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

Percentage Accurate: 61.5% → 92.5%

Time: 12.7s

Alternatives: 8

Speedup: 7.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 10^{+37}:\\ \;\;\;\;\frac{z\_m \cdot y\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\mathsf{fma}\left(a, t \cdot \frac{-0.5}{z\_m}, z\_m\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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+37)
      (* (/ (* z_m y_m) (sqrt (- (* z_m z_m) (* t a)))) x_m)
      (* x_m (* y_m (/ z_m (fma a (* t (/ -0.5 z_m)) z_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+37) {
		tmp = ((z_m * y_m) / sqrt(((z_m * z_m) - (t * a)))) * x_m;
	} else {
		tmp = x_m * (y_m * (z_m / fma(a, (t * (-0.5 / z_m)), z_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+37)
		tmp = Float64(Float64(Float64(z_m * y_m) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) * x_m);
	else
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / fma(a, Float64(t * Float64(-0.5 / z_m)), z_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, 1e+37], N[(N[(N[(z$95$m * y$95$m), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[(a * N[(t * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.99999999999999954e36

   1. Initial program 67.8%

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

      1. associate-*l* N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 66.8

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

   4. Applied rewrites 66.8%

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

   if 9.99999999999999954e36 < z

   1. Initial program 41.6%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.9

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

   7. Applied rewrites 69.9%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 62.5

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

      8. lift-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-fma.f64 69.3

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

   9. Applied rewrites 69.3%

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

       1. remove-double-neg N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. remove-double-neg N/A

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

       6. frac-2neg N/A

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

       7. frac-2neg N/A

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

       8. lift-/.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lift-/.f64 N/A

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

       12. div-inv N/A

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

       13. associate-*l* N/A

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

       14. associate-*l* N/A

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

       15. lower-*.f64 N/A

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

   11. Applied rewrites 95.3%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 10^{+37}:\\ \;\;\;\;\frac{z \cdot y}{\sqrt{z \cdot z - t \cdot a}} \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\mathsf{fma}\left(a, t \cdot \frac{-0.5}{z}, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{z\_m \cdot \left(y\_m \cdot x\_m\right)}{\sqrt{z\_m \cdot z\_m - t \cdot a}} \leq 10^{-267}:\\ \;\;\;\;\left(y\_m \cdot \left(z\_m \cdot x\_m\right)\right) \cdot \frac{1}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 (* y_m x_m)) (sqrt (- (* z_m z_m) (* t a)))) 1e-267)
      (* (* y_m (* z_m x_m)) (/ 1.0 z_m))
      (* y_m x_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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 1e-267) {
		tmp = (y_m * (z_m * x_m)) * (1.0 / z_m);
	} else {
		tmp = y_m * x_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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 1d-267) then
        tmp = (y_m * (z_m * x_m)) * (1.0d0 / z_m)
    else
        tmp = y_m * x_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 * (y_m * x_m)) / Math.sqrt(((z_m * z_m) - (t * a)))) <= 1e-267) {
		tmp = (y_m * (z_m * x_m)) * (1.0 / z_m);
	} else {
		tmp = y_m * x_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 * (y_m * x_m)) / math.sqrt(((z_m * z_m) - (t * a)))) <= 1e-267:
		tmp = (y_m * (z_m * x_m)) * (1.0 / z_m)
	else:
		tmp = y_m * x_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 (Float64(Float64(z_m * Float64(y_m * x_m)) / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))) <= 1e-267)
		tmp = Float64(Float64(y_m * Float64(z_m * x_m)) * Float64(1.0 / z_m));
	else
		tmp = Float64(y_m * x_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 * (y_m * x_m)) / sqrt(((z_m * z_m) - (t * a)))) <= 1e-267)
		tmp = (y_m * (z_m * x_m)) * (1.0 / z_m);
	else
		tmp = y_m * x_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[N[(N[(z$95$m * N[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-267], N[(N[(y$95$m * N[(z$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a)))) < 9.9999999999999998e-268

   1. Initial program 65.3%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 68.1

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

   4. Applied rewrites 68.1%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 35.9

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

   7. Applied rewrites 35.9%

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

      1. lift-*.f64 N/A

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

      2. associate-*l/ N/A

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

      3. lift-*.f64 N/A

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

      4. div-inv N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 49.1

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

   9. Applied rewrites 49.1%

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. lift-*.f64 N/A

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

       4. lower-*.f64 48.5

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

   11. Applied rewrites 48.5%

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

   if 9.9999999999999998e-268 < (/.f64 (*.f64 (*.f64 x y) z) (sqrt.f64 (-.f64 (*.f64 z z) (*.f64 t a))))

