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

Percentage Accurate: 61.0% → 90.8%

Time: 9.5s

Alternatives: 12

Speedup: 4.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% 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: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 48000000:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\log \left(\frac{z\_m}{\mathsf{fma}\left(-0.5 \cdot \left(\frac{t}{z\_m \cdot z\_m} \cdot a\right), z\_m, z\_m\right)}\right)} \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 48000000.0)
      (/ (* x_m (* y_m z_m)) (sqrt (- (* z_m z_m) (* a t))))
      (*
       (*
        (exp (log (/ z_m (fma (* -0.5 (* (/ t (* z_m z_m)) a)) z_m z_m))))
        x_m)
       y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 48000000.0) {
		tmp = (x_m * (y_m * z_m)) / sqrt(((z_m * z_m) - (a * t)));
	} else {
		tmp = (exp(log((z_m / fma((-0.5 * ((t / (z_m * z_m)) * a)), z_m, z_m)))) * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 48000000.0)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))));
	else
		tmp = Float64(Float64(exp(log(Float64(z_m / fma(Float64(-0.5 * Float64(Float64(t / Float64(z_m * z_m)) * a)), z_m, z_m)))) * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 48000000.0], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[Log[N[(z$95$m / N[(N[(-0.5 * N[(N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] * z$95$m + z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.8e7

   1. Initial program 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.1

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

   4. Applied rewrites 67.1%

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

   if 4.8e7 < z

   1. Initial program 43.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. inv-pow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 93.9%

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

       1. rem-log-exp N/A

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

       2. lift-exp.f64 N/A

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

       3. lower-log.f64 93.9

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

       4. lift-exp.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-log.f64 N/A

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

       7. exp-to-pow N/A

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

   11. Applied rewrites 93.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 48000000:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\log \left(\frac{z}{\mathsf{fma}\left(-0.5 \cdot \left(\frac{t}{z \cdot z} \cdot a\right), z, z\right)}\right)} \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 48000000:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z\_m}{\mathsf{fma}\left(-0.5, \frac{t}{z\_m \cdot z\_m} \cdot a, 1\right) \cdot z\_m} \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 48000000.0)
      (/ (* x_m (* y_m z_m)) (sqrt (- (* z_m z_m) (* a t))))
      (*
       (* (/ z_m (* (fma -0.5 (* (/ t (* z_m z_m)) a) 1.0) z_m)) x_m)
       y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 48000000.0) {
		tmp = (x_m * (y_m * z_m)) / sqrt(((z_m * z_m) - (a * t)));
	} else {
		tmp = ((z_m / (fma(-0.5, ((t / (z_m * z_m)) * a), 1.0) * z_m)) * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 48000000.0)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))));
	else
		tmp = Float64(Float64(Float64(z_m / Float64(fma(-0.5, Float64(Float64(t / Float64(z_m * z_m)) * a), 1.0) * z_m)) * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 48000000.0], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(z$95$m / N[(N[(-0.5 * N[(N[(t / N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision] + 1.0), $MachinePrecision] * z$95$m), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.8e7

   1. Initial program 69.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 67.1

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

   4. Applied rewrites 67.1%

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

   if 4.8e7 < z

   1. Initial program 43.2%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 93.9

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

   7. Applied rewrites 93.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 48000000:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\mathsf{fma}\left(-0.5, \frac{t}{z \cdot z} \cdot a, 1\right) \cdot z} \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 6 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{-1}{a \cdot t}} \cdot \left(x\_m \cdot \left(y\_m \cdot z\_m\right)\right)\\ \mathbf{elif}\;z\_m \leq 1.52 \cdot 10^{+41}:\\ \;\;\;\;\left(\frac{x\_m}{\sqrt{\mathsf{fma}\left(-a, t, z\_m \cdot z\_m\right)}} \cdot z\_m\right) \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 6e-212)
      (* (sqrt (/ -1.0 (* a t))) (* x_m (* y_m z_m)))
      (if (<= z_m 1.52e+41)
        (* (* (/ x_m (sqrt (fma (- a) t (* z_m z_m)))) z_m) y_m)
        (* (* 1.0 x_m) y_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 6e-212) {
		tmp = sqrt((-1.0 / (a * t))) * (x_m * (y_m * z_m));
	} else if (z_m <= 1.52e+41) {
		tmp = ((x_m / sqrt(fma(-a, t, (z_m * z_m)))) * z_m) * y_m;
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 6e-212)
		tmp = Float64(sqrt(Float64(-1.0 / Float64(a * t))) * Float64(x_m * Float64(y_m * z_m)));
	elseif (z_m <= 1.52e+41)
		tmp = Float64(Float64(Float64(x_m / sqrt(fma(Float64(-a), t, Float64(z_m * z_m)))) * z_m) * y_m);
	else
		tmp = Float64(Float64(1.0 * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 6e-212], N[(N[Sqrt[N[(-1.0 / N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z$95$m, 1.52e+41], N[(N[(N[(x$95$m / N[Sqrt[N[((-a) * t + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < 6.0000000000000005e-212

