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

Percentage Accurate: 82.5% → 99.3%

Time: 10.9s

Alternatives: 20

Speedup: 0.7×

• 25.4% 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: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 2 \cdot 10^{-245}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 1.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{1 + z}}{\frac{z}{y\_m} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 2e-245)
     (* (/ x_m z) (/ y_m z))
     (if (<= (* y_m x_m) 1.2e+148)
       (/ (/ (* y_m x_m) (fma z z z)) z)
       (/ (/ x_m (+ 1.0 z)) (* (/ z y_m) z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 2e-245) {
		tmp = (x_m / z) * (y_m / z);
	} else if ((y_m * x_m) <= 1.2e+148) {
		tmp = ((y_m * x_m) / fma(z, z, z)) / z;
	} else {
		tmp = (x_m / (1.0 + z)) / ((z / y_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 2e-245)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (Float64(y_m * x_m) <= 1.2e+148)
		tmp = Float64(Float64(Float64(y_m * x_m) / fma(z, z, z)) / z);
	else
		tmp = Float64(Float64(x_m / Float64(1.0 + z)) / Float64(Float64(z / y_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 2e-245], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1.2e+148], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] / N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 1.9999999999999999e-245

   1. Initial program 78.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   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 77.8

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

   5. Applied rewrites 77.8%

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

   if 1.9999999999999999e-245 < (*.f64 x y) < 1.19999999999999997e148

   1. Initial program 96.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 1.19999999999999997e148 < (*.f64 x y)

   1. Initial program 76.6%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-/r* N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 97.5

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

   4. Applied rewrites 97.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 2 \cdot 10^{-245}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;y \cdot x \leq 1.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + z}}{\frac{z}{y} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y\_m}{z \cdot z} \cdot x\_m}{z}\\ t_1 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (* (/ y_m (* z z)) x_m) z)) (t_1 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -2e+20)
       t_0
       (if (<= t_1 1.35e-317)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_1 5e+118) (* (/ x_m (* (fma z z z) z)) y_m) t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = ((y_m / (z * z)) * x_m) / z;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -2e+20) {
		tmp = t_0;
	} else if (t_1 <= 1.35e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 5e+118) {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(y_m / Float64(z * z)) * x_m) / z)
	t_1 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2e+20)
		tmp = t_0;
	elseif (t_1 <= 1.35e-317)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_1 <= 5e+118)
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -2e+20], t$95$0, If[LessEqual[t$95$1, 1.35e-317], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+118], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e20 or 4.99999999999999972e118 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l* N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 92.7%

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

   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 91.8

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

   9. Applied rewrites 91.8%

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

   if -2e20 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.34999979e-317

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.34999979e-317 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.99999999999999972e118

   1. Initial program 95.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 93.1

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

   4. Applied rewrites 93.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. associate-*l* N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 93.1

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

   6. Applied rewrites 93.1%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z} \cdot x}{z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y\_m}{z}}{z \cdot z} \cdot x\_m\\ t_1 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y_m z) (* z z)) x_m)) (t_1 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -2e+20)
       t_0
       (if (<= t_1 1.35e-317)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_1 5e+38) (* (/ x_m (* (fma z z z) z)) y_m) t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = ((y_m / z) / (z * z)) * x_m;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -2e+20) {
		tmp = t_0;
	} else if (t_1 <= 1.35e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 5e+38) {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(y_m / z) / Float64(z * z)) * x_m)
	t_1 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2e+20)
		tmp = t_0;
	elseif (t_1 <= 1.35e-317)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_1 <= 5e+38)
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(y$95$m / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -2e+20], t$95$0, If[LessEqual[t$95$1, 1.35e-317], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+38], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e20 or 4.9999999999999997e38 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-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 94.3

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

   4. Applied rewrites 94.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot \frac{z}{x}}} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 89.3%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.5

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

   9. Applied rewrites 88.5%

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

   if -2e20 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.34999979e-317

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.34999979e-317 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.9999999999999997e38

   1. Initial program 95.8%

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

      1. lift-/.f64 N/A

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

      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 92.7

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

   4. Applied rewrites 92.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. associate-*l* N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 92.7

