_m (sqrt (- 0.0 (* t a))))))
      (* y_m (/ x_m (/ (+ z_m (* t (* a (/ -0.5 z_m)))) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.1e-111) {
		tmp = y_m * (x_m * (z_m / sqrt((0.0 - (t * a)))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.1d-111) then
        tmp = y_m * (x_m * (z_m / sqrt((0.0d0 - (t * a)))))
    else
        tmp = y_m * (x_m / ((z_m + (t * (a * ((-0.5d0) / z_m)))) / z_m))
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.1e-111) {
		tmp = y_m * (x_m * (z_m / Math.sqrt((0.0 - (t * a)))));
	} else {
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.1e-111:
		tmp = y_m * (x_m * (z_m / math.sqrt((0.0 - (t * a)))))
	else:
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m))
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.1e-111)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(Float64(0.0 - Float64(t * a))))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z_m + Float64(t * Float64(a * Float64(-0.5 / z_m)))) / z_m)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.1e-111)
		tmp = y_m * (x_m * (z_m / sqrt((0.0 - (t * a)))));
	else
		tmp = y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / z_m));
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.1e-111], N[(y$95$m * N[(x$95$m * N[(z$95$m / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(N[(z$95$m + N[(t * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.0999999999999999e-111

   1. Initial program 65.8%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 67.2%

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

   3. Simplified 67.2%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 46.8%

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

   7. Simplified 46.8%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

      6. pow1/2 N/A

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

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

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

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

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

      9. *-lowering-*.f64 45.5%

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

   9. Applied egg-rr 45.5%

      \[\leadsto \color{blue}{\left(x \cdot y\right) \cdot \frac{z}{{\left(0 - a \cdot t\right)}^{0.5}}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

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

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

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

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

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

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

       6. unpow1/2 N/A

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

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

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

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

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

       9. *-lowering-*.f64 46.1%

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

   11. Applied egg-rr 46.1%

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

   if 2.0999999999999999e-111 < z

   1. Initial program 57.2%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 74.3%

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

   5. Simplified 74.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

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

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

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

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

      15. /-lowering-/.f64 85.6%

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

   7. Applied egg-rr 85.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. div-inv N/A

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

      5. clear-num N/A

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

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

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

      7. /-lowering-/.f64 87.3%

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

   9. Applied egg-rr 87.3%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.1 \cdot 10^{-111}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{0 - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{\frac{z + t \cdot \left(a \cdot \frac{-0.5}{z}\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.2 \cdot 10^{+63}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{z\_m + \frac{t \cdot a}{\frac{z\_m}{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.2e+63)
      (* x_m (/ (* z_m y_m) (+ z_m (/ (* t a) (/ z_m -0.5)))))
      (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.2e+63) {
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * a) / (z_m / -0.5))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.2d+63) then
        tmp = x_m * ((z_m * y_m) / (z_m + ((t * a) / (z_m / (-0.5d0)))))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.2e+63) {
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * a) / (z_m / -0.5))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.2e+63:
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * a) / (z_m / -0.5))))
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.2e+63)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(Float64(t * a) / Float64(z_m / -0.5)))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.2e+63)
		tmp = x_m * ((z_m * y_m) / (z_m + ((t * a) / (z_m / -0.5))));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.2e+63], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(N[(t * a), $MachinePrecision] / N[(z$95$m / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.2e63

   1. Initial program 71.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 38.7%

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

   5. Simplified 38.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. associate-/l* N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. /-lowering-/.f64 39.7%

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

   7. Applied egg-rr 39.7%

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

   if 1.2e63 < z

   1. Initial program 36.5%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 35.0%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 94.8%

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

   7. Simplified 94.8%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.2 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + \frac{t \cdot a}{\frac{z}{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 5.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{z\_m + a \cdot \frac{t}{\frac{z\_m}{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.15e+63)
      (* x_m (/ (* z_m y_m) (+ z_m (* a (/ t (/ z_m -0.5))))))
      (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e+63) {
		tmp = x_m * ((z_m * y_m) / (z_m + (a * (t / (z_m / -0.5)))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.15d+63) then
        tmp = x_m * ((z_m * y_m) / (z_m + (a * (t / (z_m / (-0.5d0))))))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.15e+63) {
		tmp = x_m * ((z_m * y_m) / (z_m + (a * (t / (z_m / -0.5)))));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.15e+63:
		tmp = x_m * ((z_m * y_m) / (z_m + (a * (t / (z_m / -0.5)))))
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.15e+63)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(z_m + Float64(a * Float64(t / Float64(z_m / -0.5))))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.15e+63)
		tmp = x_m * ((z_m * y_m) / (z_m + (a * (t / (z_m / -0.5)))));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.15e+63], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(z$95$m + N[(a * N[(t / N[(z$95$m / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.14999999999999997e63