   1. Initial program 63.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 61.5

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

   5. Applied rewrites 61.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 33.2%

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

   if 6.0000000000000005e-212 < z < 1.52000000000000002e41

   1. Initial program 89.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 59.1

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

   7. Applied rewrites 59.1%

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

   8. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 83.4

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

   10. Applied rewrites 83.4%

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

   12. Applied rewrites 88.7%

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

   if 1.52000000000000002e41 < z

   1. Initial program 39.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 94.9

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 94.9%

       \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 6 \cdot 10^{-212}:\\ \;\;\;\;\sqrt{\frac{-1}{a \cdot t}} \cdot \left(x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{+41}:\\ \;\;\;\;\left(\frac{x}{\sqrt{\mathsf{fma}\left(-a, t, z \cdot z\right)}} \cdot z\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5800000:\\ \;\;\;\;\frac{x\_m \cdot \left(y\_m \cdot z\_m\right)}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 5800000.0)
      (/ (* x_m (* y_m z_m)) (sqrt (- (* z_m z_m) (* a t))))
      (* (* 1.0 x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5800000.0) {
		tmp = (x_m * (y_m * z_m)) / sqrt(((z_m * z_m) - (a * t)));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 5800000.0d0) then
        tmp = (x_m * (y_m * z_m)) / sqrt(((z_m * z_m) - (a * t)))
    else
        tmp = (1.0d0 * x_m) * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 5800000.0) {
		tmp = (x_m * (y_m * z_m)) / Math.sqrt(((z_m * z_m) - (a * t)));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 5800000.0:
		tmp = (x_m * (y_m * z_m)) / math.sqrt(((z_m * z_m) - (a * t)))
	else:
		tmp = (1.0 * x_m) * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 5800000.0)
		tmp = Float64(Float64(x_m * Float64(y_m * z_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))));
	else
		tmp = Float64(Float64(1.0 * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 5800000.0)
		tmp = (x_m * (y_m * z_m)) / sqrt(((z_m * z_m) - (a * t)));
	else
		tmp = (1.0 * x_m) * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 5800000.0], N[(N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.8e6

   1. Initial program 69.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 66.9

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

   4. Applied rewrites 66.9%

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

   if 5.8e6 < z

   1. Initial program 43.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 77.5

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

   5. Applied rewrites 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 92.8

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 94.0%

       \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5800000:\\ \;\;\;\;\frac{x \cdot \left(y \cdot z\right)}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.3 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{z\_m \cdot z\_m - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.3e+99)
      (/ (* (* x_m y_m) z_m) (sqrt (- (* z_m z_m) (* a t))))
      (* (* 1.0 x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.3e+99) {
		tmp = ((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (a * t)));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.3d+99) then
        tmp = ((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (a * t)))
    else
        tmp = (1.0d0 * x_m) * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.3e+99) {
		tmp = ((x_m * y_m) * z_m) / Math.sqrt(((z_m * z_m) - (a * t)));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.3e+99:
		tmp = ((x_m * y_m) * z_m) / math.sqrt(((z_m * z_m) - (a * t)))
	else:
		tmp = (1.0 * x_m) * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.3e+99)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(a * t))));
	else
		tmp = Float64(Float64(1.0 * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.3e+99)
		tmp = ((x_m * y_m) * z_m) / sqrt(((z_m * z_m) - (a * t)));
	else
		tmp = (1.0 * x_m) * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.3e+99], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.30000000000000019e99

   1. Initial program 72.4%

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

   if 2.30000000000000019e99 < z

   1. Initial program 22.1%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 95.1%

       \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.3 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{z \cdot z - a \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.32 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\frac{-1}{a \cdot t}} \cdot \left(x\_m \cdot \left(y\_m \cdot z\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.32e-95)
      (* (sqrt (/ -1.0 (* a t))) (* x_m (* y_m z_m)))
      (* (* 1.0 x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.32e-95) {
		tmp = sqrt((-1.0 / (a * t))) * (x_m * (y_m * z_m));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1.32d-95) then
        tmp = sqrt(((-1.0d0) / (a * t))) * (x_m * (y_m * z_m))
    else
        tmp = (1.0d0 * x_m) * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.32e-95) {
		tmp = Math.sqrt((-1.0 / (a * t))) * (x_m * (y_m * z_m));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.32e-95:
		tmp = math.sqrt((-1.0 / (a * t))) * (x_m * (y_m * z_m))
	else:
		tmp = (1.0 * x_m) * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.32e-95)
		tmp = Float64(sqrt(Float64(-1.0 / Float64(a * t))) * Float64(x_m * Float64(y_m * z_m)));
	else
		tmp = Float64(Float64(1.0 * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.32e-95)
		tmp = sqrt((-1.0 / (a * t))) * (x_m * (y_m * z_m));
	else
		tmp = (1.0 * x_m) * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.32e-95], N[(N[Sqrt[N[(-1.0 / N[(a * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.31999999999999996e-95