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

   6. Applied rewrites 92.7%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot z} \cdot x\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{+38}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot z\\ t_1 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t\_0} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) z)) (t_1 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -2e+20)
       (/ (* y_m x_m) t_0)
       (if (<= t_1 1.35e-317)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_1 2e-14) (* (/ x_m (* z z)) y_m) (* (/ x_m t_0) y_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * z;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -2e+20) {
		tmp = (y_m * x_m) / t_0;
	} else if (t_1 <= 1.35e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 2e-14) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = (x_m / t_0) * 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) :: t_1
    real(8) :: tmp
    t_0 = (z * z) * z
    t_1 = (1.0d0 + z) * (z * z)
    if (t_1 <= (-2d+20)) then
        tmp = (y_m * x_m) / t_0
    else if (t_1 <= 1.35d-317) then
        tmp = (x_m / z) * (y_m / z)
    else if (t_1 <= 2d-14) then
        tmp = (x_m / (z * z)) * y_m
    else
        tmp = (x_m / t_0) * 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;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -2e+20) {
		tmp = (y_m * x_m) / t_0;
	} else if (t_1 <= 1.35e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 2e-14) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = (x_m / t_0) * 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
	t_1 = (1.0 + z) * (z * z)
	tmp = 0
	if t_1 <= -2e+20:
		tmp = (y_m * x_m) / t_0
	elif t_1 <= 1.35e-317:
		tmp = (x_m / z) * (y_m / z)
	elif t_1 <= 2e-14:
		tmp = (x_m / (z * z)) * y_m
	else:
		tmp = (x_m / t_0) * y_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(z * z) * z)
	t_1 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2e+20)
		tmp = Float64(Float64(y_m * x_m) / t_0);
	elseif (t_1 <= 1.35e-317)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	else
		tmp = Float64(Float64(x_m / t_0) * 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;
	t_1 = (1.0 + z) * (z * z);
	tmp = 0.0;
	if (t_1 <= -2e+20)
		tmp = (y_m * x_m) / t_0;
	elseif (t_1 <= 1.35e-317)
		tmp = (x_m / z) * (y_m / z);
	elseif (t_1 <= 2e-14)
		tmp = (x_m / (z * z)) * y_m;
	else
		tmp = (x_m / t_0) * 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] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -2e+20], N[(N[(y$95$m * x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 1.35e-317], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-14], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(x$95$m / t$95$0), $MachinePrecision] * y$95$m), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e20

   1. Initial program 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 90.7

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

   4. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l* N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 94.4%

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

   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 92.9

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

   9. Applied rewrites 92.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lift-/.f64 N/A

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

       5. frac-times N/A

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

       6. *-commutative N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if -2e20 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.34999979e-317

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.34999979e-317 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2e-14

   1. Initial program 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 92.1

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

   4. Applied rewrites 92.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 90.5

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

   7. Applied rewrites 90.5%

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

   if 2e-14 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 88.5

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

   4. Applied rewrites 88.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. associate-*l* N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 84.5

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

   6. Applied rewrites 84.5%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 81.6

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

   9. Applied rewrites 81.6%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{y \cdot x}{\left(z \cdot z\right) \cdot z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{\frac{y\_m}{z}}{z} \cdot x\_m}{z}\\ t_1 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{y\_m}{\left(\frac{z}{x\_m} \cdot z\right) \cdot \left(1 + z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (* (/ (/ y_m z) z) x_m) z)) (t_1 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -4e+50)
       t_0
       (if (<= t_1 5e+132) (/ y_m (* (* (/ z x_m) z) (+ 1.0 z))) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (((y_m / z) / z) * x_m) / z;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -4e+50) {
		tmp = t_0;
	} else if (t_1 <= 5e+132) {
		tmp = y_m / (((z / x_m) * z) * (1.0 + z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((y_m / z) / z) * x_m) / z
    t_1 = (1.0d0 + z) * (z * z)
    if (t_1 <= (-4d+50)) then
        tmp = t_0
    else if (t_1 <= 5d+132) then
        tmp = y_m / (((z / x_m) * z) * (1.0d0 + z))
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (((y_m / z) / z) * x_m) / z;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -4e+50) {
		tmp = t_0;
	} else if (t_1 <= 5e+132) {
		tmp = y_m / (((z / x_m) * z) * (1.0 + z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = (((y_m / z) / z) * x_m) / z
	t_1 = (1.0 + z) * (z * z)
	tmp = 0
	if t_1 <= -4e+50:
		tmp = t_0
	elif t_1 <= 5e+132:
		tmp = y_m / (((z / x_m) * z) * (1.0 + z))
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(y_m / z) / z) * x_m) / z)
	t_1 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -4e+50)
		tmp = t_0;
	elseif (t_1 <= 5e+132)
		tmp = Float64(y_m / Float64(Float64(Float64(z / x_m) * z) * Float64(1.0 + z)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = (((y_m / z) / z) * x_m) / z;
	t_1 = (1.0 + z) * (z * z);
	tmp = 0.0;
	if (t_1 <= -4e+50)
		tmp = t_0;
	elseif (t_1 <= 5e+132)
		tmp = y_m / (((z / x_m) * z) * (1.0 + z));
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -4e+50], t$95$0, If[LessEqual[t$95$1, 5e+132], N[(y$95$m / N[(N[(N[(z / x$95$m), $MachinePrecision] * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -4.0000000000000003e50 or 5.0000000000000001e132 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 81.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 84.6

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

   if -4.0000000000000003e50 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000001e132

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 92.1

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 92.1

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

   4. Applied rewrites 92.1%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq -4 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{z} \cdot x}{z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{+132}:\\ \;\;\;\;\frac{y}{\left(\frac{z}{x} \cdot z\right) \cdot \left(1 + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{z} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)} \cdot x\_m}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y\_m}{z}}{z} \cdot x\_m}{z}\\ \end{array}\right) \end{array} \end{array} \]

