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

Percentage Accurate: 60.8% → 92.4%

Time: 14.1s

Alternatives: 9

Speedup: 37.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.9 \cdot 10^{+135}:\\ \;\;\;\;\frac{y\_m}{\frac{\sqrt{z\_m \cdot z\_m - t \cdot a}}{z\_m}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{1 + a \cdot \left(\frac{-0.5}{\frac{1}{t}} \cdot \frac{\frac{1}{z\_m}}{z\_m}\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 4.9e+135)
      (* (/ y_m (/ (sqrt (- (* z_m z_m) (* t a))) z_m)) x_m)
      (*
       y_m
       (/ x_m (+ 1.0 (* a (* (/ -0.5 (/ 1.0 t)) (/ (/ 1.0 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 <= 4.9e+135) {
		tmp = (y_m / (sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m;
	} else {
		tmp = y_m * (x_m / (1.0 + (a * ((-0.5 / (1.0 / t)) * ((1.0 / 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 <= 4.9d+135) then
        tmp = (y_m / (sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m
    else
        tmp = y_m * (x_m / (1.0d0 + (a * (((-0.5d0) / (1.0d0 / t)) * ((1.0d0 / 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 <= 4.9e+135) {
		tmp = (y_m / (Math.sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m;
	} else {
		tmp = y_m * (x_m / (1.0 + (a * ((-0.5 / (1.0 / t)) * ((1.0 / 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 <= 4.9e+135:
		tmp = (y_m / (math.sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m
	else:
		tmp = y_m * (x_m / (1.0 + (a * ((-0.5 / (1.0 / t)) * ((1.0 / 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 <= 4.9e+135)
		tmp = Float64(Float64(y_m / Float64(sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))) / z_m)) * x_m);
	else
		tmp = Float64(y_m * Float64(x_m / Float64(1.0 + Float64(a * Float64(Float64(-0.5 / Float64(1.0 / t)) * Float64(Float64(1.0 / 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 <= 4.9e+135)
		tmp = (y_m / (sqrt(((z_m * z_m) - (t * a))) / z_m)) * x_m;
	else
		tmp = y_m * (x_m / (1.0 + (a * ((-0.5 / (1.0 / t)) * ((1.0 / 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, 4.9e+135], N[(N[(y$95$m / N[(N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(1.0 + N[(a * N[(N[(-0.5 / N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.9000000000000001e135

   1. Initial program 72.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 77.1%

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

   4. Applied egg-rr 77.1%

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

   if 4.9000000000000001e135 < z

   1. Initial program 25.0%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 74.4%

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

   5. Simplified 74.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lft-identity N/A

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

      5. times-frac N/A

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

      6. /-rgt-identity N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

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

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

   7. Applied egg-rr 100.0%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.9 \cdot 10^{+135}:\\ \;\;\;\;\frac{y}{\frac{\sqrt{z \cdot z - t \cdot a}}{z}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{1 + a \cdot \left(\frac{-0.5}{\frac{1}{t}} \cdot \frac{\frac{1}{z}}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{+133}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{1 + a \cdot \left(\frac{-0.5}{\frac{1}{t}} \cdot \frac{\frac{1}{z\_m}}{z\_m}\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 5e+133)
      (* y_m (* x_m (/ z_m (sqrt (- (* z_m z_m) (* t a))))))
      (*
       y_m
       (/ x_m (+ 1.0 (* a (* (/ -0.5 (/ 1.0 t)) (/ (/ 1.0 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 <= 5e+133) {
		tmp = y_m * (x_m * (z_m / sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = y_m * (x_m / (1.0 + (a * ((-0.5 / (1.0 / t)) * ((1.0 / 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 <= 5d+133) then
        tmp = y_m * (x_m * (z_m / sqrt(((z_m * z_m) - (t * a)))))
    else
        tmp = y_m * (x_m / (1.0d0 + (a * (((-0.5d0) / (1.0d0 / t)) * ((1.0d0 / 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 <= 5e+133) {
		tmp = y_m * (x_m * (z_m / Math.sqrt(((z_m * z_m) - (t * a)))));
	} else {
		tmp = y_m * (x_m / (1.0 + (a * ((-0.5 / (1.0 / t)) * ((1.0 / 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 <= 5e+133:
		tmp = y_m * (x_m * (z_m / math.sqrt(((z_m * z_m) - (t * a)))))
	else:
		tmp = y_m * (x_m / (1.0 + (a * ((-0.5 / (1.0 / t)) * ((1.0 / 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 <= 5e+133)
		tmp = Float64(y_m * Float64(x_m * Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a))))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(1.0 + Float64(a * Float64(Float64(-0.5 / Float64(1.0 / t)) * Float64(Float64(1.0 / 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 <= 5e+133)
		tmp = y_m * (x_m * (z_m / sqrt(((z_m * z_m) - (t * a)))));
	else
		tmp = y_m * (x_m / (1.0 + (a * ((-0.5 / (1.0 / t)) * ((1.0 / 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, 5e+133], N[(y$95$m * N[(x$95$m * N[(z$95$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[(1.0 + N[(a * N[(N[(-0.5 / N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / z$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.99999999999999961e133

