Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Percentage Accurate: 83.0% → 97.7%

Time: 9.5s

Alternatives: 16

Speedup: 0.8×

• 26.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 83.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x y) (* (* z z) (+ z 1.0))))

C

double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * y) / ((z * z) * (z + 1.0d0))
end function

Java

public static double code(double x, double y, double z) {
	return (x * y) / ((z * z) * (z + 1.0));
}

Python

def code(x, y, z):
	return (x * y) / ((z * z) * (z + 1.0))

Julia

function code(x, y, z)
	return Float64(Float64(x * y) / Float64(Float64(z * z) * Float64(z + 1.0)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * y) / ((z * z) * (z + 1.0));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\frac{z}{y\_m} \cdot z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(\frac{z - -1}{y\_m} \cdot z\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z -1e+20)
     (/ (/ x_m z) (* (/ z y_m) z))
     (if (<= z 3.3e-71)
       (/ (* (/ x_m (fma z z z)) y_m) z)
       (/ x_m (* (* (/ (- z -1.0) y_m) z) z)))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -1e+20) {
		tmp = (x_m / z) / ((z / y_m) * z);
	} else if (z <= 3.3e-71) {
		tmp = ((x_m / fma(z, z, z)) * y_m) / z;
	} else {
		tmp = x_m / ((((z - -1.0) / y_m) * z) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= -1e+20)
		tmp = Float64(Float64(x_m / z) / Float64(Float64(z / y_m) * z));
	elseif (z <= 3.3e-71)
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) * y_m) / z);
	else
		tmp = Float64(x_m / Float64(Float64(Float64(Float64(z - -1.0) / y_m) * z) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, -1e+20], N[(N[(x$95$m / z), $MachinePrecision] / N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.3e-71], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(N[(N[(N[(z - -1.0), $MachinePrecision] / y$95$m), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1e20

   1. Initial program 77.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. clear-num N/A

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

      12. inv-pow N/A

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

      13. unpow-prod-down N/A

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

      14. associate-/l* N/A

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

      15. *-lft-identity N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 96.7

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if -1e20 < z < 3.3000000000000002e-71

   1. Initial program 84.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 36.6

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

   7. Applied rewrites 36.6%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 96.9

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

   10. Applied rewrites 96.9%

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

   if 3.3000000000000002e-71 < z

   1. Initial program 83.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. clear-num N/A

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

      12. inv-pow N/A

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

      13. unpow-prod-down N/A

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

      14. associate-/l* N/A

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

      15. *-lft-identity N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 94.7

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 94.7

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

   4. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. times-frac N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l/ N/A

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

      15. associate-/r* N/A

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

      16. frac-2neg N/A

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

      17. distribute-frac-neg2 N/A

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

      18. lift-neg.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 95.8%

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

   8. Applied rewrites 93.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{x}{z}}{\frac{z}{y} \cdot z}\\ \mathbf{elif}\;z \leq 3.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{z - -1}{y} \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y\_m}{z \cdot z}}{z} \cdot x\_m\\ t_1 := \left(z \cdot z\right) \cdot \left(z - -1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y_m (* z z)) z) x_m)) (t_1 (* (* z z) (- z -1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -5e+36)
       t_0
       (if (<= t_1 0.0)
         (* (/ y_m z) (/ x_m z))
         (if (<= t_1 5e+37) (* (/ x_m (* (fma z z z) z)) y_m) t_0)))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = ((y_m / (z * z)) / z) * x_m;
	double t_1 = (z * z) * (z - -1.0);
	double tmp;
	if (t_1 <= -5e+36) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (y_m / z) * (x_m / z);
	} else if (t_1 <= 5e+37) {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(y_m / Float64(z * z)) / z) * x_m)
	t_1 = Float64(Float64(z * z) * Float64(z - -1.0))
	tmp = 0.0
	if (t_1 <= -5e+36)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (t_1 <= 5e+37)
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z - -1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -5e+36], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+37], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.99999999999999977e36 or 4.99999999999999989e37 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.7

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

   7. Applied rewrites 87.7%

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

   if -4.99999999999999977e36 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 85.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 0.0 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.99999999999999989e37