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 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 0.0)
       (/ (* (/ y_m (fma z z z)) x_m) z)
       (if (<= t_0 5e+118)
         (* (/ (/ 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 t_0 = (1.0 + z) * (z * z);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = ((y_m / fma(z, z, z)) * x_m) / z;
	} else if (t_0 <= 5e+118) {
		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)
	t_0 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(y_m / fma(z, z, z)) * x_m) / z);
	elseif (t_0 <= 5e+118)
		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_] := Block[{t$95$0 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e+118], 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 3 regimes

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

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 97.0

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

   4. Applied rewrites 97.0%

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

   if 0.0 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.99999999999999972e118

   1. Initial program 94.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 93.2

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

   4. Applied rewrites 93.2%

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

   if 4.99999999999999972e118 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 76.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 76.8

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

   4. Applied rewrites 76.8%

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

   5. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 97.5

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

   7. Applied rewrites 97.5%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 0:\\ \;\;\;\;\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)} \cdot x}{z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{z} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(1 + z\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{-215}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y\_m}{1 + z}}{\frac{z}{x\_m} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (+ 1.0 z) (* z z))) 2e-215)
     (* (/ (/ y_m z) (fma z z z)) x_m)
     (/ (/ y_m (+ 1.0 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 (((y_m * x_m) / ((1.0 + z) * (z * z))) <= 2e-215) {
		tmp = ((y_m / z) / fma(z, z, z)) * x_m;
	} else {
		tmp = (y_m / (1.0 + 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(y_m * x_m) / Float64(Float64(1.0 + z) * Float64(z * z))) <= 2e-215)
		tmp = Float64(Float64(Float64(y_m / z) / fma(z, z, z)) * x_m);
	else
		tmp = Float64(Float64(y_m / Float64(1.0 + z)) / Float64(Float64(z / x_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-215], N[(N[(N[(y$95$m / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(y$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] / N[(N[(z / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2.00000000000000008e-215

   1. Initial program 91.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      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 93.1

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

   4. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot \frac{z}{x}}} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 94.9%

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

   if 2.00000000000000008e-215 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 71.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      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 92.0

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

   4. Applied rewrites 92.0%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(1 + z\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{-215}:\\ \;\;\;\;\frac{\frac{y}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{1 + z}}{\frac{z}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(1 + z\right) \cdot \left(z \cdot z\right)} \leq 10^{-206}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= (/ (* y_m x_m) (* (+ 1.0 z) (* z z))) 1e-206)
     (* (/ (/ y_m z) (fma 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 (((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e-206) {
		tmp = ((y_m / z) / fma(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(y_m * x_m) / Float64(Float64(1.0 + z) * Float64(z * z))) <= 1e-206)
		tmp = Float64(Float64(Float64(y_m / z) / fma(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[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-206], N[(N[(N[(y$95$m / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1.00000000000000003e-206

   1. Initial program 91.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      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 93.1

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

   4. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot \frac{z}{x}}} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 94.9%

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

   if 1.00000000000000003e-206 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 86.2

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

   4. Applied rewrites 86.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(1 + z\right) \cdot \left(z \cdot z\right)} \leq 10^{-206}:\\ \;\;\;\;\frac{\frac{y}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, z\right) \cdot z\\ t_1 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{y\_m}{t\_0} \cdot x\_m\\ \mathbf{elif}\;t\_1 \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t\_0} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (fma z z z) z)) (t_1 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -2e+20)
       (* (/ y_m t_0) x_m)
       (if (<= t_1 1.35e-317) (* (/ x_m z) (/ y_m z)) (* (/ x_m t_0) y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(z, z, z) * z;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -2e+20) {
		tmp = (y_m / t_0) * x_m;
	} else if (t_1 <= 1.35e-317) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / t_0) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(fma(z, z, z) * z)
	t_1 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2e+20)
		tmp = Float64(Float64(y_m / t_0) * x_m);
	elseif (t_1 <= 1.35e-317)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(x_m / t_0) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -2e+20], N[(N[(y$95$m / t$95$0), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$1, 1.35e-317], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t$95$0), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e20

   1. Initial program 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      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 93.3

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

   4. Applied rewrites 93.3%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot \frac{z}{x}}} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-*.f64 82.8

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

   8. Applied rewrites 82.8%

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

   if -2e20 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.34999979e-317

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.34999979e-317 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 89.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 90.6

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

   4. Applied rewrites 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. associate-*l* N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 89.1

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

   6. Applied rewrites 89.1%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 90.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\left(z \cdot z\right) \cdot z}\\ \mathbf{elif}\;t\_0 \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

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 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 -2e+20)
       (/ (* y_m x_m) (* (* z z) z))
       (if (<= t_0 1.35e-317)
         (* (/ x_m z) (/ y_m z))
         (* (/ 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 t_0 = (1.0 + z) * (z * z);
	double tmp;
	if (t_0 <= -2e+20) {
		tmp = (y_m * x_m) / ((z * z) * z);
	} else if (t_0 <= 1.35e-317) {
		tmp = (x_m / z) * (y_m / z);
	} 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)
	t_0 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2e+20)
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(z * z) * z));
	elseif (t_0 <= 1.35e-317)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(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_] := Block[{t$95$0 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, -2e+20], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1.35e-317], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e20

