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

Percentage Accurate: 83.9% → 98.1%

Time: 10.7s

Alternatives: 15

Speedup: 0.9×

• 26.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.9% 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: 98.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{\frac{y\_m}{z}}{z} \cdot x\_m}{z}\\ t_1 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\frac{y\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{x\_m} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (* (/ (/ y_m z) z) x_m) z)) (t_1 (* (* z z) (+ 1.0 z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -2e+69)
       t_0
       (if (<= t_1 0.005) (/ y_m (* (/ (fma z z z) x_m) z)) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (((y_m / z) / z) * x_m) / z;
	double t_1 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -2e+69) {
		tmp = t_0;
	} else if (t_1 <= 0.005) {
		tmp = y_m / ((fma(z, z, z) / x_m) * z);
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(Float64(y_m / z) / z) * x_m) / z)
	t_1 = Float64(Float64(z * z) * Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= -2e+69)
		tmp = t_0;
	elseif (t_1 <= 0.005)
		tmp = Float64(y_m / Float64(Float64(fma(z, z, z) / x_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[(N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -2e+69], t$95$0, If[LessEqual[t$95$1, 0.005], N[(y$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2.0000000000000001e69 or 0.0050000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 78.6%

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

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 90.1

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

   6. Applied rewrites 90.1%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 96.9

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

   9. Applied rewrites 96.9%

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

   if -2.0000000000000001e69 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0050000000000000001

   1. Initial program 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. times-frac N/A

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

      11. *-rgt-identity N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 90.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 90.0

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

   6. Applied rewrites 90.0%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{z} \cdot x}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 0.005:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(z, z, z\right)}{x} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{z} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot z\\ t_1 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -40:\\ \;\;\;\;\frac{x\_m}{t\_0} \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{t\_0}\\ \end{array}\right) \end{array} \end{array} \]

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 (* (* z z) z)) (t_1 (* (* z z) (+ 1.0 z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -40.0)
       (* (/ x_m t_0) y_m)
       (if (<= t_1 5e-307)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_1 0.005) (* (/ x_m (* z z)) y_m) (/ (* x_m 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 = (z * z) * z;
	double t_1 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -40.0) {
		tmp = (x_m / t_0) * y_m;
	} else if (t_1 <= 5e-307) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 0.005) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = (x_m * y_m) / 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 = (z * z) * z
    t_1 = (z * z) * (1.0d0 + z)
    if (t_1 <= (-40.0d0)) then
        tmp = (x_m / t_0) * y_m
    else if (t_1 <= 5d-307) then
        tmp = (x_m / z) * (y_m / z)
    else if (t_1 <= 0.005d0) then
        tmp = (x_m / (z * z)) * y_m
    else
        tmp = (x_m * y_m) / 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 = (z * z) * z;
	double t_1 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -40.0) {
		tmp = (x_m / t_0) * y_m;
	} else if (t_1 <= 5e-307) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 0.005) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = (x_m * y_m) / 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 = (z * z) * z
	t_1 = (z * z) * (1.0 + z)
	tmp = 0
	if t_1 <= -40.0:
		tmp = (x_m / t_0) * y_m
	elif t_1 <= 5e-307:
		tmp = (x_m / z) * (y_m / z)
	elif t_1 <= 0.005:
		tmp = (x_m / (z * z)) * y_m
	else:
		tmp = (x_m * y_m) / 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(z * z) * z)
	t_1 = Float64(Float64(z * z) * Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= -40.0)
		tmp = Float64(Float64(x_m / t_0) * y_m);
	elseif (t_1 <= 5e-307)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_1 <= 0.005)
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	else
		tmp = Float64(Float64(x_m * y_m) / 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 = (z * z) * z;
	t_1 = (z * z) * (1.0 + z);
	tmp = 0.0;
	if (t_1 <= -40.0)
		tmp = (x_m / t_0) * y_m;
	elseif (t_1 <= 5e-307)
		tmp = (x_m / z) * (y_m / z);
	elseif (t_1 <= 0.005)
		tmp = (x_m / (z * z)) * y_m;
	else
		tmp = (x_m * y_m) / 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[(z * z), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -40.0], N[(N[(x$95$m / t$95$0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e-307], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.005], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -40

