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

Percentage Accurate: 61.7% → 92.8%

Time: 13.3s

Alternatives: 11

Speedup: 7.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{\mathsf{fma}\left(z\_m, z\_m, 0 - t \cdot a\right)}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(\frac{a \cdot -0.5}{z\_m}, \frac{t}{z\_m}, 1\right)}\\ \end{array}\right)\right) \end{array} \]

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 4.5e+46)
      (* (/ (* z_m y_m) (sqrt (fma z_m z_m (- 0.0 (* t a))))) x_m)
      (/ (* y_m x_m) (fma (/ (* a -0.5) z_m) (/ t z_m) 1.0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.5e+46) {
		tmp = ((z_m * y_m) / sqrt(fma(z_m, z_m, (0.0 - (t * a))))) * x_m;
	} else {
		tmp = (y_m * x_m) / fma(((a * -0.5) / z_m), (t / z_m), 1.0);
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 4.5e+46)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(fma(z_m, z_m, Float64(0.0 - Float64(t * a))))) * x_m);
	else
		tmp = Float64(Float64(y_m * x_m) / fma(Float64(Float64(a * -0.5) / z_m), Float64(t / z_m), 1.0));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4.5e+46], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(z$95$m * z$95$m + N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(N[(a * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision] * N[(t / z$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.5000000000000001e46

   1. Initial program 71.0%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. neg-sub0 N/A

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

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

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

      12. *-lowering-*.f64 68.5

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

   4. Applied egg-rr 68.5%

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

   if 4.5000000000000001e46 < z

   1. Initial program 37.5%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 80.3

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   5. Simplified 80.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{z \cdot z}, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   7. Step-by-step derivation

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

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

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

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 96.6

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(-0.5 \cdot a, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   8. Simplified 96.6%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 96.6

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\frac{-0.5 \cdot a}{z}, \color{blue}{\frac{t}{z}}, 1\right)} \]

   10. Applied egg-rr 96.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{\mathsf{fma}\left(z, z, 0 - t \cdot a\right)}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(\frac{a \cdot -0.5}{z}, \frac{t}{z}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\frac{y\_m \cdot \left(z\_m \cdot x\_m\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= (/ (* z_m (* y_m x_m)) (sqrt (- (* z_m z_m) (* t a)))) 2e-170)
      (/ (* 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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-170) {
		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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 2d-170) 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 * (y_m * x_m)) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-170) {
		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 * (y_m * x_m)) / math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-170:
		tmp = (y_m * (z_m * x_m)) / z_m
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(Float64(z_m * Float64(y_m * x_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 2e-170)
		tmp = Float64(Float64(y_m * Float64(z_m * x_m)) / z_m);
	else
		tmp = Float64(y_m * x_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (((z_m * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-170)
		tmp = (y_m * (z_m * x_m)) / z_m;
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-170], N[(N[(y$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 1.99999999999999997e-170

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 48.0%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

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

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

      4. *-lowering-*.f64 45.7

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

   7. Applied egg-rr 45.7%

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

   if 1.99999999999999997e-170 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 48.7%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 32.0

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

   5. Simplified 32.0%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}} \leq 2 \cdot 10^{-170}:\\ \;\;\;\;\frac{y \cdot \left(z \cdot x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 2 \cdot 10^{-170}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 (* y_m x_m)) (sqrt (- (* z_m z_m) (* t a)))) 2e-170)
      (* 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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-170) {
		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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 2d-170) 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 * (y_m * x_m)) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-170) {
		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 * (y_m * x_m)) / math.sqrt(((z_m * z_m) - (t * a)))) <= 2e-170:
		tmp = y_m * ((z_m * x_m) / z_m)
	else:
		tmp = y_m * x_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (Float64(Float64(z_m * Float64(y_m * x_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 2e-170)
		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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 2e-170)
		tmp = y_m * ((z_m * x_m) / z_m);
	else
		tmp = y_m * x_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-170], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 1.99999999999999997e-170

