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

Percentage Accurate: 62.2% → 94.5%

Time: 13.5s

Alternatives: 9

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: 62.2% 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: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.15 \cdot 10^{-150}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{-t} \cdot \sqrt{a}}\\ \mathbf{elif}\;z\_m \leq 1.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \cdot \left(x\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z\_m}, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 2.15e-150)
      (* x_m (/ (* z_m y_m) (* (sqrt (- t)) (sqrt a))))
      (if (<= z_m 1.5e+120)
        (* (/ z_m (sqrt (- (* z_m z_m) (* t a)))) (* x_m y_m))
        (* (* x_m y_m) (/ z_m (fma a (/ (* t -0.5) z_m) z_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 2.15e-150) {
		tmp = x_m * ((z_m * y_m) / (sqrt(-t) * sqrt(a)));
	} else if (z_m <= 1.5e+120) {
		tmp = (z_m / sqrt(((z_m * z_m) - (t * a)))) * (x_m * y_m);
	} else {
		tmp = (x_m * y_m) * (z_m / fma(a, ((t * -0.5) / z_m), z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
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 <= 2.15e-150)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / Float64(sqrt(Float64(-t)) * sqrt(a))));
	elseif (z_m <= 1.5e+120)
		tmp = Float64(Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) * Float64(x_m * y_m));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(a, Float64(Float64(t * -0.5) / z_m), z_m)));
	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.
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, 2.15e-150], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[(N[Sqrt[(-t)], $MachinePrecision] * N[Sqrt[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.5e+120], N[(N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(a * N[(N[(t * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 2.15000000000000002e-150

   1. Initial program 64.6%

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

   3. Taylor expanded in z around 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. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 40.1

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

   5. Simplified 40.1%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. distribute-rgt-neg-out N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 39.1

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

   7. Applied egg-rr 39.1%

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

      1. distribute-rgt-neg-out N/A

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

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

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

      3. sqrt-prod N/A

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

      4. pow1/2 N/A

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

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

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

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

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

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

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

      8. pow1/2 N/A

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

      9. sqrt-lowering-sqrt.f64 20.5

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

   9. Applied egg-rr 20.5%

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

   if 2.15000000000000002e-150 < z < 1.5e120

   1. Initial program 89.2%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. --lowering--.f64 N/A

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

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

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

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

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

      9. *-lowering-*.f64 92.5

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

   4. Applied egg-rr 92.5%

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

   if 1.5e120 < z

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 71.2

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

   5. Simplified 71.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.15 \cdot 10^{-150}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{-t} \cdot \sqrt{a}}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z}, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ \begin{array}{l} t_1 := \sqrt{z\_m \cdot z\_m - t \cdot a}\\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9.6 \cdot 10^{-60}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{t\_1}\\ \mathbf{elif}\;z\_m \leq 1.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{z\_m}{t\_1} \cdot \left(x\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z\_m}, z\_m\right)}\\ \end{array}\right)\right) \end{array} \end{array} \]

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.
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
 (let* ((t_1 (sqrt (- (* z_m z_m) (* t a)))))
   (*
    z_s
    (*
     y_s
     (*
      x_s
      (if (<= z_m 9.6e-60)
        (* x_m (/ (* z_m y_m) t_1))
        (if (<= z_m 1.5e+120)
          (* (/ z_m t_1) (* x_m y_m))
          (* (* x_m y_m) (/ z_m (fma a (/ (* t -0.5) z_m) z_m))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 t_1 = sqrt(((z_m * z_m) - (t * a)));
	double tmp;
	if (z_m <= 9.6e-60) {
		tmp = x_m * ((z_m * y_m) / t_1);
	} else if (z_m <= 1.5e+120) {
		tmp = (z_m / t_1) * (x_m * y_m);
	} else {
		tmp = (x_m * y_m) * (z_m / fma(a, ((t * -0.5) / z_m), z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
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)
	t_1 = sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))
	tmp = 0.0
	if (z_m <= 9.6e-60)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / t_1));
	elseif (z_m <= 1.5e+120)
		tmp = Float64(Float64(z_m / t_1) * Float64(x_m * y_m));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(a, Float64(Float64(t * -0.5) / z_m), z_m)));
	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.
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_] := Block[{t$95$1 = N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 9.6e-60], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.5e+120], N[(N[(z$95$m / t$95$1), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(a * N[(N[(t * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < 9.60000000000000038e-60