   1. Initial program 71.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r/ N/A

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

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

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

      12. *-commutative N/A

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

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

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

      14. *-lowering-*.f64 38.7%

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

   5. Simplified 38.7%

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

      1. associate-*r* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. associate-/l* N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

      12. /-lowering-/.f64 39.7%

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

   7. Applied egg-rr 39.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. /-lowering-/.f64 39.7%

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

   9. Applied egg-rr 39.7%

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

   if 1.14999999999999997e63 < z

   1. Initial program 36.5%

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

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

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

      2. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 35.0%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 94.8%

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

   7. Simplified 94.8%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{z + a \cdot \frac{t}{\frac{z}{-0.5}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.4% accurate, 7.5× speedup

Math

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

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 (/ (+ z_m (* t (* a (/ -0.5 z_m)))) z_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (y_m * (x_m / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 / ((z_m + (t * (a * ((-0.5d0) / z_m)))) / 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 / ((z_m + (t * (a * (-0.5 / z_m)))) / 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 / ((z_m + (t * (a * (-0.5 / z_m)))) / 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(y_m * Float64(x_m / Float64(Float64(z_m + Float64(t * Float64(a * Float64(-0.5 / z_m)))) / 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 / ((z_m + (t * (a * (-0.5 / z_m)))) / 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[(y$95$m * N[(x$95$m / N[(N[(z$95$m + N[(t * N[(a * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r/ N/A

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

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

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

   12. *-commutative N/A

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

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

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

   14. *-lowering-*.f64 49.2%

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

5. Simplified 49.2%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

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

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

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

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

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

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

   10. associate-/l* N/A

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

   11. clear-num N/A

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

   12. un-div-inv N/A

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

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

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

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

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

   15. /-lowering-/.f64 54.4%

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

7. Applied egg-rr 54.4%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(\frac{a}{\frac{z}{\frac{-1}{2}}}\right)\right)\right), z\right)\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(a \cdot \frac{1}{\frac{z}{\frac{-1}{2}}}\right)\right)\right), z\right)\right)\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \left(a \cdot \frac{\frac{-1}{2}}{z}\right)\right)\right), z\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \left(\frac{\frac{-1}{2}}{z}\right)\right)\right)\right), z\right)\right)\right) \]

   7. /-lowering-/.f64 55.2%

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(a, \mathsf{/.f64}\left(\frac{-1}{2}, z\right)\right)\right)\right), z\right)\right)\right) \]

9. Applied egg-rr 55.2%

   \[\leadsto y \cdot \frac{x}{\frac{z + \color{blue}{t \cdot \left(a \cdot \frac{-0.5}{z}\right)}}{z}} \]
10. Add Preprocessing

Alternative 10: 76.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \left(y\_m \cdot \left(x\_m \cdot \frac{z\_m}{z\_m + \frac{t \cdot a}{\frac{z\_m}{-0.5}}}\right)\right)\right)\right) \end{array} \]

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 (/ z_m (+ z_m (/ (* t a) (/ z_m -0.5))))))))))

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 * (z_m / (z_m + ((t * a) / (z_m / -0.5))))))));
}

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 * (z_m / (z_m + ((t * a) / (z_m / (-0.5d0)))))))))
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 * (z_m / (z_m + ((t * a) / (z_m / -0.5))))))));
}

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 * (z_m / (z_m + ((t * a) / (z_m / -0.5))))))))

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(z_m / Float64(z_m + Float64(Float64(t * a) / Float64(z_m / -0.5)))))))))
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 * (z_m / (z_m + ((t * a) / (z_m / -0.5))))))));
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[(z$95$m / N[(z$95$m + N[(N[(t * a), $MachinePrecision] / N[(z$95$m / -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \color{blue}{\left(z + \frac{-1}{2} \cdot \frac{a \cdot t}{z}\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \left(z + \frac{a \cdot t}{z} \cdot \color{blue}{\frac{-1}{2}}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \left(z + \left(a \cdot \frac{t}{z}\right) \cdot \frac{-1}{2}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \left(z + a \cdot \color{blue}{\left(\frac{t}{z} \cdot \frac{-1}{2}\right)}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \left(z + a \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{t}{z}}\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \color{blue}{\left(a \cdot \left(\frac{-1}{2} \cdot \frac{t}{z}\right)\right)}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \left(a \cdot \left(\frac{t}{z} \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right)\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \left(\left(a \cdot \frac{t}{z}\right) \cdot \color{blue}{\frac{-1}{2}}\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \left(\frac{a \cdot t}{z} \cdot \frac{-1}{2}\right)\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \left(\frac{-1}{2} \cdot \color{blue}{\frac{a \cdot t}{z}}\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \left(\frac{\frac{-1}{2} \cdot \left(a \cdot t\right)}{\color{blue}{z}}\right)\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \left(a \cdot t\right)\right), \color{blue}{z}\right)\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(\left(a \cdot t\right) \cdot \frac{-1}{2}\right), z\right)\right)\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(a \cdot t\right), \frac{-1}{2}\right), z\right)\right)\right) \]