   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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 88.1

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

   4. Applied rewrites 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

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

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

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

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

      17. lift-fma.f64 N/A

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

      18. associate-/r* N/A

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

      19. neg-mul-1 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 87.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}} \cdot z} \cdot y \]
   10. 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 81.0

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

   11. Applied rewrites 81.0%

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

   if -40 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.00000000000000014e-307

   1. Initial program 78.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   if 5.00000000000000014e-307 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0050000000000000001

   1. Initial program 91.0%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 85.0

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

   5. Applied rewrites 85.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 84.1

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

   7. Applied rewrites 84.1%

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

   if 0.0050000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.2

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

      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 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.2

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

   7. Applied rewrites 80.2%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -40:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot z} \cdot y\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 0.005:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y\_m}{z \cdot z} \cdot x\_m}{z}\\ t_1 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\frac{y\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{x\_m} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (* (/ y_m (* z z)) x_m) z)) (t_1 (* (* z z) (+ 1.0 z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -2e+69)
       t_0
       (if (<= t_1 0.005) (/ y_m (* (/ (fma z z z) x_m) z)) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = ((y_m / (z * z)) * x_m) / z;
	double t_1 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -2e+69) {
		tmp = t_0;
	} else if (t_1 <= 0.005) {
		tmp = y_m / ((fma(z, z, z) / x_m) * z);
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(y_m / Float64(z * z)) * x_m) / z)
	t_1 = Float64(Float64(z * z) * Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= -2e+69)
		tmp = t_0;
	elseif (t_1 <= 0.005)
		tmp = Float64(y_m / Float64(Float64(fma(z, z, z) / x_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[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -2e+69], t$95$0, If[LessEqual[t$95$1, 0.005], N[(y$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2.0000000000000001e69 or 0.0050000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 78.6%

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

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 90.1

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

   6. Applied rewrites 90.1%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.1

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

   9. Applied rewrites 90.1%

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

   if -2.0000000000000001e69 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0050000000000000001

   1. Initial program 86.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 96.2

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

   4. Applied rewrites 96.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. clear-num N/A

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

      9. lift-/.f64 N/A

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

      10. times-frac N/A

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

      11. *-rgt-identity N/A

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

      12. lift-*.f64 N/A

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

      13. lower-/.f64 90.0

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 90.0

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

   6. Applied rewrites 90.0%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -2 \cdot 10^{+69}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z} \cdot x}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 0.005:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(z, z, z\right)}{x} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, z\right) \cdot z\\ t_1 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -40:\\ \;\;\;\;\frac{y\_m}{t\_0} \cdot x\_m\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t\_0} \cdot y\_m\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (fma z z z) z)) (t_1 (* (* z z) (+ 1.0 z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -40.0)
       (* (/ y_m t_0) x_m)
       (if (<= t_1 5e-307) (* (/ x_m z) (/ y_m z)) (* (/ x_m t_0) y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(z, z, z) * z;
	double t_1 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -40.0) {
		tmp = (y_m / t_0) * x_m;
	} else if (t_1 <= 5e-307) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / t_0) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(fma(z, z, z) * z)
	t_1 = Float64(Float64(z * z) * Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= -40.0)
		tmp = Float64(Float64(y_m / t_0) * x_m);
	elseif (t_1 <= 5e-307)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(x_m / t_0) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -40.0], N[(N[(y$95$m / t$95$0), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$1, 5e-307], 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))) < -40

   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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 88.1

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

   4. Applied rewrites 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

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

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

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

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

      17. lift-fma.f64 N/A

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

      18. associate-/r* N/A

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

      19. neg-mul-1 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 84.6

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

   10. Applied rewrites 84.6%

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

   if -40 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.00000000000000014e-307

   1. Initial program 78.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

   if 5.00000000000000014e-307 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   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. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 91.8

