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

Percentage Accurate: 83.1% → 95.6%

Time: 8.7s

Alternatives: 15

Speedup: 0.9×

• 26.1% 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: 83.1% 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: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y\_m}{z}}{z} \cdot x\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (* z z) (+ z 1.0)) 2e+72)
     (* (/ (/ x_m (fma z z z)) z) y_m)
     (/ (* (/ (/ y_m 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 (((z * z) * (z + 1.0)) <= 2e+72) {
		tmp = ((x_m / fma(z, z, z)) / z) * y_m;
	} else {
		tmp = (((y_m / 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(z * z) * Float64(z + 1.0)) <= 2e+72)
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * y_m);
	else
		tmp = Float64(Float64(Float64(Float64(y_m / 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[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], 2e+72], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999989e72

   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 95.5

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

   4. Applied rewrites 95.5%

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

   if 1.99999999999999989e72 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 94.6

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

   6. Applied rewrites 94.6%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.5

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

   9. Applied rewrites 99.5%

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

Alternative 2: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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))) 2e+25)
     (/ 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))) <= 2e+25) {
		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))) <= 2e+25)
		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], 2e+25], 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)))) < 2.00000000000000018e25

   1. Initial program 92.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 94.0

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

   4. Applied rewrites 94.0%

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

   if 2.00000000000000018e25 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 66.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 89.9

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

   4. Applied rewrites 89.9%

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

Alternative 3: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{-128}:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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-128)
     (* (/ (/ 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))) <= 1e-128) {
		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))) <= 1e-128)
		tmp = Float64(Float64(Float64(y_m / 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], 1e-128], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / 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)))) < 1.00000000000000005e-128

   1. Initial program 92.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 94.1

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

   4. Applied rewrites 94.1%

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

   if 1.00000000000000005e-128 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 68.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 90.5

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

   4. Applied rewrites 90.5%

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

Alternative 4: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -500000 \lor \neg \left(t\_0 \leq 0.1\right):\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\left(z \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

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 (* (* z z) (+ z 1.0))))
   (*
    x_s
    (*
     y_s
     (if (or (<= t_0 -500000.0) (not (<= t_0 0.1)))
       (* y_m (/ x_m (* (* z z) 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 t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -500000.0) || !(t_0 <= 0.1)) {
		tmp = y_m * (x_m / ((z * z) * 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) :: t_0
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    if ((t_0 <= (-500000.0d0)) .or. (.not. (t_0 <= 0.1d0))) then
        tmp = y_m * (x_m / ((z * z) * 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 t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -500000.0) || !(t_0 <= 0.1)) {
		tmp = y_m * (x_m / ((z * z) * 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):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if (t_0 <= -500000.0) or not (t_0 <= 0.1):
		tmp = y_m * (x_m / ((z * z) * 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)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -500000.0) || !(t_0 <= 0.1))
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z * z) * z)));
	else
		tmp = Float64(Float64(Float64(x_m / 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)
	t_0 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if ((t_0 <= -500000.0) || ~((t_0 <= 0.1)))
		tmp = y_m * (x_m / ((z * z) * 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_] := 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, -500000.0], N[Not[LessEqual[t$95$0, 0.1]], $MachinePrecision]], N[(y$95$m * N[(x$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $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 (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -5e5 or 0.10000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. neg-mul-1 N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

   9. Applied rewrites 86.3%

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

   if -5e5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 95.9

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

   4. Applied rewrites 95.9%

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

   5. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 92.7

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

   7. Applied rewrites 92.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -500000 \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 0.1\right):\\ \;\;\;\;y \cdot \frac{x}{\left(z \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -500000 \lor \neg \left(t\_0 \leq 0.1\right):\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\left(z \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \end{array}\right) \end{array} \end{array} \]

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 -500000.0) (not (<= t_0 0.1)))
       (* y_m (/ x_m (* (* z z) z)))
       (* (/ 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 <= -500000.0) || !(t_0 <= 0.1)) {
		tmp = y_m * (x_m / ((z * z) * z));
	} else {
		tmp = (x_m / z) * (y_m / 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) :: t_0
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    if ((t_0 <= (-500000.0d0)) .or. (.not. (t_0 <= 0.1d0))) then
        tmp = y_m * (x_m / ((z * z) * z))
    else
        tmp = (x_m / z) * (y_m / 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 t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -500000.0) || !(t_0 <= 0.1)) {
		tmp = y_m * (x_m / ((z * z) * z));
	} else {
		tmp = (x_m / z) * (y_m / 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):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if (t_0 <= -500000.0) or not (t_0 <= 0.1):
		tmp = y_m * (x_m / ((z * z) * z))
	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 <= -500000.0) || !(t_0 <= 0.1))
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z * z) * z)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(y_m / 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)
	t_0 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if ((t_0 <= -500000.0) || ~((t_0 <= 0.1)))
		tmp = y_m * (x_m / ((z * z) * z));
	else
		tmp = (x_m / z) * (y_m / 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_] := 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, -500000.0], N[Not[LessEqual[t$95$0, 0.1]], $MachinePrecision]], N[(y$95$m * N[(x$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $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))) < -5e5 or 0.10000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. neg-mul-1 N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