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 35.3%

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

   if 1.31999999999999996e-95 < z

   1. Initial program 56.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 85.4%

       \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.32 \cdot 10^{-95}:\\ \;\;\;\;\sqrt{\frac{-1}{a \cdot t}} \cdot \left(x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.32 \cdot 10^{-95}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.32e-95)
      (/ (* (* x_m y_m) z_m) (sqrt (* (- a) t)))
      (* (* 1.0 x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.32e-95) {
		tmp = ((x_m * y_m) * z_m) / sqrt((-a * t));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1.32d-95) then
        tmp = ((x_m * y_m) * z_m) / sqrt((-a * t))
    else
        tmp = (1.0d0 * x_m) * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.32e-95) {
		tmp = ((x_m * y_m) * z_m) / Math.sqrt((-a * t));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.32e-95:
		tmp = ((x_m * y_m) * z_m) / math.sqrt((-a * t))
	else:
		tmp = (1.0 * x_m) * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.32e-95)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / sqrt(Float64(Float64(-a) * t)));
	else
		tmp = Float64(Float64(1.0 * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.32e-95)
		tmp = ((x_m * y_m) * z_m) / sqrt((-a * t));
	else
		tmp = (1.0 * x_m) * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.32e-95], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / N[Sqrt[N[((-a) * t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.31999999999999996e-95

   1. Initial program 65.1%

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

   3. Taylor expanded in z around 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. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 40.2

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

   5. Applied rewrites 40.2%

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

   if 1.31999999999999996e-95 < z

   1. Initial program 56.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 85.4%

       \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.32 \cdot 10^{-95}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{\sqrt{\left(-a\right) \cdot t}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.95 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{z\_m} \cdot \left(x\_m \cdot \left(y\_m \cdot z\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1.95e-98)
      (* (/ 1.0 z_m) (* x_m (* y_m z_m)))
      (* (* 1.0 x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e-98) {
		tmp = (1.0 / z_m) * (x_m * (y_m * z_m));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 1.95d-98) then
        tmp = (1.0d0 / z_m) * (x_m * (y_m * z_m))
    else
        tmp = (1.0d0 * x_m) * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1.95e-98) {
		tmp = (1.0 / z_m) * (x_m * (y_m * z_m));
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1.95e-98:
		tmp = (1.0 / z_m) * (x_m * (y_m * z_m))
	else:
		tmp = (1.0 * x_m) * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1.95e-98)
		tmp = Float64(Float64(1.0 / z_m) * Float64(x_m * Float64(y_m * z_m)));
	else
		tmp = Float64(Float64(1.0 * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1.95e-98)
		tmp = (1.0 / z_m) * (x_m * (y_m * z_m));
	else
		tmp = (1.0 * x_m) * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1.95e-98], N[(N[(1.0 / z$95$m), $MachinePrecision] * N[(x$95$m * N[(y$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.94999999999999986e-98

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. mul-1-neg N/A

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

      11. lower-neg.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 27.8%

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

   if 1.94999999999999986e-98 < z

   1. Initial program 56.9%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 73.2

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

   5. Applied rewrites 73.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 84.0

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

   7. Applied rewrites 84.0%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 85.4%

       \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 1.95 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{z} \cdot \left(x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 2.55 \cdot 10^{-172}:\\ \;\;\;\;\frac{\left(x\_m \cdot z\_m\right) \cdot y\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 2.55e-172)
      (/ (* (* x_m z_m) y_m) (- z_m))
      (* (* 1.0 x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.55e-172) {
		tmp = ((x_m * z_m) * y_m) / -z_m;
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z_m
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z_m <= 2.55d-172) then
        tmp = ((x_m * z_m) * y_m) / -z_m
    else
        tmp = (1.0d0 * x_m) * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 2.55e-172) {
		tmp = ((x_m * z_m) * y_m) / -z_m;
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 2.55e-172:
		tmp = ((x_m * z_m) * y_m) / -z_m
	else:
		tmp = (1.0 * x_m) * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 2.55e-172)
		tmp = Float64(Float64(Float64(x_m * z_m) * y_m) / Float64(-z_m));
	else
		tmp = Float64(Float64(1.0 * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 2.55e-172)
		tmp = ((x_m * z_m) * y_m) / -z_m;
	else
		tmp = (1.0 * x_m) * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 2.55e-172], N[(N[(N[(x$95$m * z$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.5499999999999999e-172