   1. Initial program 85.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 90.7

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

   4. Applied rewrites 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l* N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 94.4%

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

   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 92.9

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

   9. Applied rewrites 92.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lift-/.f64 N/A

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

       5. frac-times N/A

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

       6. *-commutative N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 84.3

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

   11. Applied rewrites 84.3%

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

   if -2e20 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.34999979e-317

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   if 1.34999979e-317 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 89.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 90.6

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

   4. Applied rewrites 90.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. associate-*l* N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 89.1

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

   6. Applied rewrites 89.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{y \cdot x}{\left(z \cdot z\right) \cdot z}\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 1.35 \cdot 10^{-317}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{\left(z \cdot z\right) \cdot z} \cdot y\_m\\ t_1 := \left(1 + z\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ x_m (* (* z z) z)) y_m)) (t_1 (* (+ 1.0 z) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -2e+20)
       t_0
       (if (<= t_1 2e-14) (* (/ x_m (* z z)) y_m) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / ((z * z) * z)) * y_m;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -2e+20) {
		tmp = t_0;
	} else if (t_1 <= 2e-14) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_m / ((z * z) * z)) * y_m
    t_1 = (1.0d0 + z) * (z * z)
    if (t_1 <= (-2d+20)) then
        tmp = t_0
    else if (t_1 <= 2d-14) then
        tmp = (x_m / (z * z)) * y_m
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / ((z * z) * z)) * y_m;
	double t_1 = (1.0 + z) * (z * z);
	double tmp;
	if (t_1 <= -2e+20) {
		tmp = t_0;
	} else if (t_1 <= 2e-14) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = (x_m / ((z * z) * z)) * y_m
	t_1 = (1.0 + z) * (z * z)
	tmp = 0
	if t_1 <= -2e+20:
		tmp = t_0
	elif t_1 <= 2e-14:
		tmp = (x_m / (z * z)) * y_m
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m / Float64(Float64(z * z) * z)) * y_m)
	t_1 = Float64(Float64(1.0 + z) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2e+20)
		tmp = t_0;
	elseif (t_1 <= 2e-14)
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = (x_m / ((z * z) * z)) * y_m;
	t_1 = (1.0 + z) * (z * z);
	tmp = 0.0;
	if (t_1 <= -2e+20)
		tmp = t_0;
	elseif (t_1 <= 2e-14)
		tmp = (x_m / (z * z)) * y_m;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -2e+20], t$95$0, If[LessEqual[t$95$1, 2e-14], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e20 or 2e-14 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 87.6

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

   4. Applied rewrites 87.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. *-rgt-identity N/A

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

      7. distribute-lft-in N/A

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

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

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

      9. associate-*l* N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 84.9

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

   6. Applied rewrites 84.9%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.7

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

   9. Applied rewrites 82.7%

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

   if -2e20 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2e-14

   1. Initial program 83.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 90.4

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

   4. Applied rewrites 90.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 80.9

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

   7. Applied rewrites 80.9%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot z} \cdot y\\ \mathbf{elif}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(1 + z\right) \cdot \left(z \cdot z\right)} \leq 10^{-206}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\left(z \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (+ 1.0 z) (* z z))) 1e-206)
     (/ (* 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 tmp;
	if (((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e-206) {
		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) :: tmp
    if (((y_m * x_m) / ((1.0d0 + z) * (z * z))) <= 1d-206) 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 tmp;
	if (((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e-206) {
		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):
	tmp = 0
	if ((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e-206:
		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)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(1.0 + z) * Float64(z * z))) <= 1e-206)
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(z * z) * z));
	else
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e-206)
		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_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-206], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1.00000000000000003e-206

   1. Initial program 91.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 95.8

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

   4. Applied rewrites 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. associate-*r* N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft1-in N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-/l* N/A

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

      13. div-inv N/A

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

      14. lift-/.f64 N/A

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

      15. lower-*.f64 N/A

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

   6. Applied rewrites 95.3%

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

   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 67.8

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

   9. Applied rewrites 67.8%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lift-/.f64 N/A

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

       5. frac-times N/A

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

       6. *-commutative N/A

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

       7. lower-/.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-*.f64 66.2

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

   11. Applied rewrites 66.2%

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

   if 1.00000000000000003e-206 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 67.5