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

   4. Applied rewrites 91.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

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

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

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

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

      17. lift-fma.f64 N/A

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

      18. associate-/r* N/A

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

      19. neg-mul-1 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 85.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -40:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ t_1 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -40:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (* (/ x_m (* (fma z z z) z)) y_m)) (t_1 (* (* z z) (+ 1.0 z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -40.0)
       t_0
       (if (<= t_1 5e-307) (* (/ 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 = (x_m / (fma(z, z, z) * z)) * y_m;
	double t_1 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -40.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-307) {
		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(x_m / Float64(fma(z, z, z) * z)) * y_m)
	t_1 = Float64(Float64(z * z) * Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= -40.0)
		tmp = t_0;
	elseif (t_1 <= 5e-307)
		tmp = Float64(Float64(x_m / z) * Float64(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[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -40.0], t$95$0, If[LessEqual[t$95$1, 5e-307], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -40 or 5.00000000000000014e-307 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 90.8

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

   4. Applied rewrites 90.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

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

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

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

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

      17. lift-fma.f64 N/A

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

      18. associate-/r* N/A

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

      19. neg-mul-1 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 86.5%

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

   if -40 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.00000000000000014e-307

   1. Initial program 78.7%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -40:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 5 \cdot 10^{-307}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot z\\ t_1 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -40:\\ \;\;\;\;\frac{x\_m}{t\_0} \cdot y\_m\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{t\_0}\\ \end{array}\right) \end{array} \end{array} \]

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 (* (* z z) z)) (t_1 (* (* z z) (+ 1.0 z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -40.0)
       (* (/ x_m t_0) y_m)
       (if (<= t_1 0.005) (* (/ x_m (* z z)) y_m) (/ (* x_m 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 = (z * z) * z;
	double t_1 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -40.0) {
		tmp = (x_m / t_0) * y_m;
	} else if (t_1 <= 0.005) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = (x_m * y_m) / 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 = (z * z) * z
    t_1 = (z * z) * (1.0d0 + z)
    if (t_1 <= (-40.0d0)) then
        tmp = (x_m / t_0) * y_m
    else if (t_1 <= 0.005d0) then
        tmp = (x_m / (z * z)) * y_m
    else
        tmp = (x_m * y_m) / 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 = (z * z) * z;
	double t_1 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -40.0) {
		tmp = (x_m / t_0) * y_m;
	} else if (t_1 <= 0.005) {
		tmp = (x_m / (z * z)) * y_m;
	} else {
		tmp = (x_m * y_m) / 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 = (z * z) * z
	t_1 = (z * z) * (1.0 + z)
	tmp = 0
	if t_1 <= -40.0:
		tmp = (x_m / t_0) * y_m
	elif t_1 <= 0.005:
		tmp = (x_m / (z * z)) * y_m
	else:
		tmp = (x_m * y_m) / 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(z * z) * z)
	t_1 = Float64(Float64(z * z) * Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= -40.0)
		tmp = Float64(Float64(x_m / t_0) * y_m);
	elseif (t_1 <= 0.005)
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	else
		tmp = Float64(Float64(x_m * y_m) / 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 = (z * z) * z;
	t_1 = (z * z) * (1.0 + z);
	tmp = 0.0;
	if (t_1 <= -40.0)
		tmp = (x_m / t_0) * y_m;
	elseif (t_1 <= 0.005)
		tmp = (x_m / (z * z)) * y_m;
	else
		tmp = (x_m * y_m) / 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[(z * z), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -40.0], N[(N[(x$95$m / t$95$0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[t$95$1, 0.005], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(x$95$m * y$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -40

   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

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 88.1

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

   4. Applied rewrites 88.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.2

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

   6. Applied rewrites 92.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

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

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

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

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

      17. lift-fma.f64 N/A

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

      18. associate-/r* N/A

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

      19. neg-mul-1 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 87.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}} \cdot z} \cdot y \]
   10. 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 81.0