   9. Applied rewrites 86.3%

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

   if -5e5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 86.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -500000 \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 0.1\right):\\ \;\;\;\;y \cdot \frac{x}{\left(z \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -500000 \lor \neg \left(t\_0 \leq 0.1\right):\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\left(z \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    x_s
    (*
     y_s
     (if (or (<= t_0 -500000.0) (not (<= t_0 0.1)))
       (* y_m (/ x_m (* (* z z) z)))
       (* y_m (/ 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 t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -500000.0) || !(t_0 <= 0.1)) {
		tmp = y_m * (x_m / ((z * z) * z));
	} else {
		tmp = y_m * (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) :: t_0
    real(8) :: tmp
    t_0 = (z * z) * (z + 1.0d0)
    if ((t_0 <= (-500000.0d0)) .or. (.not. (t_0 <= 0.1d0))) then
        tmp = y_m * (x_m / ((z * z) * z))
    else
        tmp = y_m * (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 t_0 = (z * z) * (z + 1.0);
	double tmp;
	if ((t_0 <= -500000.0) || !(t_0 <= 0.1)) {
		tmp = y_m * (x_m / ((z * z) * z));
	} else {
		tmp = y_m * (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):
	t_0 = (z * z) * (z + 1.0)
	tmp = 0
	if (t_0 <= -500000.0) or not (t_0 <= 0.1):
		tmp = y_m * (x_m / ((z * z) * z))
	else:
		tmp = y_m * (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)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if ((t_0 <= -500000.0) || !(t_0 <= 0.1))
		tmp = Float64(y_m * Float64(x_m / Float64(Float64(z * z) * z)));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(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)
	t_0 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if ((t_0 <= -500000.0) || ~((t_0 <= 0.1)))
		tmp = y_m * (x_m / ((z * z) * z));
	else
		tmp = y_m * (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_] := 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, -500000.0], N[Not[LessEqual[t$95$0, 0.1]], $MachinePrecision]], N[(y$95$m * N[(x$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -5e5 or 0.10000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.3%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.2

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

   7. Applied rewrites 85.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. associate-*l/ N/A

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

      10. neg-mul-1 N/A

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

      11. lift-neg.f64 N/A

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

      12. remove-double-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lift-*.f64 N/A

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

   9. Applied rewrites 86.3%

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

   if -5e5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.10000000000000001

   1. Initial program 86.1%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 83.4

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

   5. Applied rewrites 83.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.8

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

   7. Applied rewrites 86.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -500000 \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 0.1\right):\\ \;\;\;\;y \cdot \frac{x}{\left(z \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 5e+305)
     (* y_m (/ x_m (* (fma z z z) 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) / ((z * z) * (z + 1.0))) <= 5e+305) {
		tmp = y_m * (x_m / (fma(z, z, z) * 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(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 5e+305)
		tmp = Float64(y_m * Float64(x_m / Float64(fma(z, z, z) * z)));
	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[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+305], N[(y$95$m * N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $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 (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 5.00000000000000009e305

   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

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 96.9

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

   6. Applied rewrites 96.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. distribute-lft-in N/A

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

      15. *-rgt-identity N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-fma.f64 N/A

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

      18. *-commutative N/A

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

      19. lower-*.f64 95.9

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

   8. Applied rewrites 95.9%

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

   if 5.00000000000000009e305 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 60.0%

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

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 77.9

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

   7. Applied rewrites 77.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;y \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{-x\_m}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-x\_m\right) \cdot \frac{y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 2e+247)
     (* (/ (- x_m) z) y_m)
     (* (- x_m) (/ y_m z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 2e+247) {
		tmp = (-x_m / z) * y_m;
	} else {
		tmp = -x_m * (y_m / 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) / ((z * z) * (z + 1.0d0))) <= 2d+247) then
        tmp = (-x_m / z) * y_m
    else
        tmp = -x_m * (y_m / 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) / ((z * z) * (z + 1.0))) <= 2e+247) {
		tmp = (-x_m / z) * y_m;
	} else {
		tmp = -x_m * (y_m / 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) / ((z * z) * (z + 1.0))) <= 2e+247:
		tmp = (-x_m / z) * y_m
	else:
		tmp = -x_m * (y_m / z)
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 2e+247)
		tmp = Float64(Float64(Float64(-x_m) / z) * y_m);
	else
		tmp = Float64(Float64(-x_m) * Float64(y_m / 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) / ((z * z) * (z + 1.0))) <= 2e+247)
		tmp = (-x_m / z) * y_m;
	else
		tmp = -x_m * (y_m / 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[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+247], N[(N[((-x$95$m) / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[((-x$95$m) * N[(y$95$m / 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)))) < 1.9999999999999999e247