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

   7. Applied rewrites 67.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(1 + z\right) \cdot \left(z \cdot z\right)} \leq 10^{-206}:\\ \;\;\;\;\frac{y \cdot x}{\left(z \cdot z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(1 + z\right) \cdot \left(z \cdot z\right)} \leq 10^{+274}:\\ \;\;\;\;\frac{-x\_m}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{-y\_m}{z} \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (+ 1.0 z) (* z z))) 1e+274)
     (* (/ (- x_m) z) y_m)
     (* (/ (- y_m) z) x_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e+274) {
		tmp = (-x_m / z) * y_m;
	} else {
		tmp = (-y_m / z) * x_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((y_m * x_m) / ((1.0d0 + z) * (z * z))) <= 1d+274) then
        tmp = (-x_m / z) * y_m
    else
        tmp = (-y_m / z) * x_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e+274) {
		tmp = (-x_m / z) * y_m;
	} else {
		tmp = (-y_m / z) * x_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if ((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e+274:
		tmp = (-x_m / z) * y_m
	else:
		tmp = (-y_m / z) * x_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(1.0 + z) * Float64(z * z))) <= 1e+274)
		tmp = Float64(Float64(Float64(-x_m) / z) * y_m);
	else
		tmp = Float64(Float64(Float64(-y_m) / z) * x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((y_m * x_m) / ((1.0 + z) * (z * z))) <= 1e+274)
		tmp = (-x_m / z) * y_m;
	else
		tmp = (-y_m / z) * x_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+274], N[(N[((-x$95$m) / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[((-y$95$m) / z), $MachinePrecision] * x$95$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)))) < 9.99999999999999921e273

   1. Initial program 92.6%

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

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv 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. distribute-lft-out-- N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. unpow2 N/A

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

      23. lower-*.f64 69.9

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

   5. Applied rewrites 69.9%

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

   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 32.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 34.9%

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

   if 9.99999999999999921e273 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 59.0%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv 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. distribute-lft-out-- N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-out N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. sub-neg N/A

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

      19. lower--.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. unpow2 N/A

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

      23. lower-*.f64 59.1

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

   5. Applied rewrites 59.1%

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

   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 30.6%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(1 + z\right) \cdot \left(z \cdot z\right)} \leq 10^{+274}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 94.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(1 + z\right) \cdot \left(z \cdot z\right) \leq 0:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)} \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= (* (+ 1.0 z) (* z z)) 0.0)
     (/ (* (/ y_m (fma z z z)) x_m) z)
     (* (/ (/ 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 (((1.0 + z) * (z * z)) <= 0.0) {
		tmp = ((y_m / fma(z, z, z)) * x_m) / z;
	} 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(1.0 + z) * Float64(z * z)) <= 0.0)
		tmp = Float64(Float64(Float64(y_m / fma(z, z, z)) * x_m) / z);
	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[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $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 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 77.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 97.0

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

   4. Applied rewrites 97.0%

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

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

   1. Initial program 89.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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 90.8

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

   4. Applied rewrites 90.8%

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

4. Final simplification 94.0%

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

Alternative 15: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 10^{-209}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\left(1 + z\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 1e-209)
     (* (/ x_m z) (/ y_m z))
     (if (<= (* y_m x_m) 2e+223)
       (/ (* y_m x_m) (* (+ 1.0 z) (* z z)))
       (* (/ y_m (* (fma z z z) z)) x_m))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 1e-209) {
		tmp = (x_m / z) * (y_m / z);
	} else if ((y_m * x_m) <= 2e+223) {
		tmp = (y_m * x_m) / ((1.0 + z) * (z * z));
	} else {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 1e-209)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (Float64(y_m * x_m) <= 2e+223)
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(1.0 + z) * Float64(z * z)));
	else
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e-209], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 2e+223], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(1.0 + z), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 1e-209

   1. Initial program 78.4%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if 1e-209 < (*.f64 x y) < 2.00000000000000009e223

   1. Initial program 99.7%

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

   if 2.00000000000000009e223 < (*.f64 x y)

   1. Initial program 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot \frac{z}{x}}} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-*.f64 N/A

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

   6. Applied rewrites 87.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-fma.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. distribute-lft1-in N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-209}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{y \cdot x}{\left(1 + z\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 91.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, z\right) \cdot z\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 10^{-209}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{t\_0} \cdot x\_m\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (fma z z z) z)))
   (*
    x_s
    (*
     y_s
     (if (<= (* y_m x_m) 1e-209)
       (* (/ x_m z) (/ y_m z))
       (if (<= (* y_m x_m) 2e+223)
         (/ (* y_m x_m) t_0)
         (* (/ y_m t_0) x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(z, z, z) * z;
	double tmp;
	if ((y_m * x_m) <= 1e-209) {
		tmp = (x_m / z) * (y_m / z);
	} else if ((y_m * x_m) <= 2e+223) {
		tmp = (y_m * x_m) / t_0;
	} else {
		tmp = (y_m / t_0) * x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(fma(z, z, z) * z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 1e-209)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (Float64(y_m * x_m) <= 2e+223)
		tmp = Float64(Float64(y_m * x_m) / t_0);
	else
		tmp = Float64(Float64(y_m / t_0) * x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e-209], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 2e+223], N[(N[(y$95$m * x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(y$95$m / t$95$0), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 1e-209

   1. Initial program 78.4%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if 1e-209 < (*.f64 x y) < 2.00000000000000009e223

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-lft1-in N/A

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

      12. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 2.00000000000000009e223 < (*.f64 x y)