   1. Initial program 83.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 83.9

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 93.6

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

   7. Applied rewrites 93.6%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z}}{z} \cdot x\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z}}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{y\_m}{\left(z \cdot z\right) \cdot z} \cdot x\_m\\ t_1 := \left(z \cdot z\right) \cdot \left(z - -1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ y_m (* (* z z) z)) x_m)) (t_1 (* (* z z) (- z -1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -5e+36)
       t_0
       (if (<= t_1 0.0)
         (* (/ y_m z) (/ x_m z))
         (if (<= t_1 5e+37) (* (/ x_m (* (fma z z z) z)) y_m) t_0)))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / ((z * z) * z)) * x_m;
	double t_1 = (z * z) * (z - -1.0);
	double tmp;
	if (t_1 <= -5e+36) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (y_m / z) * (x_m / z);
	} else if (t_1 <= 5e+37) {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(y_m / Float64(Float64(z * z) * z)) * x_m)
	t_1 = Float64(Float64(z * z) * Float64(z - -1.0))
	tmp = 0.0
	if (t_1 <= -5e+36)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (t_1 <= 5e+37)
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(y$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z - -1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -5e+36], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+37], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.99999999999999977e36 or 4.99999999999999989e37 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 88.0

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

   4. Applied rewrites 88.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.2

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

   6. Applied rewrites 85.2%

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

   7. Taylor expanded in z around inf

      \[\leadsto \frac{y}{\color{blue}{{z}^{2}} \cdot z} \cdot x \]
   8. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 84.9

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

   9. Applied rewrites 84.9%

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

   if -4.99999999999999977e36 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 85.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if 0.0 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.99999999999999989e37

   1. Initial program 83.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 83.9

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 93.6

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

   7. Applied rewrites 93.6%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot z} \cdot x\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z - -1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\frac{y\_m}{z} \cdot \frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(\frac{z - -1}{y\_m} \cdot z\right) \cdot z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (- z -1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 -5e+202)
       (/ (* (/ y_m z) (/ x_m z)) z)
       (if (<= t_0 2e-113)
         (/ (* y_m (/ x_m z)) (fma z z z))
         (/ x_m (* (* (/ (- z -1.0) y_m) z) z))))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z - -1.0);
	double tmp;
	if (t_0 <= -5e+202) {
		tmp = ((y_m / z) * (x_m / z)) / z;
	} else if (t_0 <= 2e-113) {
		tmp = (y_m * (x_m / z)) / fma(z, z, z);
	} else {
		tmp = x_m / ((((z - -1.0) / y_m) * z) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z - -1.0))
	tmp = 0.0
	if (t_0 <= -5e+202)
		tmp = Float64(Float64(Float64(y_m / z) * Float64(x_m / z)) / z);
	elseif (t_0 <= 2e-113)
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / fma(z, z, z));
	else
		tmp = Float64(x_m / Float64(Float64(Float64(Float64(z - -1.0) / y_m) * z) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z - -1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, -5e+202], N[(N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 2e-113], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(N[(z - -1.0), $MachinePrecision] / y$95$m), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.9999999999999999e202

   1. Initial program 74.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 76.3

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

   4. Applied rewrites 76.3%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 76.3

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{{z}^{2}}}}{z} \]
   9. Step-by-step derivation

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 98.0

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

   10. Applied rewrites 98.0%

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

   if -4.9999999999999999e202 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999996e-113

   1. Initial program 85.3%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 96.3

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

   4. Applied rewrites 96.3%

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

   if 1.99999999999999996e-113 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. clear-num N/A

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

      12. inv-pow N/A

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

      13. unpow-prod-down N/A

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

      14. associate-/l* N/A

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

      15. *-lft-identity N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 94.7

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 94.7

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

   4. Applied rewrites 94.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. times-frac N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l/ N/A