   1. Initial program 92.9%

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

   3. Taylor expanded in z around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 63.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.1%

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

   if 1.9999999999999999e247 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0

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

      1. remove-double-neg N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 21.8%

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

   10. Applied rewrites 20.1%

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

Alternative 9: 94.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{z \cdot z} \cdot x\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z -1.5e-16)
     (* (/ y_m (* (fma z z z) z)) x_m)
     (if (<= z 1.0) (* (/ (/ x_m z) z) y_m) (/ (* (/ y_m (* 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 (z <= -1.5e-16) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else if (z <= 1.0) {
		tmp = ((x_m / z) / z) * y_m;
	} else {
		tmp = ((y_m / (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 (z <= -1.5e-16)
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(x_m / z) / z) * y_m);
	else
		tmp = Float64(Float64(Float64(y_m / Float64(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[z, -1.5e-16], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.49999999999999997e-16

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-fma.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 91.6%

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

   if -1.49999999999999997e-16 < z < 1

   1. Initial program 85.7%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.3

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

   7. Applied rewrites 94.3%

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

   if 1 < z

   1. Initial program 82.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 96.1

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

   4. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r/ N/A

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

      12. associate-*l/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 92.1

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

   9. Applied rewrites 92.1%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(z \cdot z\right) \cdot \frac{z}{y\_m}}\\ \end{array}\right) \end{array} \]

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 (<= z -1.5e-16)
     (* (/ y_m (* (fma z z z) z)) x_m)
     (if (<= z 1.0) (* (/ (/ x_m z) z) y_m) (/ x_m (* (* 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 (z <= -1.5e-16) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else if (z <= 1.0) {
		tmp = ((x_m / z) / z) * y_m;
	} else {
		tmp = x_m / ((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 (z <= -1.5e-16)
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(x_m / z) / z) * y_m);
	else
		tmp = Float64(x_m / Float64(Float64(z * 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[z, -1.5e-16], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[(x$95$m / N[(N[(z * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.49999999999999997e-16

   1. Initial program 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. frac-times N/A

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

      8. lift-fma.f64 N/A

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

      9. *-rgt-identity N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. associate-*l* N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lower-*.f64 N/A

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

   6. Applied rewrites 91.6%

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

   if -1.49999999999999997e-16 < z < 1

   1. Initial program 85.7%

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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 95.8

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

   4. Applied rewrites 95.8%

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

   5. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 94.3

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

   7. Applied rewrites 94.3%

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

   if 1 < z

   1. Initial program 82.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. frac-2neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

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

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 84.4

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

   4. Applied rewrites 84.4%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.5

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

   7. Applied rewrites 81.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lift-neg.f64 N/A

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

      6. distribute-frac-neg N/A

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

      7. distribute-neg-frac2 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lift-neg.f64 N/A

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

      13. distribute-frac-neg N/A

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

      14. lift-/.f64 N/A

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

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

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

   9. Applied rewrites 89.3%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.1% accurate, 0.7× speedup

Math

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

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 (/ (/ y_m (+ 1.0 z)) (* z (/ z x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((y_m / (1.0 + z)) / (z * (z / x_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 * ((y_m / (1.0d0 + z)) / (z * (z / x_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 * ((y_m / (1.0 + z)) / (z * (z / x_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 * ((y_m / (1.0 + z)) / (z * (z / x_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(y_m / Float64(1.0 + z)) / Float64(z * Float64(z / x_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 * ((y_m / (1.0 + z)) / (z * (z / x_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[(y$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.7%

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

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot \frac{z}{x}}} \]
5. Add Preprocessing

Alternative 12: 94.7% accurate, 0.9× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ (* (/ 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) {
	return x_s * (y_s * (((y_m / fma(z, z, z)) * x_m) / z));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/r/ N/A

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

   12. associate-*l/ N/A

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

   13. lower-/.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 96.0

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

6. Applied rewrites 96.0%

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

Alternative 13: 92.4% accurate, 0.9× speedup

Math

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

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 (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) {
	return x_s * (y_s * (((x_m / fma(z, z, 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(Float64(x_m / fma(z, z, 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[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.7%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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 94.5

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

4. Applied rewrites 94.5%

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

Alternative 14: 75.3% accurate, 1.4× speedup

Math

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

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 (* y_m (/ 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) {
	return x_s * (y_s * (y_m * (x_m / (z * z))));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in z around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 73.9

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

5. Applied rewrites 73.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 77.5

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

7. Applied rewrites 77.5%

   \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot z}} \]
8. Add Preprocessing

Alternative 15: 28.3% accurate, 1.6× speedup

Math

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

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) (/ y_m z)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

3. Taylor expanded in z around 0

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

   1. remove-double-neg N/A

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

   2. mul-1-neg N/A

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

   3. sub-neg N/A

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

   4. lower-/.f64 N/A

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

5. Applied rewrites 63.0%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 34.2%

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

10. Applied rewrites 34.1%

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

Developer Target 1: 97.2% 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 2024296 
(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))))