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

   11. Applied rewrites 81.0%

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

   if -40 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0050000000000000001

   1. Initial program 85.1%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.0

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

   7. Applied rewrites 82.0%

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

   if 0.0050000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 80.2

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

      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 80.2

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

   4. Applied rewrites 80.2%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.2

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

   7. Applied rewrites 80.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -40:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot z} \cdot y\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 0.005:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{\left(z \cdot z\right) \cdot z} \cdot y\_m\\ t_1 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -40:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.005:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (* (* z z) (+ 1.0 z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -40.0)
       t_0
       (if (<= t_1 0.005) (* (/ 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 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -40.0) {
		tmp = t_0;
	} else if (t_1 <= 0.005) {
		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 = (z * z) * (1.0d0 + z)
    if (t_1 <= (-40.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.005d0) 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 = (z * z) * (1.0 + z);
	double tmp;
	if (t_1 <= -40.0) {
		tmp = t_0;
	} else if (t_1 <= 0.005) {
		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 = (z * z) * (1.0 + z)
	tmp = 0
	if t_1 <= -40.0:
		tmp = t_0
	elif t_1 <= 0.005:
		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(z * z) * Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= -40.0)
		tmp = t_0;
	elseif (t_1 <= 0.005)
		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 = (z * z) * (1.0 + z);
	tmp = 0.0;
	if (t_1 <= -40.0)
		tmp = t_0;
	elseif (t_1 <= 0.005)
		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[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -40.0], t$95$0, If[LessEqual[t$95$1, 0.005], 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))) < -40 or 0.0050000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 89.7

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

   4. Applied rewrites 89.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 91.7

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

   6. Applied rewrites 91.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

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

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

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

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

      17. lift-fma.f64 N/A

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

      18. associate-/r* N/A

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

      19. neg-mul-1 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 84.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}} \cdot z} \cdot y \]
   10. 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 80.9

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

   11. Applied rewrites 80.9%

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

   if -40 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0050000000000000001

   1. Initial program 85.1%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 82.0

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

   5. Applied rewrites 82.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 82.0

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

   7. Applied rewrites 82.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -40:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot z} \cdot y\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 0.005:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 10^{+121}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ 1.0 z))) 1e+121)
     (* (/ y_m (* (fma z z z) z)) x_m)
     (/ y_m (* (/ z x_m) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (1.0 + z))) <= 1e+121) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = y_m / ((z / x_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(1.0 + z))) <= 1e+121)
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(y_m / Float64(Float64(z / x_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+121], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y$95$m / N[(N[(z / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1.00000000000000004e121

   1. Initial program 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 95.1

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

   4. Applied rewrites 95.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 96.1

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

   6. Applied rewrites 96.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

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

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

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

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

      17. lift-fma.f64 N/A

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

      18. associate-/r* N/A

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

      19. neg-mul-1 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 90.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 92.0

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

   10. Applied rewrites 92.0%

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

   if 1.00000000000000004e121 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 63.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 87.7

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

   4. Applied rewrites 87.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 87.8

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

   6. Applied rewrites 87.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. remove-double-neg N/A

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

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

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

      9. lift-fma.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. remove-double-neg N/A

