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

Percentage Accurate: 83.6% → 99.5%

Time: 7.5s

Alternatives: 13

Speedup: 0.7×

• 25.7% 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: 83.6% 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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 10^{-317}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;x\_m \cdot y\_m \leq 10^{+220}:\\ \;\;\;\;\frac{\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{1 + z}}{z \cdot \frac{z}{y\_m}}\\ \end{array}\right) \end{array} \]

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 (<= (* x_m y_m) 1e-317)
     (* (/ x_m z) (/ y_m z))
     (if (<= (* x_m y_m) 1e+220)
       (/ (/ (* y_m x_m) (fma z z z)) z)
       (/ (/ x_m (+ 1.0 z)) (* 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) {
	double tmp;
	if ((x_m * y_m) <= 1e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else if ((x_m * y_m) <= 1e+220) {
		tmp = ((y_m * x_m) / fma(z, z, z)) / z;
	} else {
		tmp = (x_m / (1.0 + z)) / (z * (z / 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)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 1e-317)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (Float64(x_m * y_m) <= 1e+220)
		tmp = Float64(Float64(Float64(y_m * x_m) / fma(z, z, z)) / z);
	else
		tmp = Float64(Float64(x_m / Float64(1.0 + z)) / Float64(z * Float64(z / 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_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 1e-317], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 1e+220], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 1.00000023e-317

   1. Initial program 82.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   if 1.00000023e-317 < (*.f64 x y) < 1e220

   1. Initial program 93.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   if 1e220 < (*.f64 x y)

   1. Initial program 71.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-/r* N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{-65}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{x\_m} \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 1e-65)
     (/ x_m (* z (/ (fma z z z) y_m)))
     (/ y_m (* (/ (fma z z z) x_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 (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 1e-65) {
		tmp = x_m / (z * (fma(z, z, z) / y_m));
	} else {
		tmp = y_m / ((fma(z, z, z) / x_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(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e-65)
		tmp = Float64(x_m / Float64(z * Float64(fma(z, z, z) / y_m)));
	else
		tmp = Float64(y_m / Float64(Float64(fma(z, z, z) / x_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[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-65], N[(x$95$m / N[(z * N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $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)))) < 9.99999999999999923e-66

   1. Initial program 91.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 92.5

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

   4. Applied rewrites 92.5%

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

   if 9.99999999999999923e-66 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 71.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 89.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 89.6

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

   4. Applied rewrites 89.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 89.7

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 89.7

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

   6. Applied rewrites 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l/ N/A

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

      18. lift-fma.f64 N/A

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

      19. clear-num N/A

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

   8. Applied rewrites 83.6%

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

Alternative 3: 96.3% accurate, 0.4× speedup

Math

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

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 (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 1e-37)
     (/ x_m (* z (/ (fma z z z) y_m)))
     (* (/ (/ x_m (fma z z 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) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 1e-37) {
		tmp = x_m / (z * (fma(z, z, z) / y_m));
	} else {
		tmp = ((x_m / fma(z, z, z)) / z) * 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)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e-37)
		tmp = Float64(x_m / Float64(z * Float64(fma(z, z, z) / y_m)));
	else
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * 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_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-37], N[(x$95$m / N[(z * N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $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)))) < 1.00000000000000007e-37

   1. Initial program 91.3%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 92.6

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

   4. Applied rewrites 92.6%

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

   if 1.00000000000000007e-37 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 83.0

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

   4. Applied rewrites 83.0%

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

Alternative 4: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 2e-22)
     (* (/ y_m (* (fma z z z) z)) x_m)
     (* (/ (/ x_m (fma z z 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) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e-22) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = ((x_m / fma(z, z, z)) / z) * 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)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e-22)
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * 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_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-22], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $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)))) < 2.0000000000000001e-22

   1. Initial program 91.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 95.7

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.8

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

   6. Applied rewrites 96.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l/ N/A

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

      18. lift-fma.f64 N/A

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

      19. clear-num N/A

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

   8. Applied rewrites 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. remove-double-neg N/A

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

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

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

      8. lift-fma.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.1

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

   10. Applied rewrites 89.1%

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

   if 2.0000000000000001e-22 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 82.6

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

   4. Applied rewrites 82.6%

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

Alternative 5: 94.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 2e-22)
     (* (/ y_m (* (fma z z z) z)) x_m)
     (* (/ (/ x_m z) (fma z 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) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e-22) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = ((x_m / z) / fma(z, z, z)) * 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)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e-22)
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m / z) / fma(z, z, z)) * 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_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-22], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * y$95$m), $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)))) < 2.0000000000000001e-22

   1. Initial program 91.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 95.7

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 95.7

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

   4. Applied rewrites 95.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.8

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

   6. Applied rewrites 96.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l/ N/A

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

      18. lift-fma.f64 N/A

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

      19. clear-num N/A

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

   8. Applied rewrites 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. remove-double-neg N/A

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

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

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

      8. lift-fma.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 89.1

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

   10. Applied rewrites 89.1%

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

   if 2.0000000000000001e-22 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 82.6