   1. Initial program 72.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      13. *-lowering-*.f64 77.1%

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

   4. Applied egg-rr 77.1%

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

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 77.2%

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

   6. Applied egg-rr 77.2%

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

   if 4.99999999999999961e133 < z

   1. Initial program 25.0%

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

   3. Taylor expanded in z around inf

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

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

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

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

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

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

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

      4. associate-/l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 74.4%

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

   5. Simplified 74.4%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. times-frac N/A

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

      4. *-lft-identity N/A

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

      5. times-frac N/A

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

      6. /-rgt-identity N/A

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

      7. *-inverses N/A

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

      8. *-rgt-identity N/A

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

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

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

   7. Applied egg-rr 100.0%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. times-frac N/A

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

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

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

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

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

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

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

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

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

      10. /-lowering-/.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5 \cdot 10^{+133}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{z}{\sqrt{z \cdot z - t \cdot a}}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{1 + a \cdot \left(\frac{-0.5}{\frac{1}{t}} \cdot \frac{\frac{1}{z}}{z}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.2e-84)
      (* x_m (/ (* z_m y_m) (sqrt (- 0.0 (* t a)))))
      (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.2e-84) {
		tmp = x_m * ((z_m * y_m) / sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.2d-84) then
        tmp = x_m * ((z_m * y_m) / sqrt((0.0d0 - (t * a))))
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.2e-84) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt((0.0 - (t * a))));
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.2e-84:
		tmp = x_m * ((z_m * y_m) / math.sqrt((0.0 - (t * a))))
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.2e-84)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(0.0 - Float64(t * a)))));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.2e-84)
		tmp = x_m * ((z_m * y_m) / sqrt((0.0 - (t * a))));
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.2e-84], 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 * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.1999999999999999e-84

   1. Initial program 64.5%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 39.0%

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

   5. Simplified 39.0%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 43.5%

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

   7. Applied egg-rr 43.5%

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

   if 3.1999999999999999e-84 < z

   1. Initial program 63.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 93.0%

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

   5. Simplified 93.0%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{-84}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.9% accurate, 7.5× speedup

Math

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

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 (* x_m (/ y_m (/ (+ z_m (* a (* t (/ -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 * (x_m * (y_m / ((z_m + (a * (t * (-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 * (x_m * (y_m / ((z_m + (a * (t * ((-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 * (x_m * (y_m / ((z_m + (a * (t * (-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 * (x_m * (y_m / ((z_m + (a * (t * (-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(x_m * Float64(y_m / Float64(Float64(z_m + Float64(a * Float64(t * 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 * (x_m * (y_m / ((z_m + (a * (t * (-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[(x$95$m * N[(y$95$m / N[(N[(z$95$m + N[(a * N[(t * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.0%

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

   1. associate-*l* N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

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

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

   5. associate-*r/ N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

   13. *-lowering-*.f64 68.9%

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

4. Applied egg-rr 68.9%

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

5. Taylor expanded in t around 0

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

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

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

   2. associate-*r/ N/A

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

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

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

   4. *-commutative N/A

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

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

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

   6. *-lowering-*.f64 50.1%

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

7. Simplified 50.1%

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

   1. associate-/l* N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. /-lowering-/.f64 51.3%

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

9. Applied egg-rr 51.3%

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

10. Final simplification 51.3%

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

Alternative 5: 78.9% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in z around inf

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

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

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

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

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 47.0%

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

5. Simplified 47.0%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-lft-identity N/A

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

   5. times-frac N/A

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

   6. /-rgt-identity N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

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

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

7. Applied egg-rr 51.2%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

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

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

   5. associate-/r* N/A

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

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

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

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

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

   8. *-commutative N/A

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

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

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

   10. metadata-eval N/A

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

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

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

   12. /-lowering-/.f64 51.2%

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

9. Applied egg-rr 51.2%

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

10. Final simplification 51.2%

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

Alternative 6: 78.8% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in z around inf