   1. Initial program 72.4%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 44.6

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

   5. Simplified 44.6%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. *-rgt-identity N/A

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

      5. times-frac N/A

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

      6. /-rgt-identity N/A

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

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

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

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

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

      9. *-lowering-*.f64 38.8

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

   7. Applied egg-rr 38.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r* N/A

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

      5. times-frac N/A

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

      6. /-rgt-identity N/A

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

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

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

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

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

      9. *-lowering-*.f64 43.4

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

   9. Applied egg-rr 43.4%

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

   if 1.99999999999999997e-170 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 48.7%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 32.0

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

   5. Simplified 32.0%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}} \leq 2 \cdot 10^{-170}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{\mathsf{fma}\left(z\_m, z\_m, 0 - t \cdot a\right)}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z\_m \cdot z\_m}, 1\right)}\\ \end{array}\right)\right) \end{array} \]

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 6.5e+45)
      (* (/ (* z_m y_m) (sqrt (fma z_m z_m (- 0.0 (* t a))))) x_m)
      (/ (* y_m x_m) (fma (* a -0.5) (/ t (* z_m z_m)) 1.0)))))))

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 <= 6.5e+45) {
		tmp = ((z_m * y_m) / sqrt(fma(z_m, z_m, (0.0 - (t * a))))) * x_m;
	} else {
		tmp = (y_m * x_m) / fma((a * -0.5), (t / (z_m * z_m)), 1.0);
	}
	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 <= 6.5e+45)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(fma(z_m, z_m, Float64(0.0 - Float64(t * a))))) * x_m);
	else
		tmp = Float64(Float64(y_m * x_m) / fma(Float64(a * -0.5), Float64(t / Float64(z_m * z_m)), 1.0));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 6.5e+45], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(z$95$m * z$95$m + N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(a * -0.5), $MachinePrecision] * N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 6.50000000000000034e45

   1. Initial program 71.0%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

      8. sub-neg N/A

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

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

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

      10. neg-sub0 N/A

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

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

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

      12. *-lowering-*.f64 68.5

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

   4. Applied egg-rr 68.5%

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

   if 6.50000000000000034e45 < z

   1. Initial program 37.5%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 80.3

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   5. Simplified 80.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{z \cdot z}, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   7. Step-by-step derivation

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

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

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

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 96.6

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(-0.5 \cdot a, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   8. Simplified 96.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{\mathsf{fma}\left(z, z, 0 - t \cdot a\right)}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z \cdot z}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.15 \cdot 10^{+47}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z\_m \cdot z\_m}, 1\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 3.15e+47)
      (* x_m (/ (* z_m y_m) (sqrt (- (* z_m z_m) (* t a)))))
      (/ (* y_m x_m) (fma (* a -0.5) (/ t (* z_m z_m)) 1.0)))))))

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 <= 3.15e+47) {
		tmp = x_m * ((z_m * y_m) / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = (y_m * x_m) / fma((a * -0.5), (t / (z_m * z_m)), 1.0);
	}
	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 <= 3.15e+47)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(Float64(y_m * x_m) / fma(Float64(a * -0.5), Float64(t / Float64(z_m * z_m)), 1.0));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 3.15e+47], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(a * -0.5), $MachinePrecision] * N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.15000000000000002e47

   1. Initial program 71.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 67.4

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

   4. Applied egg-rr 67.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. *-commutative N/A

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

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

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 68.5

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

   6. Applied egg-rr 68.5%

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

   if 3.15000000000000002e47 < z

   1. Initial program 37.5%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 80.3

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   5. Simplified 80.3%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{z \cdot z}, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   7. Step-by-step derivation

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

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

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

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 96.6

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(-0.5 \cdot a, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   8. Simplified 96.6%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.15 \cdot 10^{+47}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z \cdot z}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{-85}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\_m\right) \cdot x\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z\_m \cdot z\_m}, 1\right)}\\ \end{array}\right)\right) \end{array} \]