   1. Initial program 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-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 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot \frac{z}{x}}} \]
   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. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z\right)} \cdot \left(1 + z\right)} \cdot x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(z \cdot z\right) \cdot \color{blue}{\left(1 + z\right)}} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{y}{\left(z \cdot z\right) \cdot \color{blue}{\left(z + 1\right)}} \cdot x \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \cdot x} \]

   6. Applied rewrites 87.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\mathsf{fma}\left(z, z, z\right)}} \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{\mathsf{fma}\left(z, z, z\right)} \cdot x \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \cdot x \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z + z\right)} \cdot z} \cdot x \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(z + 1\right) \cdot z\right)} \cdot z} \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z + 1\right) \cdot \left(z \cdot z\right)}} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \cdot x \]

      9. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z + 1\right) \cdot \left(z \cdot z\right)}} \cdot x \]

      10. associate-*r* N/A

          \[\leadsto \frac{y}{\color{blue}{\left(\left(z + 1\right) \cdot z\right) \cdot z}} \cdot x \]

      11. distribute-lft1-in N/A

          \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z + z\right)} \cdot z} \cdot x \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z\right)} \cdot z} \cdot x \]

      13. lower-*.f64 81.6

          \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \cdot x \]

   8. Applied rewrites 81.6%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \cdot x \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-209}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{+223}:\\ \;\;\;\;\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 92.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ y_m (* (fma z z z) z)) x_m)))
   (*
    x_s
    (*
     y_s
     (if (<= z -1.05e-17)
       t_0
       (if (<= z 1.95e-87) (/ (* (/ x_m z) y_m) z) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / (fma(z, z, z) * z)) * x_m;
	double tmp;
	if (z <= -1.05e-17) {
		tmp = t_0;
	} else if (z <= 1.95e-87) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m)
	tmp = 0.0
	if (z <= -1.05e-17)
		tmp = t_0;
	elseif (z <= 1.95e-87)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -1.05e-17], t$95$0, If[LessEqual[z, 1.95e-87], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.04999999999999996e-17 or 1.9499999999999999e-87 < z

   1. Initial program 85.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{z + 1} \cdot \frac{x}{z \cdot z}} \]

      6. clear-num N/A

         \[\leadsto \frac{y}{z + 1} \cdot \color{blue}{\frac{1}{\frac{z \cdot z}{x}}} \]

      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 95.4

          \[\leadsto \frac{\frac{y}{1 + z}}{z \cdot \color{blue}{\frac{z}{x}}} \]

   4. Applied rewrites 95.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot \frac{z}{x}}} \]
   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. lift-/.f64 N/A

         \[\leadsto \frac{\frac{y}{1 + z}}{z \cdot \color{blue}{\frac{z}{x}}} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\frac{y}{1 + z}}{\color{blue}{\frac{z \cdot z}{x}}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\frac{y}{1 + z}}{\frac{\color{blue}{z \cdot z}}{x}} \]

      6. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{1 + z}}{z \cdot z} \cdot x} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + z}}}{z \cdot z} \cdot x \]

      8. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{y}{\left(1 + z\right) \cdot \left(z \cdot z\right)}} \cdot x \]

      9. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z\right) \cdot \left(1 + z\right)}} \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z\right)} \cdot \left(1 + z\right)} \cdot x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(z \cdot z\right) \cdot \color{blue}{\left(1 + z\right)}} \cdot x \]

      12. +-commutative N/A

          \[\leadsto \frac{y}{\left(z \cdot z\right) \cdot \color{blue}{\left(z + 1\right)}} \cdot x \]

      13. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \cdot x} \]

   6. Applied rewrites 91.4%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\mathsf{fma}\left(z, z, z\right)} \cdot x} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{\mathsf{fma}\left(z, z, z\right)}} \cdot x \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{\mathsf{fma}\left(z, z, z\right)} \cdot x \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \cdot x \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z + z\right)} \cdot z} \cdot x \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(z + 1\right) \cdot z\right)} \cdot z} \cdot x \]

      6. associate-*r* N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z + 1\right) \cdot \left(z \cdot z\right)}} \cdot x \]

      7. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \cdot x \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \cdot x \]

      9. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z + 1\right) \cdot \left(z \cdot z\right)}} \cdot x \]

      10. associate-*r* N/A

          \[\leadsto \frac{y}{\color{blue}{\left(\left(z + 1\right) \cdot z\right) \cdot z}} \cdot x \]

      11. distribute-lft1-in N/A

          \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z + z\right)} \cdot z} \cdot x \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z\right)} \cdot z} \cdot x \]

      13. lower-*.f64 86.9

          \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \cdot x \]

   8. Applied rewrites 86.9%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \cdot x \]

   if -1.04999999999999996e-17 < z < 1.9499999999999999e-87

   1. Initial program 80.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot \left(z + 1\right)\right) \cdot z}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}{z}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}{z}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}}{z} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{z \cdot \left(z + 1\right)}}{z} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z \cdot \left(z + 1\right)}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z \cdot \left(z + 1\right)}}{z} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(z + 1\right) \cdot z}}}{z} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(z + 1\right)} \cdot z}}{z} \]