   14. *-lowering-*.f64 49.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(x, y\right), z\right), \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(a, t\right), \frac{-1}{2}\right), z\right)\right)\right) \]

5. Simplified 49.2%

   \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z + \frac{\left(a \cdot t\right) \cdot -0.5}{z}}} \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{\frac{z}{z + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}}} \]

   2. *-commutative N/A

      \[\leadsto \left(y \cdot x\right) \cdot \frac{\color{blue}{z}}{z + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}} \]

   3. associate-*l* N/A

      \[\leadsto y \cdot \color{blue}{\left(x \cdot \frac{z}{z + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \color{blue}{\left(x \cdot \frac{z}{z + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}}\right)}\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(x \cdot \frac{1}{\color{blue}{\frac{z + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}}{z}}}\right)\right) \]

   6. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{x}{\color{blue}{\frac{z + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}}{z}}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}}{z}\right)}\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\left(z + \frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}\right), \color{blue}{z}\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \left(\frac{\left(a \cdot t\right) \cdot \frac{-1}{2}}{z}\right)\right), z\right)\right)\right) \]

   10. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \left(\left(a \cdot t\right) \cdot \frac{\frac{-1}{2}}{z}\right)\right), z\right)\right)\right) \]

   11. clear-num N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \left(\left(a \cdot t\right) \cdot \frac{1}{\frac{z}{\frac{-1}{2}}}\right)\right), z\right)\right)\right) \]

   12. un-div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \left(\frac{a \cdot t}{\frac{z}{\frac{-1}{2}}}\right)\right), z\right)\right)\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(a \cdot t\right), \left(\frac{z}{\frac{-1}{2}}\right)\right)\right), z\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, t\right), \left(\frac{z}{\frac{-1}{2}}\right)\right)\right), z\right)\right)\right) \]

   15. /-lowering-/.f64 54.4%

       \[\leadsto \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(\mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, t\right), \mathsf{/.f64}\left(z, \frac{-1}{2}\right)\right)\right), z\right)\right)\right) \]

7. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{y \cdot \frac{x}{\frac{z + \frac{a \cdot t}{\frac{z}{-0.5}}}{z}}} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{\color{blue}{\frac{\frac{z + \frac{a \cdot t}{\frac{z}{\frac{-1}{2}}}}{z}}{x}}}\right)\right) \]

   2. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{1}{\frac{z + \frac{a \cdot t}{\frac{z}{\frac{-1}{2}}}}{z}} \cdot \color{blue}{x}\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(y, \left(\frac{z}{z + \frac{a \cdot t}{\frac{z}{\frac{-1}{2}}}} \cdot x\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(\frac{z}{z + \frac{a \cdot t}{\frac{z}{\frac{-1}{2}}}}\right), \color{blue}{x}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \left(z + \frac{a \cdot t}{\frac{z}{\frac{-1}{2}}}\right)\right), x\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \left(\frac{a \cdot t}{\frac{z}{\frac{-1}{2}}}\right)\right)\right), x\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\left(a \cdot t\right), \left(\frac{z}{\frac{-1}{2}}\right)\right)\right)\right), x\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, t\right), \left(\frac{z}{\frac{-1}{2}}\right)\right)\right)\right), x\right)\right) \]

   9. /-lowering-/.f64 54.4%

      \[\leadsto \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{/.f64}\left(z, \mathsf{+.f64}\left(z, \mathsf{/.f64}\left(\mathsf{*.f64}\left(a, t\right), \mathsf{/.f64}\left(z, \frac{-1}{2}\right)\right)\right)\right), x\right)\right) \]

9. Applied egg-rr 54.4%

   \[\leadsto y \cdot \color{blue}{\left(\frac{z}{z + \frac{a \cdot t}{\frac{z}{-0.5}}} \cdot x\right)} \]

10. Final simplification 54.4%

    \[\leadsto y \cdot \left(x \cdot \frac{z}{z + \frac{t \cdot a}{\frac{z}{-0.5}}}\right) \]
11. Add Preprocessing