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 4e-85)
      (/ (* (* z_m y_m) x_m) (sqrt (- 0.0 (* t a))))
      (/ (* y_m x_m) (fma (* a -0.5) (/ t (* z_m z_m)) 1.0)))))))

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 <= 4e-85) {
		tmp = ((z_m * y_m) * x_m) / sqrt((0.0 - (t * a)));
	} else {
		tmp = (y_m * x_m) / fma((a * -0.5), (t / (z_m * z_m)), 1.0);
	}
	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 <= 4e-85)
		tmp = Float64(Float64(Float64(z_m * y_m) * x_m) / sqrt(Float64(0.0 - Float64(t * a))));
	else
		tmp = Float64(Float64(y_m * x_m) / fma(Float64(a * -0.5), Float64(t / Float64(z_m * z_m)), 1.0));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 4e-85], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(a * -0.5), $MachinePrecision] * N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.9999999999999999e-85

   1. Initial program 67.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 63.9

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

   4. Applied egg-rr 63.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 39.6

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

   7. Simplified 39.6%

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

   if 3.9999999999999999e-85 < z

   1. Initial program 56.1%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 76.0

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   5. Simplified 76.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{z \cdot z}, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   7. Step-by-step derivation

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

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

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

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 86.6

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(-0.5 \cdot a, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   8. Simplified 86.6%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{-85}:\\ \;\;\;\;\frac{\left(z \cdot y\right) \cdot x}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(a \cdot -0.5, \frac{t}{z \cdot z}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.9 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\_m\right) \cdot x\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z\_m \cdot z\_m}, 1\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 1.9e-84)
      (/ (* (* z_m y_m) x_m) (sqrt (- 0.0 (* t a))))
      (* x_m (/ y_m (fma -0.5 (* a (/ t (* z_m z_m))) 1.0))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.9e-84) {
		tmp = ((z_m * y_m) * x_m) / sqrt((0.0 - (t * a)));
	} else {
		tmp = x_m * (y_m / fma(-0.5, (a * (t / (z_m * z_m))), 1.0));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.9e-84)
		tmp = Float64(Float64(Float64(z_m * y_m) * x_m) / sqrt(Float64(0.0 - Float64(t * a))));
	else
		tmp = Float64(x_m * Float64(y_m / fma(-0.5, Float64(a * Float64(t / Float64(z_m * z_m))), 1.0)));
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 1.9e-84], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(-0.5 * N[(a * N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.89999999999999993e-84

   1. Initial program 67.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 63.9

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

   4. Applied egg-rr 63.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 39.6

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

   7. Simplified 39.6%

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

   if 1.89999999999999993e-84 < z

   1. Initial program 56.1%

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

   3. Taylor expanded in z around inf

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

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

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 76.0

          \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   5. Simplified 76.0%

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\color{blue}{z \cdot \mathsf{fma}\left(a \cdot -0.5, \frac{t}{z \cdot z}, 1\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{1 + \frac{-1}{2} \cdot \frac{a \cdot t}{{z}^{2}}}} \]
   7. Step-by-step derivation

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

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

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

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

      3. +-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 86.6

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(-0.5 \cdot a, \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \]

   8. Simplified 86.6%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 86.6

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\frac{-0.5 \cdot a}{z}, \color{blue}{\frac{t}{z}}, 1\right)} \]

   10. Applied egg-rr 86.6%

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

       1. associate-/l* N/A

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

       2. *-commutative N/A

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

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

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

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

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

       5. associate-/l* N/A

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

       6. associate-*l* N/A

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

       7. times-frac N/A

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

       8. associate-*r/ N/A

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

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

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

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

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

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

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

       12. *-lowering-*.f64 85.6

           \[\leadsto \frac{y}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{\color{blue}{z \cdot z}}, 1\right)} \cdot x \]