      14. distribute-lft1-in N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{z \cdot z + z}}}{z} \]

      15. lower-fma.f64 89.4

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}}{z} \]

   4. Applied rewrites 89.4%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right)}}{z}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{z}}}{z} \]
   6. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{z} \]

      3. lower-/.f64 95.8

         \[\leadsto \frac{\color{blue}{\frac{x}{z}} \cdot y}{z} \]

   7. Applied rewrites 95.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 94.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{y\_m}{z \cdot z} \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

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 (<= z -3.2e+16)
     (/ (* (/ y_m (* z z)) x_m) z)
     (* (/ (/ 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 (z <= -3.2e+16) {
		tmp = ((y_m / (z * z)) * x_m) / z;
	} 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 (z <= -3.2e+16)
		tmp = Float64(Float64(Float64(y_m / Float64(z * z)) * x_m) / z);
	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[z, -3.2e+16], N[(N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $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 z < -3.2e16

   1. Initial program 85.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      5. *-commutative N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot \left(z + 1\right)\right) \cdot z}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}{z}} \]

      7. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}{z}} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{z \cdot \left(z + 1\right)}}}{z} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{x \cdot y}}{z \cdot \left(z + 1\right)}}{z} \]

      10. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z \cdot \left(z + 1\right)}}{z} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{z \cdot \left(z + 1\right)}}{z} \]

      12. *-commutative N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(z + 1\right) \cdot z}}}{z} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\left(z + 1\right)} \cdot z}}{z} \]

      14. distribute-lft1-in N/A

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{z \cdot z + z}}}{z} \]

      15. lower-fma.f64 90.2

          \[\leadsto \frac{\frac{y \cdot x}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}}{z} \]

   4. Applied rewrites 90.2%

      \[\leadsto \color{blue}{\frac{\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right)}}{z}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right)}}}{z} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(z, z, z\right)}}{z} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}}}{z} \]

      4. clear-num N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}}}{z} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(z, z, z\right)} \cdot x\right)}}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{\mathsf{fma}\left(z, z, z\right)}\right) \cdot x}}{z} \]

      7. lift-fma.f64 N/A

         \[\leadsto \frac{\left(y \cdot \frac{1}{\color{blue}{z \cdot z + z}}\right) \cdot x}{z} \]

      8. distribute-lft1-in N/A

         \[\leadsto \frac{\left(y \cdot \frac{1}{\color{blue}{\left(z + 1\right) \cdot z}}\right) \cdot x}{z} \]

      9. +-commutative N/A

         \[\leadsto \frac{\left(y \cdot \frac{1}{\color{blue}{\left(1 + z\right)} \cdot z}\right) \cdot x}{z} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\left(y \cdot \frac{1}{\color{blue}{\left(1 + z\right)} \cdot z}\right) \cdot x}{z} \]

      11. associate-/r* N/A

          \[\leadsto \frac{\left(y \cdot \color{blue}{\frac{\frac{1}{1 + z}}{z}}\right) \cdot x}{z} \]

      12. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{1}{1 + z}}{z}} \cdot x}{z} \]

      13. div-inv N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{y}{1 + z}}}{z} \cdot x}{z} \]

      14. lift-/.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\frac{y}{1 + z}}}{z} \cdot x}{z} \]

      15. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{\frac{y}{1 + z}}{z} \cdot x}}{z} \]

   6. Applied rewrites 94.1%

      \[\leadsto \frac{\color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right)} \cdot x}}{z} \]

   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 94.1

         \[\leadsto \frac{\frac{y}{\color{blue}{z \cdot z}} \cdot x}{z} \]

   9. Applied rewrites 94.1%

      \[\leadsto \frac{\frac{y}{\color{blue}{z \cdot z}} \cdot x}{z} \]

   if -3.2e16 < 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

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \cdot y} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \cdot y} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \cdot y \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)} \cdot y \]

      9. associate-*l* N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \cdot y \]

      10. *-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(z \cdot \left(z + 1\right)\right) \cdot z}} \cdot y \]

      11. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot \left(z + 1\right)}}{z}} \cdot y \]

      12. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot \left(z + 1\right)}}{z}} \cdot y \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{z \cdot \left(z + 1\right)}}}{z} \cdot y \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(z + 1\right) \cdot z}}}{z} \cdot y \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{\left(z + 1\right)} \cdot z}}{z} \cdot y \]

      16. distribute-lft1-in N/A

          \[\leadsto \frac{\frac{x}{\color{blue}{z \cdot z + z}}}{z} \cdot y \]

      17. lower-fma.f64 90.1

          \[\leadsto \frac{\frac{x}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}}{z} \cdot y \]

   4. Applied rewrites 90.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \left(\frac{x\_m}{z \cdot z} \cdot y\_m\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (* (/ x_m (* z z)) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / (z * z)) * y_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * ((x_m / (z * z)) * y_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / (z * z)) * y_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * ((x_m / (z * z)) * y_m))