Alternative 11: 76.6% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.05 \cdot 10^{-175}:\\ \;\;\;\;\frac{y\_m \cdot \left(z\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= z_m 1.05e-175) (/ (* y_m (* z_m x_m)) z_m) (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.05e-175) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.05d-175) then
        tmp = (y_m * (z_m * x_m)) / z_m
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.05e-175) {
		tmp = (y_m * (z_m * x_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.05e-175:
		tmp = (y_m * (z_m * x_m)) / z_m
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.05e-175)
		tmp = Float64(Float64(y_m * Float64(z_m * x_m)) / z_m);
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.05e-175)
		tmp = (y_m * (z_m * x_m)) / z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.05e-175], N[(N[(y$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.05e-175

   1. Initial program 64.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 65.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{z}\right) \]
   6. Step-by-step derivation

   7. Simplified 23.5%

      \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), z\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\left(z \cdot \left(x \cdot y\right)\right), z\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(z \cdot x\right) \cdot y\right), z\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(z \cdot x\right), y\right), z\right) \]

      5. *-lowering-*.f64 23.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(z, x\right), y\right), z\right) \]

   9. Applied egg-rr 23.5%

      \[\leadsto \frac{\color{blue}{\left(z \cdot x\right) \cdot y}}{z} \]

   if 1.05e-175 < z

   1. Initial program 59.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 80.9%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.05 \cdot 10^{-175}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.7 \cdot 10^{-176}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= z_m 2.7e-176) (/ (* x_m (* z_m y_m)) z_m) (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.7e-176) {
		tmp = (x_m * (z_m * y_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.7d-176) then
        tmp = (x_m * (z_m * y_m)) / z_m
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.7e-176) {
		tmp = (x_m * (z_m * y_m)) / z_m;
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.7e-176:
		tmp = (x_m * (z_m * y_m)) / z_m
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.7e-176)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / z_m);
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.7e-176)
		tmp = (x_m * (z_m * y_m)) / z_m;
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2.7e-176], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.6999999999999998e-176

   1. Initial program 64.4%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 65.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 65.9%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \color{blue}{z}\right) \]
   6. Step-by-step derivation

   7. Simplified 23.5%

      \[\leadsto \frac{x \cdot \left(y \cdot z\right)}{\color{blue}{z}} \]

   if 2.6999999999999998e-176 < z

   1. Initial program 59.3%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   2. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

      8. *-lowering-*.f64 59.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

   3. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x \cdot y} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 80.9%

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

   7. Simplified 80.9%

      \[\leadsto \color{blue}{x \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-176}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.9% accurate, 37.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \left(x\_m \cdot y\_m\right)\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (* z_s (* y_s (* x_s (* x_m y_m)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (x_m * y_m)));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = z_s * (y_s * (x_s * (x_m * y_m)))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	return z_s * (y_s * (x_s * (x_m * y_m)));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	return z_s * (y_s * (x_s * (x_m * y_m)))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	return Float64(z_s * Float64(y_s * Float64(x_s * Float64(x_m * y_m))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = z_s * (y_s * (x_s * (x_m * y_m)));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
2. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\left(x \cdot y\right) \cdot z\right), \color{blue}{\left(\sqrt{z \cdot z - t \cdot a}\right)}\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \left(y \cdot z\right)\right), \left(\sqrt{\color{blue}{z \cdot z - t \cdot a}}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \left(\sqrt{z \cdot z - \color{blue}{t \cdot a}}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\left(z \cdot z - t \cdot a\right)\right)\right) \]

   6. --lowering--.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\left(z \cdot z\right), \left(t \cdot a\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \left(t \cdot a\right)\right)\right)\right) \]

   8. *-lowering-*.f64 62.4%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(y, z\right)\right), \mathsf{sqrt.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(z, z\right), \mathsf{*.f64}\left(t, a\right)\right)\right)\right) \]

3. Simplified 62.4%

   \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - t \cdot a}}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x \cdot y} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 48.8%

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{y}\right) \]

7. Simplified 48.8%

   \[\leadsto \color{blue}{x \cdot y} \]
8. Add Preprocessing

Developer Target 1: 87.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -3.1921305903852764 \cdot 10^{+46}:\\ \;\;\;\;-y \cdot x\\ \mathbf{elif}\;z < 5.976268120920894 \cdot 10^{+90}:\\ \;\;\;\;\frac{x \cdot z}{\frac{\sqrt{z \cdot z - a \cdot t}}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (< z -3.1921305903852764e+46)
   (- (* y x))
   (if (< z 5.976268120920894e+90)
     (/ (* x z) (/ (sqrt (- (* z z) (* a t))) y))
     (* y x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z < -3.1921305903852764e+46) {
		tmp = -(y * x);
	} else if (z < 5.976268120920894e+90) {
		tmp = (x * z) / (Math.sqrt(((z * z) - (a * t))) / y);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024191 
(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)))))