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

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

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

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

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

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

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

   4. associate-/l* N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 47.0%

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

5. Simplified 47.0%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. *-lft-identity N/A

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

   5. times-frac N/A

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

   6. /-rgt-identity N/A

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

   7. *-inverses N/A

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

   8. *-rgt-identity N/A

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

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

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

7. Applied egg-rr 51.2%

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

8. Final simplification 51.2%

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

Alternative 7: 76.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.5 \cdot 10^{-126}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (* x_s (if (<= z_m 3.5e-126) (* y_m (/ (* z_m x_m) z_m)) (* y_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.5e-126) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.5d-126) then
        tmp = y_m * ((z_m * x_m) / z_m)
    else
        tmp = y_m * x_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.5e-126) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = y_m * x_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.5e-126:
		tmp = y_m * ((z_m * x_m) / z_m)
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 3.5e-126)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / z_m));
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.5e-126)
		tmp = y_m * ((z_m * x_m) / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.5e-126], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.5e-126

   1. Initial program 63.0%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 13.5%

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

   5. Simplified 13.5%

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

      1. *-rgt-identity N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-rgt-identity N/A

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

      7. times-frac N/A

         \[\leadsto \frac{z \cdot x}{z} \cdot \color{blue}{\frac{y}{1}} \]

      8. /-rgt-identity N/A

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

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

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

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

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

      11. *-lowering-*.f64 22.1%

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

   7. Applied egg-rr 22.1%

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

   if 3.5e-126 < z

   1. Initial program 65.7%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 89.8%

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

   5. Simplified 89.8%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{-126}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.8% accurate, 9.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-185}:\\ \;\;\;\;y\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\left(z\_m \cdot x\_m\right) \cdot \frac{y\_m}{z\_m}\\ \end{array}\right)\right) \end{array} \]

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 (<= a 4.8e-185) (* y_m x_m) (* (* z_m x_m) (/ y_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 (a <= 4.8e-185) {
		tmp = y_m * x_m;
	} else {
		tmp = (z_m * x_m) * (y_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 (a <= 4.8d-185) then
        tmp = y_m * x_m
    else
        tmp = (z_m * x_m) * (y_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 (a <= 4.8e-185) {
		tmp = y_m * x_m;
	} else {
		tmp = (z_m * x_m) * (y_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 a <= 4.8e-185:
		tmp = y_m * x_m
	else:
		tmp = (z_m * x_m) * (y_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 (a <= 4.8e-185)
		tmp = Float64(y_m * x_m);
	else
		tmp = Float64(Float64(z_m * x_m) * Float64(y_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 (a <= 4.8e-185)
		tmp = y_m * x_m;
	else
		tmp = (z_m * x_m) * (y_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[a, 4.8e-185], N[(y$95$m * x$95$m), $MachinePrecision], N[(N[(z$95$m * x$95$m), $MachinePrecision] * N[(y$95$m / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if a < 4.8000000000000002e-185

   1. Initial program 65.4%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 44.1%

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

   5. Simplified 44.1%

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

   if 4.8000000000000002e-185 < a

   1. Initial program 61.6%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 42.0%

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

   5. Simplified 42.0%

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

      1. *-rgt-identity N/A

         \[\leadsto \left(x \cdot y\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. *-rgt-identity N/A

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

      5. times-frac N/A

         \[\leadsto \frac{x \cdot y}{z} \cdot \color{blue}{\frac{z}{1}} \]

      6. /-rgt-identity N/A

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

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

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

      8. associate-/l* N/A

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

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

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

      10. /-lowering-/.f64 37.3%

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

   7. Applied egg-rr 37.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. *-lowering-*.f64 40.6%

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

   9. Applied egg-rr 40.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 4.8 \cdot 10^{-185}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot x\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 37.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.0%

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

3. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 43.3%

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

5. Simplified 43.3%

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

6. Final simplification 43.3%

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

Developer Target 1: 87.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z < (-3.1921305903852764d+46)) then
        tmp = -(y * x)
    else if (z < 5.976268120920894d+90) then
        tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t, a):
	tmp = 0
	if z < -3.1921305903852764e+46:
		tmp = -(y * x)
	elif z < 5.976268120920894e+90:
		tmp = (x * z) / (math.sqrt(((z * z) - (a * t))) / y)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z < -3.1921305903852764e+46)
		tmp = Float64(-Float64(y * x));
	elseif (z < 5.976268120920894e+90)
		tmp = Float64(Float64(x * z) / Float64(sqrt(Float64(Float64(z * z) - Float64(a * t))) / y));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z < -3.1921305903852764e+46)
		tmp = -(y * x);
	elseif (z < 5.976268120920894e+90)
		tmp = (x * z) / (sqrt(((z * z) - (a * t))) / y);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Less[z, -3.1921305903852764e+46], (-N[(y * x), $MachinePrecision]), If[Less[z, 5.976268120920894e+90], N[(N[(x * z), $MachinePrecision] / N[(N[Sqrt[N[(N[(z * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024161 
(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)))))