   1. Initial program 64.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 62.7

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

   4. Applied rewrites 62.7%

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

   5. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if 2.5499999999999999e-172 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 82.3%

       \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.55 \cdot 10^{-172}:\\ \;\;\;\;\frac{\left(x \cdot z\right) \cdot y}{-z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{-179}:\\ \;\;\;\;\frac{\left(x\_m \cdot y\_m\right) \cdot z\_m}{-z\_m}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\_m\right) \cdot y\_m\\ \end{array}\right)\right) \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
(FPCore (x_s y_s z_s x_m y_m z_m t a)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (*
    z_s
    (if (<= z_m 1e-179)
      (/ (* (* x_m y_m) z_m) (- z_m))
      (* (* 1.0 x_m) y_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z_m && z_m < t && t < a);
double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-179) {
		tmp = ((x_m * y_m) * z_m) / -z_m;
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: z_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 <= 1d-179) then
        tmp = ((x_m * y_m) * z_m) / -z_m
    else
        tmp = (1.0d0 * x_m) * y_m
    end if
    code = x_s * (y_s * (z_s * tmp))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z_m && z_m < t && t < a;
public static double code(double x_s, double y_s, double z_s, double x_m, double y_m, double z_m, double t, double a) {
	double tmp;
	if (z_m <= 1e-179) {
		tmp = ((x_m * y_m) * z_m) / -z_m;
	} else {
		tmp = (1.0 * x_m) * y_m;
	}
	return x_s * (y_s * (z_s * tmp));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z_m, t, a] = sort([x_m, y_m, z_m, t, a])
def code(x_s, y_s, z_s, x_m, y_m, z_m, t, a):
	tmp = 0
	if z_m <= 1e-179:
		tmp = ((x_m * y_m) * z_m) / -z_m
	else:
		tmp = (1.0 * x_m) * y_m
	return x_s * (y_s * (z_s * tmp))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z_m, t, a = sort([x_m, y_m, z_m, t, a])
function code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0
	if (z_m <= 1e-179)
		tmp = Float64(Float64(Float64(x_m * y_m) * z_m) / Float64(-z_m));
	else
		tmp = Float64(Float64(1.0 * x_m) * y_m);
	end
	return Float64(x_s * Float64(y_s * Float64(z_s * tmp)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z_m, t, a = num2cell(sort([x_m, y_m, z_m, t, a])){:}
function tmp_2 = code(x_s, y_s, z_s, x_m, y_m, z_m, t, a)
	tmp = 0.0;
	if (z_m <= 1e-179)
		tmp = ((x_m * y_m) * z_m) / -z_m;
	else
		tmp = (1.0 * x_m) * y_m;
	end
	tmp_2 = x_s * (y_s * (z_s * tmp));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, 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]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, z_m, t, and a should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, z$95$s_, x$95$m_, y$95$m_, z$95$m_, t_, a_] := N[(x$95$s * N[(y$95$s * N[(z$95$s * If[LessEqual[z$95$m, 1e-179], N[(N[(N[(x$95$m * y$95$m), $MachinePrecision] * z$95$m), $MachinePrecision] / (-z$95$m)), $MachinePrecision], N[(N[(1.0 * x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1e-179

   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 -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 69.7

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

   5. Applied rewrites 69.7%

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

   if 1e-179 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 70.7

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

   5. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 80.9

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

   7. Applied rewrites 80.9%

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

   8. Taylor expanded in z around inf

      \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 82.3%

       \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{-179}:\\ \;\;\;\;\frac{\left(x \cdot y\right) \cdot z}{-z}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in z around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. associate-/l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 46.7

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

5. Applied rewrites 46.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 51.2

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

7. Applied rewrites 51.2%

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

8. Taylor expanded in z around inf

   \[\leadsto y \cdot \left(x \cdot \color{blue}{1}\right) \]
9. Step-by-step derivation

10. Applied rewrites 47.7%

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

11. Final simplification 47.7%

    \[\leadsto \left(1 \cdot x\right) \cdot y \]
12. Add Preprocessing

Alternative 12: 13.5% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

3. Taylor expanded in z around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 44.3

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

5. Applied rewrites 44.3%

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

6. Final simplification 44.3%

   \[\leadsto \left(-x\right) \cdot y \]
7. Add Preprocessing

Developer Target 1: 87.5% 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 2024298 
(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)))))