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

   4. Applied rewrites 82.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 82.5

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

   6. Applied rewrites 82.5%

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

Alternative 6: 90.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+18} \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \end{array}\right) \end{array} \end{array} \]

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 (or (<= t_0 -2e+18) (not (<= t_0 0.0)))
       (* (/ x_m (* (fma z z z) z)) y_m)
       (* (/ x_m 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 t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -2e+18) || !(t_0 <= 0.0)) {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	} else {
		tmp = (x_m / 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)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -2e+18) || !(t_0 <= 0.0))
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	else
		tmp = Float64(Float64(x_m / z) * Float64(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_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[Or[LessEqual[t$95$0, -2e+18], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e18 or 0.0 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 89.1

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

   6. Applied rewrites 89.1%

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

   if -2e18 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 76.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -2 \cdot 10^{+18} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 0\right):\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, z\right) \cdot z\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{y\_m}{t\_0} \cdot x\_m\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t\_0} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

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)) (t_1 (* (* z z) (+ z 1.0))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -2e+18)
       (* (/ y_m t_0) x_m)
       (if (<= t_1 0.0) (* (/ x_m z) (/ y_m z)) (* (/ 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 t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -2e+18) {
		tmp = (y_m / t_0) * x_m;
	} else if (t_1 <= 0.0) {
		tmp = (x_m / z) * (y_m / z);
	} 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)
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -2e+18)
		tmp = Float64(Float64(y_m / t_0) * x_m);
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	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]}, 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, -2e+18], N[(N[(y$95$m / t$95$0), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $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 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e18

   1. Initial program 88.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 96.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l/ N/A

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

      18. lift-fma.f64 N/A

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

      19. clear-num N/A

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

   8. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. remove-double-neg N/A

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

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

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

      8. lift-fma.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 85.1

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

   10. Applied rewrites 85.1%

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

   if -2e18 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 76.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 96.9

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

   5. Applied rewrites 96.9%

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

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

   1. Initial program 87.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 88.6

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

   6. Applied rewrites 88.6%

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

Alternative 8: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 10^{-317}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;x\_m \cdot y\_m \leq 10^{+220}:\\ \;\;\;\;\frac{\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(\frac{z + 1}{y\_m} \cdot z\right) \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= (* x_m y_m) 1e-317)
     (* (/ x_m z) (/ y_m z))
     (if (<= (* x_m y_m) 1e+220)
       (/ (/ (* y_m x_m) (fma z 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 ((x_m * y_m) <= 1e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else if ((x_m * y_m) <= 1e+220) {
		tmp = ((y_m * x_m) / fma(z, 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(x_m * y_m) <= 1e-317)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (Float64(x_m * y_m) <= 1e+220)
		tmp = Float64(Float64(Float64(y_m * x_m) / fma(z, 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[(x$95$m * y$95$m), $MachinePrecision], 1e-317], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 1e+220], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $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 (*.f64 x y) < 1.00000023e-317

   1. Initial program 82.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   if 1.00000023e-317 < (*.f64 x y) < 1e220

   1. Initial program 93.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   if 1e220 < (*.f64 x y)

   1. Initial program 71.5%

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

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 91.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 91.4

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

   4. Applied rewrites 91.4%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. frac-times N/A

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

      9. frac-times N/A

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

      10. *-lft-identity N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 96.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 96.9

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

   6. Applied rewrites 96.9%

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

Alternative 9: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 10^{-317}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y\_m}{1 + z} \cdot x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

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 (<= (* x_m y_m) 1e-317)
     (* (/ x_m z) (/ y_m z))
     (/ (/ (* (/ y_m (+ 1.0 z)) x_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 ((x_m * y_m) <= 1e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (((y_m / (1.0 + z)) * x_m) / z) / z;
	}
	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) :: tmp
    if ((x_m * y_m) <= 1d-317) then
        tmp = (x_m / z) * (y_m / z)
    else
        tmp = (((y_m / (1.0d0 + z)) * x_m) / z) / z
    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 tmp;
	if ((x_m * y_m) <= 1e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (((y_m / (1.0 + z)) * x_m) / z) / z;
	}
	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):
	tmp = 0
	if (x_m * y_m) <= 1e-317:
		tmp = (x_m / z) * (y_m / z)
	else:
		tmp = (((y_m / (1.0 + z)) * x_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(x_m * y_m) <= 1e-317)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(Float64(Float64(y_m / Float64(1.0 + z)) * x_m) / z) / z);
	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)
	tmp = 0.0;
	if ((x_m * y_m) <= 1e-317)
		tmp = (x_m / z) * (y_m / z);
	else
		tmp = (((y_m / (1.0 + z)) * x_m) / z) / z;
	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_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 1e-317], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.00000023e-317