   1. Initial program 65.8%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. 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. --lowering--.f64 N/A

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

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

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

      10. *-lowering-*.f64 64.2

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

   4. Applied egg-rr 64.2%

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

   if 9.60000000000000038e-60 < z < 1.5e120

   1. Initial program 92.5%

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

      1. associate-/l* N/A

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

      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. --lowering--.f64 N/A

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

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

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

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

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

      9. *-lowering-*.f64 94.9

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

   4. Applied egg-rr 94.9%

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

   if 1.5e120 < z

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 71.2

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

   5. Simplified 71.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.6 \cdot 10^{-60}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+120}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z}, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z\_m \leq 2.05 \cdot 10^{+120}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z\_m}, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 5.5e-153)
      (/ (* x_m (* z_m y_m)) (sqrt (* t (- a))))
      (if (<= z_m 2.05e+120)
        (* y_m (/ (* z_m x_m) (sqrt (- (* z_m z_m) (* t a)))))
        (* (* x_m y_m) (/ z_m (fma a (/ (* t -0.5) z_m) z_m)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 5.5e-153) {
		tmp = (x_m * (z_m * y_m)) / sqrt((t * -a));
	} else if (z_m <= 2.05e+120) {
		tmp = y_m * ((z_m * x_m) / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = (x_m * y_m) * (z_m / fma(a, ((t * -0.5) / z_m), z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
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 <= 5.5e-153)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(t * Float64(-a))));
	elseif (z_m <= 2.05e+120)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(a, Float64(Float64(t * -0.5) / z_m), z_m)));
	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.
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, 5.5e-153], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 2.05e+120], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(a * N[(N[(t * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 5.49999999999999962e-153

   1. Initial program 65.0%

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

   3. Taylor expanded in z around 0

      \[\leadsto \frac{\left(x \cdot y\right) \cdot z}{\sqrt{\color{blue}{-1 \cdot \left(a \cdot t\right)}}} \]
   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. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 40.3

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

   5. Simplified 40.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 38.4

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

   7. Applied egg-rr 38.4%

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

   if 5.49999999999999962e-153 < z < 2.05e120

   1. Initial program 87.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. *-lowering-*.f64 86.8

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

   4. Applied egg-rr 86.8%

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

      1. associate-*l/ N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 85.7

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

   6. Applied egg-rr 85.7%

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

   if 2.05e120 < z

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 71.2

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

   5. Simplified 71.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+120}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z}, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\frac{z\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \cdot \left(x\_m \cdot y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z\_m}, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (z_s y_s x_s x_m y_m z_m t a)
 :precision binary64
 (*
  z_s
  (*
   y_s
   (*
    x_s
    (if (<= z_m 2e+120)
      (* (/ z_m (sqrt (- (* z_m z_m) (* t a)))) (* x_m y_m))
      (* (* x_m y_m) (/ z_m (fma a (/ (* t -0.5) z_m) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2e+120) {
		tmp = (z_m / sqrt(((z_m * z_m) - (t * a)))) * (x_m * y_m);
	} else {
		tmp = (x_m * y_m) * (z_m / fma(a, ((t * -0.5) / z_m), z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2e+120)
		tmp = Float64(Float64(z_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) * Float64(x_m * y_m));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(a, Float64(Float64(t * -0.5) / z_m), z_m)));
	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.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[z$95$s_, y$95$s_, x$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(z$95$s * N[(y$95$s * N[(x$95$s * If[LessEqual[z$95$m, 2e+120], N[(N[(z$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(x$95$m * y$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(a * N[(N[(t * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2e120