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

      15. associate-/r* N/A

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

      16. frac-2neg N/A

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

      17. distribute-frac-neg2 N/A

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

      18. lift-neg.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 95.8%

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

   8. Applied rewrites 94.0%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq -5 \cdot 10^{+202}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq 2 \cdot 10^{-113}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{z - -1}{y} \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z \cdot z\right) \cdot \left(z - -1\right)} \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)} \cdot y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (* z z) (- z -1.0))) 2e+62)
     (* (/ (/ y_m (fma z z z)) z) x_m)
     (/ (* (/ x_m (fma z z z)) y_m) z)))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z * z) * (z - -1.0))) <= 2e+62) {
		tmp = ((y_m / fma(z, z, z)) / z) * x_m;
	} else {
		tmp = ((x_m / fma(z, z, z)) * y_m) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(z * z) * Float64(z - -1.0))) <= 2e+62)
		tmp = Float64(Float64(Float64(y_m / fma(z, z, z)) / z) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) * y_m) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+62], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2.00000000000000007e62

   1. Initial program 88.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 92.5

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

   4. Applied rewrites 92.5%

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

   if 2.00000000000000007e62 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 68.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 70.8

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

   4. Applied rewrites 70.8%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 36.1

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 88.6

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

   10. Applied rewrites 88.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z \cdot z\right) \cdot \left(z - -1\right)} \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)} \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{y\_m}{\left(z \cdot z\right) \cdot z} \cdot x\_m\\ t_1 := \left(z \cdot z\right) \cdot \left(z - -1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.004:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ y_m (* (* z z) z)) x_m)) (t_1 (* (* z z) (- z -1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -5e+36)
       t_0
       (if (<= t_1 0.004) (* (/ y_m z) (/ x_m z)) t_0))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / ((z * z) * z)) * x_m;
	double t_1 = (z * z) * (z - -1.0);
	double tmp;
	if (t_1 <= -5e+36) {
		tmp = t_0;
	} else if (t_1 <= 0.004) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

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, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_m / ((z * z) * z)) * x_m
    t_1 = (z * z) * (z - (-1.0d0))
    if (t_1 <= (-5d+36)) then
        tmp = t_0
    else if (t_1 <= 0.004d0) then
        tmp = (y_m / z) * (x_m / z)
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

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;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / ((z * z) * z)) * x_m;
	double t_1 = (z * z) * (z - -1.0);
	double tmp;
	if (t_1 <= -5e+36) {
		tmp = t_0;
	} else if (t_1 <= 0.004) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = (y_m / ((z * z) * z)) * x_m
	t_1 = (z * z) * (z - -1.0)
	tmp = 0
	if t_1 <= -5e+36:
		tmp = t_0
	elif t_1 <= 0.004:
		tmp = (y_m / z) * (x_m / z)
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(y_m / Float64(Float64(z * z) * z)) * x_m)
	t_1 = Float64(Float64(z * z) * Float64(z - -1.0))
	tmp = 0.0
	if (t_1 <= -5e+36)
		tmp = t_0;
	elseif (t_1 <= 0.004)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = (y_m / ((z * z) * z)) * x_m;
	t_1 = (z * z) * (z - -1.0);
	tmp = 0.0;
	if (t_1 <= -5e+36)
		tmp = t_0;
	elseif (t_1 <= 0.004)
		tmp = (y_m / z) * (x_m / z);
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(y$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z - -1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -5e+36], t$95$0, If[LessEqual[t$95$1, 0.004], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.99999999999999977e36 or 0.0040000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.4

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

   6. Applied rewrites 85.4%

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

   7. Taylor expanded in z around inf

      \[\leadsto \frac{y}{\color{blue}{{z}^{2}} \cdot z} \cdot x \]
   8. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 85.0

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

   9. Applied rewrites 85.0%

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

   if -4.99999999999999977e36 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0040000000000000001

   1. Initial program 84.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 94.2

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

   5. Applied rewrites 94.2%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq -5 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot z} \cdot x\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z - -1\right) \leq 0.004:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{y\_m} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 5e-291)
     (* (/ y_m z) (/ x_m z))
     (if (<= (* y_m x_m) 5e+111)
       (/ (* y_m x_m) (* (fma z z z) z))
       (/ x_m (* (/ (fma z z z) y_m) z)))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 5e-291) {
		tmp = (y_m / z) * (x_m / z);
	} else if ((y_m * x_m) <= 5e+111) {
		tmp = (y_m * x_m) / (fma(z, z, z) * z);
	} else {
		tmp = x_m / ((fma(z, z, z) / y_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 5e-291)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (Float64(y_m * x_m) <= 5e+111)
		tmp = Float64(Float64(y_m * x_m) / Float64(fma(z, z, z) * z));
	else
		tmp = Float64(x_m / Float64(Float64(fma(z, z, z) / y_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e-291], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e+111], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 5.0000000000000003e-291