   1. Initial program 82.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   if 1.00000023e-317 < (*.f64 x y)

   1. Initial program 87.3%

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

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 96.7

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 96.7

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

   4. Applied rewrites 96.7%

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

Alternative 10: 93.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-10} \lor \neg \left(z \leq 3.7 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m} \cdot z}\\ \end{array}\right) \end{array} \]

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 (or (<= z -1.04e-10) (not (<= z 3.7e-34)))
     (* (/ y_m (* (fma z z z) z)) x_m)
     (/ y_m (* (/ z x_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 ((z <= -1.04e-10) || !(z <= 3.7e-34)) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = y_m / ((z / x_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 ((z <= -1.04e-10) || !(z <= 3.7e-34))
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(y_m / Float64(Float64(z / x_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[Or[LessEqual[z, -1.04e-10], N[Not[LessEqual[z, 3.7e-34]], $MachinePrecision]], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m / N[(N[(z / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.04e-10 or 3.69999999999999988e-34 < z

   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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 94.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.1

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

   6. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l/ N/A

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

      18. lift-fma.f64 N/A

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

      19. clear-num N/A

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

   8. Applied rewrites 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. remove-double-neg N/A

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

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

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

      8. lift-fma.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if -1.04e-10 < z < 3.69999999999999988e-34

   1. Initial program 84.6%

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

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 92.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 92.6

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 92.6

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

   6. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l/ N/A

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

      18. lift-fma.f64 N/A

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

      19. clear-num N/A

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

   8. Applied rewrites 89.6%

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

   9. Taylor expanded in z around 0

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

       1. lower-/.f64 89.6

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

   11. Applied rewrites 89.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-10} \lor \neg \left(z \leq 3.7 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-10} \lor \neg \left(z \leq 3.75 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (or (<= z -1.04e-10) (not (<= z 3.75e-34)))
     (* (/ y_m (* (fma z z z) z)) x_m)
     (* (/ (/ 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) {
	double tmp;
	if ((z <= -1.04e-10) || !(z <= 3.75e-34)) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = ((x_m / z) / z) * 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)
	tmp = 0.0
	if ((z <= -1.04e-10) || !(z <= 3.75e-34))
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m / z) / z) * 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_] := N[(x$95$s * N[(y$95$s * If[Or[LessEqual[z, -1.04e-10], N[Not[LessEqual[z, 3.75e-34]], $MachinePrecision]], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.04e-10 or 3.7500000000000002e-34 < z

   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. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 94.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 94.6

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

   4. Applied rewrites 94.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 96.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.1

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

   6. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. associate-*r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. lift-+.f64 N/A

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

      10. distribute-lft1-in N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. clear-num N/A

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

      14. un-div-inv N/A

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

      15. lower-/.f64 N/A

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

      16. *-lft-identity N/A

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

      17. associate-*l/ N/A

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

      18. lift-fma.f64 N/A

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

      19. clear-num N/A

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

   8. Applied rewrites 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. remove-double-neg N/A

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

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

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

      8. lift-fma.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. distribute-lft-neg-out N/A

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

      14. remove-double-neg N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 87.4

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

   10. Applied rewrites 87.4%

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

   if -1.04e-10 < z < 3.7500000000000002e-34

   1. Initial program 84.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 89.4

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

   4. Applied rewrites 89.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 89.4

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

   7. Applied rewrites 89.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.04 \cdot 10^{-10} \lor \neg \left(z \leq 3.75 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= (* x_m y_m) 2e-77)
     (* (/ x_m z) (/ y_m z))
     (* (/ 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) {
	double tmp;
	if ((x_m * y_m) <= 2e-77) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / (z * z)) * y_m;
	}
	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) :: tmp
    if ((x_m * y_m) <= 2d-77) then
        tmp = (x_m / z) * (y_m / z)
    else
        tmp = (x_m / (z * z)) * y_m
    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 tmp;
	if ((x_m * y_m) <= 2e-77) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / (z * z)) * y_m;
	}
	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):
	tmp = 0
	if (x_m * y_m) <= 2e-77:
		tmp = (x_m / z) * (y_m / z)
	else:
		tmp = (x_m / (z * z)) * 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)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 2e-77)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	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)
	tmp = 0.0;
	if ((x_m * y_m) <= 2e-77)
		tmp = (x_m / z) * (y_m / z);
	else
		tmp = (x_m / (z * z)) * y_m;
	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_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e-77], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.9999999999999999e-77

   1. Initial program 83.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 81.5

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

   5. Applied rewrites 81.5%

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

   if 1.9999999999999999e-77 < (*.f64 x y)

   1. Initial program 88.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 90.1

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 68.8

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

   7. Applied rewrites 68.8%

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

Alternative 13: 75.1% 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 84.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. distribute-lft1-in N/A

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

   17. lower-fma.f64 90.1

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

4. Applied rewrites 90.1%

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

5. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 74.3

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

7. Applied rewrites 74.3%

   \[\leadsto \color{blue}{\frac{x}{z \cdot z}} \cdot y \]
8. 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 2024313 
(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))))