   1. Initial program 54.0%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 39.8

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

   5. Applied rewrites 39.8%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z \cdot \left(y \cdot x\right)}{\sqrt{z \cdot z - t \cdot a}} \leq 10^{-267}:\\ \;\;\;\;\left(y \cdot \left(z \cdot x\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 5 \cdot 10^{+111}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\sqrt{\mathsf{fma}\left(t, -a, z\_m \cdot z\_m\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\mathsf{fma}\left(a, t \cdot \frac{-0.5}{z\_m}, z\_m\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 5e+111)
      (* x_m (* y_m (/ z_m (sqrt (fma t (- a) (* z_m z_m))))))
      (* x_m (* y_m (/ z_m (fma a (* t (/ -0.5 z_m)) z_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 <= 5e+111) {
		tmp = x_m * (y_m * (z_m / sqrt(fma(t, -a, (z_m * z_m)))));
	} else {
		tmp = x_m * (y_m * (z_m / fma(a, (t * (-0.5 / z_m)), z_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 <= 5e+111)
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / sqrt(fma(t, Float64(-a), Float64(z_m * z_m))))));
	else
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / fma(a, Float64(t * Float64(-0.5 / z_m)), z_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, 5e+111], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[Sqrt[N[(t * (-a) + N[(z$95$m * z$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[(a * N[(t * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.9999999999999997e111

   1. Initial program 70.2%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. associate-*r/ N/A

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

      13. *-commutative N/A

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

      14. lower-/.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 68.7

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

   4. Applied rewrites 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-/l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 71.7%

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

   if 4.9999999999999997e111 < z

   1. Initial program 23.9%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 26.0

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

   4. Applied rewrites 26.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 64.5

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

   7. Applied rewrites 64.5%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 58.2

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

      8. lift-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-fma.f64 67.2

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

   9. Applied rewrites 67.2%

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

       1. remove-double-neg N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. remove-double-neg N/A

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

       6. frac-2neg N/A

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

       7. frac-2neg N/A

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

       8. lift-/.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lift-/.f64 N/A

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

       12. div-inv N/A

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

       13. associate-*l* N/A

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

       14. associate-*l* N/A

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

       15. lower-*.f64 N/A

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

   11. Applied rewrites 98.3%

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

4. Final simplification 77.3%

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

Alternative 4: 91.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\left(z\_m \cdot y\_m\right) \cdot \frac{x\_m}{\sqrt{z\_m \cdot z\_m - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\mathsf{fma}\left(a, t \cdot \frac{-0.5}{z\_m}, z\_m\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 4e+34)
      (* (* z_m y_m) (/ x_m (sqrt (- (* z_m z_m) (* t a)))))
      (* x_m (* y_m (/ z_m (fma a (* t (/ -0.5 z_m)) z_m)))))))))

C

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 <= 4e+34) {
		tmp = (z_m * y_m) * (x_m / sqrt(((z_m * z_m) - (t * a))));
	} else {
		tmp = x_m * (y_m * (z_m / fma(a, (t * (-0.5 / z_m)), z_m)));
	}
	return 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 <= 4e+34)
		tmp = Float64(Float64(z_m * y_m) * Float64(x_m / sqrt(Float64(Float64(z_m * z_m) - Float64(t * a)))));
	else
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / fma(a, Float64(t * Float64(-0.5 / z_m)), z_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, 4e+34], N[(N[(z$95$m * y$95$m), $MachinePrecision] * N[(x$95$m / N[Sqrt[N[(N[(z$95$m * z$95$m), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[(a * N[(t * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 3.99999999999999978e34

   1. Initial program 67.8%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 66.1

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

   4. Applied rewrites 66.1%

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

   if 3.99999999999999978e34 < z

   1. Initial program 41.6%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 69.9

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

   7. Applied rewrites 69.9%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 62.5

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

      8. lift-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-fma.f64 69.3

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

   9. Applied rewrites 69.3%

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

       1. remove-double-neg N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. remove-double-neg N/A

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

       6. frac-2neg N/A

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

       7. frac-2neg N/A

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

       8. lift-/.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lift-/.f64 N/A

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

       12. div-inv N/A

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

       13. associate-*l* N/A

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

       14. associate-*l* N/A

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

       15. lower-*.f64 N/A

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

   11. Applied rewrites 95.3%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4 \cdot 10^{+34}:\\ \;\;\;\;\left(z \cdot y\right) \cdot \frac{x}{\sqrt{z \cdot z - t \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\mathsf{fma}\left(a, t \cdot \frac{-0.5}{z}, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 4.4 \cdot 10^{-95}:\\ \;\;\;\;z\_m \cdot \frac{y\_m \cdot x\_m}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(y\_m \cdot \frac{z\_m}{\mathsf{fma}\left(a, t \cdot \frac{-0.5}{z\_m}, z\_m\right)}\right)\\ \end{array}\right)\right) \end{array} \]

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 4.4e-95)
      (* z_m (/ (* y_m x_m) (sqrt (* t (- a)))))
      (* x_m (* y_m (/ z_m (fma a (* t (/ -0.5 z_m)) z_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 <= 4.4e-95) {
		tmp = z_m * ((y_m * x_m) / sqrt((t * -a)));
	} else {
		tmp = x_m * (y_m * (z_m / fma(a, (t * (-0.5 / z_m)), z_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 <= 4.4e-95)
		tmp = Float64(z_m * Float64(Float64(y_m * x_m) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(x_m * Float64(y_m * Float64(z_m / fma(a, Float64(t * Float64(-0.5 / z_m)), z_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, 4.4e-95], N[(z$95$m * N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m * N[(z$95$m / N[(a * N[(t * N[(-0.5 / z$95$m), $MachinePrecision]), $MachinePrecision] + z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 4.3999999999999998e-95