   1. Initial program 70.7%

      \[\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - t \cdot a}} \]
   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. --lowering--.f64 N/A

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

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

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

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

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

      9. *-lowering-*.f64 72.6

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

   4. Applied egg-rr 72.6%

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

   if 2e120 < z

   1. Initial program 25.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 71.2

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

   5. Simplified 71.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 100.0

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2 \cdot 10^{+120}:\\ \;\;\;\;\frac{z}{\sqrt{z \cdot z - t \cdot a}} \cdot \left(x \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z}, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(x\_m \cdot y\_m\right) \cdot \frac{z\_m}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z\_m}, z\_m\right)}\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 3.6e-124)
      (/ (* x_m (* z_m y_m)) (sqrt (* t (- a))))
      (* (* x_m y_m) (/ z_m (fma a (/ (* t -0.5) z_m) z_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 3.6e-124) {
		tmp = (x_m * (z_m * y_m)) / sqrt((t * -a));
	} else {
		tmp = (x_m * y_m) * (z_m / fma(a, ((t * -0.5) / z_m), z_m));
	}
	return z_s * (y_s * (x_s * tmp));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
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 <= 3.6e-124)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(t * Float64(-a))));
	else
		tmp = Float64(Float64(x_m * y_m) * Float64(z_m / fma(a, Float64(Float64(t * -0.5) / z_m), z_m)));
	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.
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, 3.6e-124], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] * N[(z$95$m / N[(a * N[(N[(t * -0.5), $MachinePrecision] / z$95$m), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.6000000000000001e-124

   1. Initial program 65.7%

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

   3. Taylor expanded in z around 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. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 41.3

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

   5. Simplified 41.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 39.0

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

   7. Applied egg-rr 39.0%

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

   if 3.6000000000000001e-124 < z

   1. Initial program 57.2%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. associate-*r/ N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 75.8

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

   5. Simplified 75.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      8. *-lowering-*.f64 92.0

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

   7. Applied egg-rr 92.0%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{z}{\mathsf{fma}\left(a, \frac{t \cdot -0.5}{z}, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 3.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x\_m \cdot \left(z\_m \cdot y\_m\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 3.6e-124)
      (/ (* x_m (* z_m y_m)) (sqrt (* t (- a))))
      (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 3.6e-124) {
		tmp = (x_m * (z_m * y_m)) / sqrt((t * -a));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 3.6d-124) then
        tmp = (x_m * (z_m * y_m)) / sqrt((t * -a))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 3.6e-124) {
		tmp = (x_m * (z_m * y_m)) / Math.sqrt((t * -a));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 3.6e-124:
		tmp = (x_m * (z_m * y_m)) / math.sqrt((t * -a))
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
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 <= 3.6e-124)
		tmp = Float64(Float64(x_m * Float64(z_m * y_m)) / sqrt(Float64(t * Float64(-a))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 3.6e-124)
		tmp = (x_m * (z_m * y_m)) / sqrt((t * -a));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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, 3.6e-124], N[(N[(x$95$m * N[(z$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.6000000000000001e-124

   1. Initial program 65.7%

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

   3. Taylor expanded in z around 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. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 41.3