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

      15. lower-*.f64 69.8

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

   8. Applied rewrites 69.8%

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

   9. Taylor expanded in z around 0

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

       1. unpow2 N/A

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

       2. associate-*l/ N/A

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

       3. lower-*.f64 N/A

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

       4. lower-/.f64 68.8

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

   11. Applied rewrites 68.8%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 10^{+121}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y\_m}{z \cdot z} \cdot x\_m}{z}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\frac{x\_m}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \cdot y\_m\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{\left(\frac{x\_m}{z} - x\_m\right) \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (* z z)) x_m) z)))
   (*
    x_s
    (*
     y_s
     (if (<= z -1.65e+21)
       t_0
       (if (<= z -1e-138)
         (* (/ x_m (* (* z z) (+ 1.0 z))) y_m)
         (if (<= z 0.75) (/ (* (- (/ x_m z) x_m) 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 / (z * z)) * x_m) / z;
	double tmp;
	if (z <= -1.65e+21) {
		tmp = t_0;
	} else if (z <= -1e-138) {
		tmp = (x_m / ((z * z) * (1.0 + z))) * y_m;
	} else if (z <= 0.75) {
		tmp = (((x_m / z) - x_m) * y_m) / z;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_m / (z * z)) * x_m) / z
    if (z <= (-1.65d+21)) then
        tmp = t_0
    else if (z <= (-1d-138)) then
        tmp = (x_m / ((z * z) * (1.0d0 + z))) * y_m
    else if (z <= 0.75d0) then
        tmp = (((x_m / z) - x_m) * y_m) / z
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = ((y_m / (z * z)) * x_m) / z;
	double tmp;
	if (z <= -1.65e+21) {
		tmp = t_0;
	} else if (z <= -1e-138) {
		tmp = (x_m / ((z * z) * (1.0 + z))) * y_m;
	} else if (z <= 0.75) {
		tmp = (((x_m / z) - x_m) * y_m) / z;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = ((y_m / (z * z)) * x_m) / z
	tmp = 0
	if z <= -1.65e+21:
		tmp = t_0
	elif z <= -1e-138:
		tmp = (x_m / ((z * z) * (1.0 + z))) * y_m
	elif z <= 0.75:
		tmp = (((x_m / z) - x_m) * 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(Float64(y_m / Float64(z * z)) * x_m) / z)
	tmp = 0.0
	if (z <= -1.65e+21)
		tmp = t_0;
	elseif (z <= -1e-138)
		tmp = Float64(Float64(x_m / Float64(Float64(z * z) * Float64(1.0 + z))) * y_m);
	elseif (z <= 0.75)
		tmp = Float64(Float64(Float64(Float64(x_m / z) - x_m) * y_m) / z);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = ((y_m / (z * z)) * x_m) / z;
	tmp = 0.0;
	if (z <= -1.65e+21)
		tmp = t_0;
	elseif (z <= -1e-138)
		tmp = (x_m / ((z * z) * (1.0 + z))) * y_m;
	elseif (z <= 0.75)
		tmp = (((x_m / z) - x_m) * y_m) / z;
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -1.65e+21], t$95$0, If[LessEqual[z, -1e-138], N[(N[(x$95$m / N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[z, 0.75], N[(N[(N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.65e21 or 0.75 < z

   1. Initial program 78.6%

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

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 88.8

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

   4. Applied rewrites 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 90.1

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

   6. Applied rewrites 90.1%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 90.1

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

   9. Applied rewrites 90.1%

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

   if -1.65e21 < z < -1.00000000000000007e-138

   1. Initial program 91.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 90.3

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

   4. Applied rewrites 90.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 95.4

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

   6. Applied rewrites 95.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. associate-*r* N/A

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

      10. frac-2neg N/A

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

      11. metadata-eval N/A

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

      12. times-frac N/A

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

      13. metadata-eval N/A

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

      14. div-inv N/A

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

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

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

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

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

      17. lift-fma.f64 N/A

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

      18. associate-/r* N/A

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

      19. neg-mul-1 N/A

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

      20. lower-*.f64 N/A

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

   8. Applied rewrites 95.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 95.4

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

   10. Applied rewrites 95.4%

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

   if -1.00000000000000007e-138 < z < 0.75

   1. Initial program 84.0%

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

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 98.9

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

   4. Applied rewrites 98.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 97.0

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

   6. Applied rewrites 97.0%

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

   7. Taylor expanded in z around 0

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

   9. Applied rewrites 98.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z} \cdot x}{z}\\ \mathbf{elif}\;z \leq -1 \cdot 10^{-138}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \cdot y\\ \mathbf{elif}\;z \leq 0.75:\\ \;\;\;\;\frac{\left(\frac{x}{z} - x\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z \cdot z} \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}}{\frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y\_m}{1 + z} \cdot x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 5e+86)
     (/ (/ y_m (fma z z z)) (/ z x_m))
     (/ (/ (* (/ y_m (+ 1.0 z)) x_m) z) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 5e+86) {
		tmp = (y_m / fma(z, z, z)) / (z / x_m);
	} else {
		tmp = (((y_m / (1.0 + z)) * x_m) / z) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 5e+86)
		tmp = Float64(Float64(y_m / fma(z, z, z)) / Float64(z / x_m));
	else
		tmp = Float64(Float64(Float64(Float64(y_m / Float64(1.0 + z)) * x_m) / z) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

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[y$95$m, 5e+86], N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / N[(z / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999998e86