   1. Initial program 63.4%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 66.7

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

   4. Applied rewrites 66.7%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 45.8

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

   7. Applied rewrites 45.8%

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

   if 4.3999999999999998e-95 < z

   1. Initial program 55.7%

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

      1. 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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 56.2

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

   4. Applied rewrites 56.2%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 74.3

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

   7. Applied rewrites 74.3%

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

      1. *-commutative N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 68.7

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

      8. lift-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r/ N/A

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

      11. lift-*.f64 N/A

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

      12. associate-*r* N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. associate-*l/ N/A

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

      16. lift-/.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-fma.f64 73.8

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

   9. Applied rewrites 73.8%

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

       1. remove-double-neg N/A

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

       2. lift-/.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. remove-double-neg N/A

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

       6. frac-2neg N/A

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

       7. frac-2neg N/A

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

       8. lift-/.f64 N/A

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

       9. associate-*l* N/A

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

       10. *-commutative N/A

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

       11. lift-/.f64 N/A

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

       12. div-inv N/A

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

       13. associate-*l* N/A

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

       14. associate-*l* N/A

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

       15. lower-*.f64 N/A

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

   11. Applied rewrites 94.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.4 \cdot 10^{-95}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot \frac{z}{\mathsf{fma}\left(a, t \cdot \frac{-0.5}{z}, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z_m, t, a] = \mathsf{sort}([x_m, y_m, z_m, t, a])\\ \\ x\_s \cdot \left(y\_s \cdot \left(z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;z\_m \cdot \frac{y\_m \cdot x\_m}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot x\_m\\ \end{array}\right)\right) \end{array} \]

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 9.5e-88)
      (* z_m (/ (* y_m x_m) (sqrt (* t (- a)))))
      (* y_m x_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 <= 9.5e-88) {
		tmp = z_m * ((y_m * x_m) / sqrt((t * -a)));
	} else {
		tmp = y_m * x_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 <= 9.5d-88) then
        tmp = z_m * ((y_m * x_m) / sqrt((t * -a)))
    else
        tmp = y_m * x_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 <= 9.5e-88) {
		tmp = z_m * ((y_m * x_m) / Math.sqrt((t * -a)));
	} else {
		tmp = y_m * x_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 <= 9.5e-88:
		tmp = z_m * ((y_m * x_m) / math.sqrt((t * -a)))
	else:
		tmp = y_m * x_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 <= 9.5e-88)
		tmp = Float64(z_m * Float64(Float64(y_m * x_m) / sqrt(Float64(t * Float64(-a)))));
	else
		tmp = Float64(y_m * x_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 <= 9.5e-88)
		tmp = z_m * ((y_m * x_m) / sqrt((t * -a)));
	else
		tmp = y_m * x_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, 9.5e-88], N[(z$95$m * N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[Sqrt[N[(t * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 9.5e-88

   1. Initial program 63.6%

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

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 66.9

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 45.5

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

   7. Applied rewrites 45.5%

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

   if 9.5e-88 < z

   1. Initial program 55.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 91.3

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

   5. Applied rewrites 91.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 9.5 \cdot 10^{-88}:\\ \;\;\;\;z \cdot \frac{y \cdot x}{\sqrt{t \cdot \left(-a\right)}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 2.0× speedup

Math

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

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 (* y_m (/ (* z_m x_m) z_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 * (y_m * ((z_m * x_m) / z_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 * (y_m * ((z_m * x_m) / z_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 * (y_m * ((z_m * x_m) / z_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 * (y_m * ((z_m * x_m) / z_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(y_m * Float64(Float64(z_m * x_m) / z_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 * (y_m * ((z_m * x_m) / z_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[(y$95$m * N[(N[(z$95$m * x$95$m), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

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

   3. lift-*.f64 N/A

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

   4. lift--.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 62.7

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

4. Applied rewrites 62.7%

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

5. Taylor expanded in z around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 37.2

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

7. Applied rewrites 37.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 36.6

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

9. Applied rewrites 36.6%

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

    1. lift-*.f64 N/A

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

    2. clear-num N/A

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

    3. associate-/r/ N/A

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

    4. lift-/.f64 N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. lift-/.f64 N/A

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

    9. associate-*l/ N/A

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

    10. *-lft-identity N/A

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

    11. lower-/.f64 41.1

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

11. Applied rewrites 41.1%

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

12. Final simplification 41.1%

    \[\leadsto y \cdot \frac{z \cdot x}{z} \]
13. Add Preprocessing

Alternative 8: 73.2% accurate, 7.5× speedup

Math

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

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 (* y_m x_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 * (y_m * x_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 * (y_m * x_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 * (y_m * x_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 * (y_m * x_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(y_m * x_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 * (y_m * x_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[(y$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.4%

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

3. Taylor expanded in z around inf

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

   1. lower-*.f64 41.5

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

5. Applied rewrites 41.5%

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

6. Final simplification 41.5%

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

Developer Target 1: 87.6% 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 2024219 
(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)))))