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

   5. Simplified 41.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-lowering-*.f64 39.0

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

   7. Applied egg-rr 39.0%

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

   if 3.6000000000000001e-124 < z

   1. Initial program 57.2%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 90.9

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

   5. Simplified 90.9%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.6 \cdot 10^{-124}:\\ \;\;\;\;\frac{x \cdot \left(z \cdot y\right)}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.5 \cdot 10^{-75}:\\ \;\;\;\;x\_m \cdot \frac{z\_m \cdot y\_m}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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-75)
      (* x_m (/ (* z_m y_m) (sqrt (* t (- a)))))
      (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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-75) {
		tmp = x_m * ((z_m * y_m) / sqrt((t * -a)));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 4.5d-75) then
        tmp = x_m * ((z_m * y_m) / sqrt((t * -a)))
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double z_s, double y_s, double x_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 4.5e-75) {
		tmp = x_m * ((z_m * y_m) / Math.sqrt((t * -a)));
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(z_s, y_s, x_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 4.5e-75:
		tmp = x_m * ((z_m * y_m) / math.sqrt((t * -a)))
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
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-75)
		tmp = Float64(x_m * Float64(Float64(z_m * y_m) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(z_s, y_s, x_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 4.5e-75)
		tmp = x_m * ((z_m * y_m) / sqrt((t * -a)));
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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-75], N[(x$95$m * N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.5000000000000003e-75

   1. Initial program 66.3%

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

   3. Taylor expanded in z around 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. distribute-rgt-neg-in N/A

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

      3. mul-1-neg N/A

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

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

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

      5. mul-1-neg N/A

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

      6. neg-lowering-neg.f64 41.0

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

   5. Simplified 41.0%

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

      1. associate-*l* N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. distribute-rgt-neg-out N/A

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

      8. *-commutative N/A

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

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

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

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

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

      11. neg-lowering-neg.f64 38.6

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

   7. Applied egg-rr 38.6%

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

   if 4.5000000000000003e-75 < z

   1. Initial program 55.5%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 93.6

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

   5. Simplified 93.6%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.5 \cdot 10^{-75}:\\ \;\;\;\;x \cdot \frac{z \cdot y}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\\\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ z\_s \cdot \left(y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.18 \cdot 10^{-132}:\\ \;\;\;\;y\_m \cdot \frac{z\_m \cdot x\_m}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 1.18e-132) (* y_m (/ (* z_m x_m) z_m)) (* x_m y_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
z\_m = fabs(z);
z\_s = copysign(1.0, z);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
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 <= 1.18e-132) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
z\_m = abs(z)
z\_s = copysign(1.0d0, z)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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 <= 1.18d-132) then
        tmp = y_m * ((z_m * x_m) / z_m)
    else
        tmp = x_m * y_m
    end if
    code = z_s * (y_s * (x_s * tmp))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
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 <= 1.18e-132) {
		tmp = y_m * ((z_m * x_m) / z_m);
	} else {
		tmp = x_m * y_m;
	}
	return z_s * (y_s * (x_s * tmp));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
[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 <= 1.18e-132:
		tmp = y_m * ((z_m * x_m) / z_m)
	else:
		tmp = x_m * y_m
	return z_s * (y_s * (x_s * tmp))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
z\_m = abs(z)
z\_s = copysign(1.0, z)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
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 <= 1.18e-132)
		tmp = Float64(y_m * Float64(Float64(z_m * x_m) / z_m));
	else
		tmp = Float64(x_m * y_m);
	end
	return Float64(z_s * Float64(y_s * Float64(x_s * tmp)))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
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 <= 1.18e-132)
		tmp = y_m * ((z_m * x_m) / z_m);
	else
		tmp = x_m * y_m;
	end
	tmp_2 = z_s * (y_s * (x_s * tmp));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
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, 1.18e-132], N[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(x$95$m * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.17999999999999993e-132

   1. Initial program 65.2%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 14.6

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

   5. Simplified 14.6%

      \[\leadsto \color{blue}{x \cdot y} \]
   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. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 18.3

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

   7. Applied egg-rr 18.3%

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

   if 1.17999999999999993e-132 < z

   1. Initial program 58.1%

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

   3. Taylor expanded in z around inf

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

      1. *-lowering-*.f64 89.1

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

   5. Simplified 89.1%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.18 \cdot 10^{-132}:\\ \;\;\;\;y \cdot \frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.2% accurate, 7.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in z around inf

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

   1. *-lowering-*.f64 42.2

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

5. Simplified 42.2%

   \[\leadsto \color{blue}{x \cdot y} \]
6. Add Preprocessing

Developer Target 1: 87.7% 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 2024199 
(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)))))