   1. Initial program 78.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if 5.0000000000000003e-291 < (*.f64 x y) < 4.9999999999999997e111

   1. Initial program 99.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-lft1-in N/A

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

      12. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 4.9999999999999997e111 < (*.f64 x y)

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 89.5

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

   4. Applied rewrites 89.5%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+111}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(z, z, z\right)}{y} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 10^{+143}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 5e-291)
     (* (/ y_m z) (/ x_m z))
     (if (<= (* y_m x_m) 1e+143)
       (/ (* y_m x_m) (* (fma z z z) z))
       (* (/ (/ y_m (fma z z z)) z) x_m))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 5e-291) {
		tmp = (y_m / z) * (x_m / z);
	} else if ((y_m * x_m) <= 1e+143) {
		tmp = (y_m * x_m) / (fma(z, z, z) * z);
	} else {
		tmp = ((y_m / fma(z, z, z)) / z) * x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 5e-291)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (Float64(y_m * x_m) <= 1e+143)
		tmp = Float64(Float64(y_m * x_m) / Float64(fma(z, z, z) * z));
	else
		tmp = Float64(Float64(Float64(y_m / fma(z, z, z)) / z) * x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e-291], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e+143], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 5.0000000000000003e-291

   1. Initial program 78.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if 5.0000000000000003e-291 < (*.f64 x y) < 1e143

   1. Initial program 99.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-lft1-in N/A

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

      12. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 1e143 < (*.f64 x y)

   1. Initial program 70.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 88.9

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

   4. Applied rewrites 88.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;y \cdot x \leq 10^{+143}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z - -1} \cdot \frac{\frac{y\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 2e+159)
     (/ (* y_m (/ x_m z)) (fma z z z))
     (* (/ x_m (- z -1.0)) (/ (/ y_m z) z))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 2e+159) {
		tmp = (y_m * (x_m / z)) / fma(z, z, z);
	} else {
		tmp = (x_m / (z - -1.0)) * ((y_m / z) / z);
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 2e+159)
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / fma(z, z, z));
	else
		tmp = Float64(Float64(x_m / Float64(z - -1.0)) * Float64(Float64(y_m / z) / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 2e+159], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.9999999999999999e159

   1. Initial program 85.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 95.1

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

   4. Applied rewrites 95.1%

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

   if 1.9999999999999999e159 < (*.f64 x y)

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{z} \cdot \frac{x}{1 + z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 2 \cdot 10^{+159}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - -1} \cdot \frac{\frac{y}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, z\right) \cdot z\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t\_0} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (fma z z z) z)))
   (*
    x_s
    (*
     y_s
     (if (<= (* y_m x_m) 5e-291)
       (* (/ y_m z) (/ x_m z))
       (if (<= (* y_m x_m) 5e+100)
         (/ (* y_m x_m) t_0)
         (* (/ x_m t_0) y_m)))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(z, z, z) * z;
	double tmp;
	if ((y_m * x_m) <= 5e-291) {
		tmp = (y_m / z) * (x_m / z);
	} else if ((y_m * x_m) <= 5e+100) {
		tmp = (y_m * x_m) / t_0;
	} else {
		tmp = (x_m / t_0) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(fma(z, z, z) * z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 5e-291)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (Float64(y_m * x_m) <= 5e+100)
		tmp = Float64(Float64(y_m * x_m) / t_0);
	else
		tmp = Float64(Float64(x_m / t_0) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e-291], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e+100], N[(N[(y$95$m * x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(x$95$m / t$95$0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 5.0000000000000003e-291

   1. Initial program 78.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if 5.0000000000000003e-291 < (*.f64 x y) < 4.9999999999999999e100

   1. Initial program 99.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-lft1-in N/A

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

      12. lower-fma.f64 99.8

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

   4. Applied rewrites 99.8%

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

   if 4.9999999999999999e100 < (*.f64 x y)