   12. Applied egg-rr 85.6%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{-84}:\\ \;\;\;\;\frac{\left(z \cdot y\right) \cdot x}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{\mathsf{fma}\left(-0.5, a \cdot \frac{t}{z \cdot z}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.9 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(z\_m \cdot y\_m\right) \cdot x\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 1.9e-88)
      (/ (* (* z_m y_m) x_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 <= 1.9e-88) {
		tmp = ((z_m * y_m) * x_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 <= 1.9d-88) then
        tmp = ((z_m * y_m) * x_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 <= 1.9e-88) {
		tmp = ((z_m * y_m) * x_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 <= 1.9e-88:
		tmp = ((z_m * y_m) * x_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 <= 1.9e-88)
		tmp = Float64(Float64(Float64(z_m * y_m) * x_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 <= 1.9e-88)
		tmp = ((z_m * y_m) * x_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, 1.9e-88], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] * x$95$m), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.90000000000000006e-88

   1. Initial program 67.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 63.9

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

   4. Applied egg-rr 63.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 39.6

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

   7. Simplified 39.6%

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

   if 1.90000000000000006e-88 < z

   1. Initial program 56.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.9 \cdot 10^{-88}:\\ \;\;\;\;\frac{\left(z \cdot y\right) \cdot x}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.45 \cdot 10^{-89}:\\ \;\;\;\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 2.45e-89)
      (/ (* z_m (* y_m x_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 <= 2.45e-89) {
		tmp = (z_m * (y_m * x_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 <= 2.45d-89) then
        tmp = (z_m * (y_m * x_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 <= 2.45e-89) {
		tmp = (z_m * (y_m * x_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 <= 2.45e-89:
		tmp = (z_m * (y_m * x_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 <= 2.45e-89)
		tmp = Float64(Float64(z_m * Float64(y_m * x_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 <= 2.45e-89)
		tmp = (z_m * (y_m * x_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, 2.45e-89], N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.0 - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.45e-89

   1. Initial program 67.8%

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

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 44.6

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

   5. Simplified 44.6%

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

   if 2.45e-89 < z

   1. Initial program 56.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.45 \cdot 10^{-89}:\\ \;\;\;\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.55 \cdot 10^{-88}:\\ \;\;\;\;\left(y\_m \cdot x\_m\right) \cdot \frac{z\_m}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 2.55e-88)
      (* (* y_m x_m) (/ z_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 <= 2.55e-88) {
		tmp = (y_m * x_m) * (z_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 <= 2.55d-88) then
        tmp = (y_m * x_m) * (z_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 <= 2.55e-88) {
		tmp = (y_m * x_m) * (z_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 <= 2.55e-88:
		tmp = (y_m * x_m) * (z_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 <= 2.55e-88)
		tmp = Float64(Float64(y_m * x_m) * Float64(z_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 <= 2.55e-88)
		tmp = (y_m * x_m) * (z_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, 2.55e-88], N[(N[(y$95$m * x$95$m), $MachinePrecision] * N[(z$95$m / 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 < 2.55000000000000023e-88

   1. Initial program 67.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. sub-neg N/A

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

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

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

      8. neg-sub0 N/A

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

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

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

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

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

      11. *-lowering-*.f64 69.7

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

   4. Applied egg-rr 69.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. *-lowering-*.f64 43.7

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

   7. Simplified 43.7%

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

   if 2.55000000000000023e-88 < z

   1. Initial program 56.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 84.4

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

   5. Simplified 84.4%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.55 \cdot 10^{-88}:\\ \;\;\;\;\left(y \cdot x\right) \cdot \frac{z}{\sqrt{0 - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.8% accurate, 7.5× speedup

Math

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

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 64.1%

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

3. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 40.2

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

5. Simplified 40.2%

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

6. Final simplification 40.2%

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

Developer Target 1: 87.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024196 
(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)))))