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(x_m / Float64(z * z)) * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * ((x_m / (z * z)) * y_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \cdot y} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \cdot y} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \cdot y \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)} \cdot y \]

   9. associate-*l* N/A

      \[\leadsto \frac{x}{\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}} \cdot y \]

   10. *-commutative N/A

       \[\leadsto \frac{x}{\color{blue}{\left(z \cdot \left(z + 1\right)\right) \cdot z}} \cdot y \]

   11. associate-/r* N/A

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot \left(z + 1\right)}}{z}} \cdot y \]

   12. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{x}{z \cdot \left(z + 1\right)}}{z}} \cdot y \]

   13. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{x}{z \cdot \left(z + 1\right)}}}{z} \cdot y \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{x}{\color{blue}{\left(z + 1\right) \cdot z}}}{z} \cdot y \]

   15. lift-+.f64 N/A

       \[\leadsto \frac{\frac{x}{\color{blue}{\left(z + 1\right)} \cdot z}}{z} \cdot y \]

   16. distribute-lft1-in N/A

       \[\leadsto \frac{\frac{x}{\color{blue}{z \cdot z + z}}}{z} \cdot y \]

   17. lower-fma.f64 89.2

       \[\leadsto \frac{\frac{x}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}}{z} \cdot y \]

4. Applied rewrites 89.2%

   \[\leadsto \color{blue}{\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y} \]

5. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{x}{{z}^{2}}} \cdot y \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{{z}^{2}}} \cdot y \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{\color{blue}{z \cdot z}} \cdot y \]

   3. lower-*.f64 73.2

      \[\leadsto \frac{x}{\color{blue}{z \cdot z}} \cdot y \]

7. Applied rewrites 73.2%

   \[\leadsto \color{blue}{\frac{x}{z \cdot z}} \cdot y \]
8. Add Preprocessing

Alternative 20: 31.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \left(\frac{-x\_m}{z} \cdot y\_m\right)\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (* (/ (- x_m) z) y_m))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((-x_m / z) * y_m));
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    code = x_s * (y_s * ((-x_m / z) * y_m))
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((-x_m / z) * y_m));
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	return x_s * (y_s * ((-x_m / z) * 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) / z) * y_m)))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp = code(x_s, y_s, x_m, y_m, z)
	tmp = x_s * (y_s * ((-x_m / z) * y_m));
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[((-x$95$m) / z), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + x \cdot y}{{z}^{2}}} \]
4. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{1 \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   3. cancel-sign-sub-inv 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. distribute-lft-out-- N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right) - x \cdot y\right)}}{{z}^{2}} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(x \cdot \left(y \cdot z\right) - x \cdot y\right)\right)}}{{z}^{2}} \]

   7. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)}\right)}{{z}^{2}} \]

   8. associate-*r* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{\left(x \cdot y\right) \cdot z} + \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right)}{{z}^{2}} \]

   9. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{-1 \cdot \left(x \cdot y\right)}\right)\right)}{{z}^{2}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{\left(x \cdot y\right) \cdot -1}\right)\right)}{{z}^{2}} \]

   11. distribute-lft-out N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot y\right) \cdot \left(z + -1\right)}\right)}{{z}^{2}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(z + -1\right) \cdot \left(x \cdot y\right)}\right)}{{z}^{2}} \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z + -1\right)\right)\right) \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z + -1\right)\right)\right) \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

   15. +-commutative N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(-1 + z\right)}\right)\right) \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   16. distribute-neg-in N/A

       \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(z\right)\right)\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\left(\color{blue}{1} + \left(\mathsf{neg}\left(z\right)\right)\right) \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   18. sub-neg N/A

       \[\leadsto \frac{\color{blue}{\left(1 - z\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   19. lower--.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(1 - z\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   20. *-commutative N/A

       \[\leadsto \frac{\left(1 - z\right) \cdot \color{blue}{\left(y \cdot x\right)}}{{z}^{2}} \]

   21. lower-*.f64 N/A

       \[\leadsto \frac{\left(1 - z\right) \cdot \color{blue}{\left(y \cdot x\right)}}{{z}^{2}} \]

   22. unpow2 N/A

       \[\leadsto \frac{\left(1 - z\right) \cdot \left(y \cdot x\right)}{\color{blue}{z \cdot z}} \]

   23. lower-*.f64 66.8

       \[\leadsto \frac{\left(1 - z\right) \cdot \left(y \cdot x\right)}{\color{blue}{z \cdot z}} \]

5. Applied rewrites 66.8%

   \[\leadsto \color{blue}{\frac{\left(1 - z\right) \cdot \left(y \cdot x\right)}{z \cdot z}} \]

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 32.3%

   \[\leadsto \frac{-y}{z} \cdot \color{blue}{x} \]

9. Taylor expanded in z around inf

   \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot y}{z}} \]
10. Step-by-step derivation

11. Applied rewrites 32.6%

    \[\leadsto \frac{-x}{z} \cdot \color{blue}{y} \]
12. Add Preprocessing

Developer Target 1: 96.6% 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 2024248 
(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))))