   1. Initial program 73.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 74.1

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 85.0

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

   7. Applied rewrites 85.0%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+100}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{-x\_m}{\left(\frac{-z}{y\_m} \cdot z\right) \cdot z}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (- x_m) (* (* (/ (- z) y_m) z) z))))
   (*
    x_s
    (*
     y_s
     (if (<= z -4e+64)
       t_0
       (if (<= z 6.5e+15) (/ (* y_m (/ x_m z)) (fma z z z)) t_0))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = -x_m / (((-z / y_m) * z) * z);
	double tmp;
	if (z <= -4e+64) {
		tmp = t_0;
	} else if (z <= 6.5e+15) {
		tmp = (y_m * (x_m / z)) / fma(z, z, z);
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(-x_m) / Float64(Float64(Float64(Float64(-z) / y_m) * z) * z))
	tmp = 0.0
	if (z <= -4e+64)
		tmp = t_0;
	elseif (z <= 6.5e+15)
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / fma(z, z, z));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[((-x$95$m) / N[(N[(N[((-z) / y$95$m), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -4e+64], t$95$0, If[LessEqual[z, 6.5e+15], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.00000000000000009e64 or 6.5e15 < z

   1. Initial program 78.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. clear-num N/A

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

      12. inv-pow N/A

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

      13. unpow-prod-down N/A

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

      14. associate-/l* N/A

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

      15. *-lft-identity N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 96.1

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. times-frac N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l/ N/A

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

      15. associate-/r* N/A

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

      16. frac-2neg N/A

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

      17. distribute-frac-neg2 N/A

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

      18. lift-neg.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 96.8%

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in z around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 92.4

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

   11. Applied rewrites 92.4%

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

   if -4.00000000000000009e64 < z < 6.5e15

   1. Initial program 86.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 97.2

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

   4. Applied rewrites 97.2%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+64}:\\ \;\;\;\;\frac{-x}{\left(\frac{-z}{y} \cdot z\right) \cdot z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\left(\frac{-z}{y} \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 10^{+114}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(\frac{z - -1}{y\_m} \cdot z\right) \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 1e+114)
     (/ (* y_m (/ x_m z)) (fma z z z))
     (/ x_m (* (* (/ (- z -1.0) y_m) z) z))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 1e+114) {
		tmp = (y_m * (x_m / z)) / fma(z, z, z);
	} else {
		tmp = x_m / ((((z - -1.0) / y_m) * z) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 1e+114)
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / fma(z, z, z));
	else
		tmp = Float64(x_m / Float64(Float64(Float64(Float64(z - -1.0) / y_m) * z) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e+114], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(N[(N[(z - -1.0), $MachinePrecision] / y$95$m), $MachinePrecision] * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1e114

   1. Initial program 85.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 95.0

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

   4. Applied rewrites 95.0%

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

   if 1e114 < (*.f64 x y)

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. frac-times N/A

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

      10. metadata-eval N/A

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

      11. clear-num N/A

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

      12. inv-pow N/A

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

      13. unpow-prod-down N/A

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

      14. associate-/l* N/A

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

      15. *-lft-identity N/A

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

      16. inv-pow N/A

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

      17. clear-num N/A

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

      18. lower-/.f64 N/A

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

      19. lower-/.f64 N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 99.9

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. times-frac N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. distribute-lft1-in N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l/ N/A

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

      15. associate-/r* N/A

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

      16. frac-2neg N/A

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

      17. distribute-frac-neg2 N/A

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

      18. lift-neg.f64 N/A

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

      19. lower-/.f64 N/A

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

   6. Applied rewrites 98.2%

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

   8. Applied rewrites 93.0%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{+114}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{z - -1}{y} \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 97.9% accurate, 0.7× speedup

Math

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

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ (/ x_m z) (* (/ (- z -1.0) y_m) z)))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / z) / (((z - -1.0) / y_m) * z)));
}

Fortran

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, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * ((x_m / z) / (((z - (-1.0d0)) / y_m) * z)))
end function

Java

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;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / z) / (((z - -1.0) / y_m) * z)));
}

Python

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] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * ((x_m / z) / (((z - -1.0) / y_m) * z)))

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(x_m / z) / Float64(Float64(Float64(z - -1.0) / y_m) * z))))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * ((x_m / z) / (((z - -1.0) / y_m) * z)));
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(x$95$m / z), $MachinePrecision] / N[(N[(N[(z - -1.0), $MachinePrecision] / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/r* N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. clear-num N/A