   1. Initial program 85.8%

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

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 95.5

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

   4. Applied rewrites 95.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 95.1

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

   6. Applied rewrites 95.1%

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

   if 4.9999999999999998e86 < y

   1. Initial program 71.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 90.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 90.6

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

   4. Applied rewrites 90.6%

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

Alternative 11: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{-x\_m}{z}}{z} \cdot \left(-y\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z -8.4e-17)
     (* (/ y_m (* (fma z z z) z)) x_m)
     (if (<= z 1.0)
       (* (/ (/ (- x_m) z) z) (- y_m))
       (/ (* (/ x_m z) y_m) (* z z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -8.4e-17) {
		tmp = (y_m / (fma(z, z, z) * z)) * x_m;
	} else if (z <= 1.0) {
		tmp = ((-x_m / z) / z) * -y_m;
	} else {
		tmp = ((x_m / z) * y_m) / (z * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= -8.4e-17)
		tmp = Float64(Float64(y_m / Float64(fma(z, z, z) * z)) * x_m);
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(Float64(-x_m) / z) / z) * Float64(-y_m));
	else
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / Float64(z * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, -8.4e-17], N[(N[(y$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(N[((-x$95$m) / z), $MachinePrecision] / z), $MachinePrecision] * (-y$95$m)), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -8.39999999999999968e-17

   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. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 89.2

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

   4. Applied rewrites 89.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 92.9

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

   6. Applied rewrites 92.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{z}{x}} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

      4. lift-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{z}{x}}} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)} \]

      5. clear-num N/A

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

      7. clear-num N/A

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z, z, z\right)}{y}}} \]

      8. associate-/r/ N/A

         \[\leadsto \frac{x}{z} \cdot \color{blue}{\left(\frac{1}{\mathsf{fma}\left(z, z, z\right)} \cdot y\right)} \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{x}{z} \cdot \frac{1}{\mathsf{fma}\left(z, z, z\right)}\right) \cdot y} \]

      10. frac-2neg N/A

          \[\leadsto \left(\frac{x}{z} \cdot \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\mathsf{fma}\left(z, z, z\right)\right)}}\right) \cdot y \]

      11. metadata-eval N/A

          \[\leadsto \left(\frac{x}{z} \cdot \frac{\color{blue}{-1}}{\mathsf{neg}\left(\mathsf{fma}\left(z, z, z\right)\right)}\right) \cdot y \]

      12. times-frac N/A

          \[\leadsto \color{blue}{\frac{x \cdot -1}{z \cdot \left(\mathsf{neg}\left(\mathsf{fma}\left(z, z, z\right)\right)\right)}} \cdot y \]

      13. metadata-eval N/A

          \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{-1}}}{z \cdot \left(\mathsf{neg}\left(\mathsf{fma}\left(z, z, z\right)\right)\right)} \cdot y \]

      14. div-inv N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{-1}}}{z \cdot \left(\mathsf{neg}\left(\mathsf{fma}\left(z, z, z\right)\right)\right)} \cdot y \]

      15. distribute-rgt-neg-in N/A

          \[\leadsto \frac{\frac{x}{-1}}{\color{blue}{\mathsf{neg}\left(z \cdot \mathsf{fma}\left(z, z, z\right)\right)}} \cdot y \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \frac{\frac{x}{-1}}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, z, z\right)}} \cdot y \]

      17. lift-fma.f64 N/A

          \[\leadsto \frac{\frac{x}{-1}}{\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(z \cdot z + z\right)}} \cdot y \]

      18. associate-/r* N/A

          \[\leadsto \color{blue}{\frac{x}{-1 \cdot \left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot z + z\right)\right)}} \cdot y \]

      19. neg-mul-1 N/A

          \[\leadsto \frac{x}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot z + z\right)\right)}} \cdot y \]

      20. lower-*.f64 N/A

          \[\leadsto \color{blue}{\frac{x}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot z + z\right)\right)} \cdot y} \]

   8. Applied rewrites 89.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \cdot y \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x} \]

      7. lower-/.f64 86.1

         \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z}} \cdot x \]