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

   12. inv-pow N/A

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

   13. unpow-prod-down N/A

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

   14. associate-/l* N/A

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

   15. *-lft-identity N/A

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

   16. inv-pow N/A

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

   17. clear-num N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lower-/.f64 96.1

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

   22. lift-+.f64 N/A

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

   23. +-commutative N/A

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

   24. lower-+.f64 96.1

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

4. Applied rewrites 96.1%

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

5. Final simplification 96.1%

   \[\leadsto \frac{\frac{x}{z}}{\frac{z - -1}{y} \cdot z} \]
6. Add Preprocessing

Alternative 14: 85.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{y\_m}{\left(z \cdot z\right) \cdot z} \cdot x\_m\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ y_m (* (* z z) z)) x_m)))
   (*
    x_s
    (*
     y_s
     (if (<= z -3000000.0) t_0 (if (<= z 1.0) (* (/ x_m (* z z)) y_m) t_0))))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / ((z * z) * z)) * x_m;
	double tmp;
	if (z <= -3000000.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

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, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_m / ((z * z) * z)) * x_m
    if (z <= (-3000000.0d0)) then
        tmp = t_0
    else if (z <= 1.0d0) then
        tmp = (x_m / (z * z)) * y_m
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

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;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / ((z * z) * z)) * x_m;
	double tmp;
	if (z <= -3000000.0) {
		tmp = t_0;
	} else if (z <= 1.0) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

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] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = (y_m / ((z * z) * z)) * x_m
	tmp = 0
	if z <= -3000000.0:
		tmp = t_0
	elif z <= 1.0:
		tmp = (x_m / (z * z)) * y_m
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(y_m / Float64(Float64(z * z) * z)) * x_m)
	tmp = 0.0
	if (z <= -3000000.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = (y_m / ((z * z) * z)) * x_m;
	tmp = 0.0;
	if (z <= -3000000.0)
		tmp = t_0;
	elseif (z <= 1.0)
		tmp = (x_m / (z * z)) * y_m;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(y$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -3000000.0], t$95$0, If[LessEqual[z, 1.0], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3e6 or 1 < z

   1. Initial program 80.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 88.2

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

   4. Applied rewrites 88.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.4

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

   6. Applied rewrites 85.4%

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

   7. Taylor expanded in z around inf

      \[\leadsto \frac{y}{\color{blue}{{z}^{2}} \cdot z} \cdot x \]
   8. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 85.0

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

   9. Applied rewrites 85.0%

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

   if -3e6 < z < 1

   1. Initial program 84.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 81.2

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

   5. Applied rewrites 81.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.2

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

   7. Applied rewrites 86.2%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3000000:\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(z \cdot z\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 5e-291)
     (* (/ y_m z) (/ x_m z))
     (* (/ y_m (* (fma z z z) z)) x_m)))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 5e-291) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 5e-291)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e-291], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 5.0000000000000003e-291

   1. Initial program 78.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if 5.0000000000000003e-291 < (*.f64 x y)

   1. Initial program 87.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 90.9

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

   4. Applied rewrites 90.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 89.2

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

   6. Applied rewrites 89.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 5 \cdot 10^{-291}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.1% accurate, 1.4× speedup

Math

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

FPCore

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, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (* (/ x_m (* z z)) y_m))))

C

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);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / (z * z)) * y_m));
}

Fortran

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, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * ((x_m / (z * z)) * y_m))
end function

Java

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;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / (z * z)) * y_m));
}

Python

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] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * ((x_m / (z * z)) * y_m))

Julia

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 = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(x_m / Float64(z * z)) * y_m)))
end

MATLAB

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 = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * ((x_m / (z * z)) * y_m));
end

Wolfram

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, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.5%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{x \cdot y}{\color{blue}{{z}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f64 70.0

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

5. Applied rewrites 70.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 76.0

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

7. Applied rewrites 76.0%

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

8. Final simplification 76.0%

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

Developer Target 1: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024268 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z 2496182814532307/10000000000000) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z)))

  (/ (* x y) (* (* z z) (+ z 1.0))))