   10. Applied rewrites 86.1%

       \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x} \]

   if -8.39999999999999968e-17 < z < 1

   1. Initial program 84.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(x \cdot y\right)}{\mathsf{neg}\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{x \cdot y}\right)}{\mathsf{neg}\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{y \cdot x}\right)}{\mathsf{neg}\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot x}}{\mathsf{neg}\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)} \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{x}{\mathsf{neg}\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{x}{\mathsf{neg}\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}} \]

      8. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-y\right)} \cdot \frac{x}{\mathsf{neg}\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)} \]

      9. lower-/.f64 N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{x}{\mathsf{neg}\left(\left(z \cdot z\right) \cdot \left(z + 1\right)\right)}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\mathsf{neg}\left(\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\mathsf{neg}\left(\color{blue}{\left(z \cdot z\right)} \cdot \left(z + 1\right)\right)} \]

      12. associate-*l* N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\mathsf{neg}\left(\color{blue}{z \cdot \left(z \cdot \left(z + 1\right)\right)}\right)} \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      14. lower-*.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot \left(z + 1\right)\right)}} \]

      15. lower-neg.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\color{blue}{\left(-z\right)} \cdot \left(z \cdot \left(z + 1\right)\right)} \]

      16. *-commutative N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\left(-z\right) \cdot \color{blue}{\left(\left(z + 1\right) \cdot z\right)}} \]

      17. lift-+.f64 N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\left(-z\right) \cdot \left(\color{blue}{\left(z + 1\right)} \cdot z\right)} \]

      18. distribute-lft1-in N/A

          \[\leadsto \left(-y\right) \cdot \frac{x}{\left(-z\right) \cdot \color{blue}{\left(z \cdot z + z\right)}} \]

      19. lower-fma.f64 84.6

          \[\leadsto \left(-y\right) \cdot \frac{x}{\left(-z\right) \cdot \color{blue}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\left(-y\right) \cdot \frac{x}{\left(-z\right) \cdot \mathsf{fma}\left(z, z, z\right)}} \]

   5. Taylor expanded in z around 0

      \[\leadsto \left(-y\right) \cdot \color{blue}{\left(-1 \cdot \frac{x}{{z}^{2}}\right)} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \left(-y\right) \cdot \left(-1 \cdot \frac{x}{\color{blue}{z \cdot z}}\right) \]

      2. associate-/r* N/A

         \[\leadsto \left(-y\right) \cdot \left(-1 \cdot \color{blue}{\frac{\frac{x}{z}}{z}}\right) \]

      3. associate-/l* N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{-1 \cdot \frac{x}{z}}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{-1 \cdot \frac{x}{z}}{z}} \]

      5. associate-*r/ N/A

         \[\leadsto \left(-y\right) \cdot \frac{\color{blue}{\frac{-1 \cdot x}{z}}}{z} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-y\right) \cdot \frac{\color{blue}{\frac{-1 \cdot x}{z}}}{z} \]

      7. mul-1-neg N/A

         \[\leadsto \left(-y\right) \cdot \frac{\frac{\color{blue}{\mathsf{neg}\left(x\right)}}{z}}{z} \]

      8. lower-neg.f64 87.2

         \[\leadsto \left(-y\right) \cdot \frac{\frac{\color{blue}{-x}}{z}}{z} \]

   7. Applied rewrites 87.2%

      \[\leadsto \left(-y\right) \cdot \color{blue}{\frac{\frac{-x}{z}}{z}} \]

   if 1 < z

   1. Initial program 80.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot z}} \cdot \frac{y}{z + 1} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z}} \cdot \frac{y}{z + 1} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{z \cdot \left(z + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{z}} \cdot y}{z \cdot \left(z + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right) \cdot z}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right)} \cdot z} \]

      13. distribute-lft1-in N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{z \cdot z + z}} \]

      14. lower-fma.f64 91.2

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}} \]

   5. Taylor expanded in z around inf

      \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{{z}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{z \cdot z}} \]

      2. lower-*.f64 91.1

         \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{z \cdot z}} \]

   7. Applied rewrites 91.1%

      \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{z \cdot z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\frac{-x}{z}}{z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 3.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\left(\frac{z}{x\_m} \cdot z\right) \cdot \left(1 + z\right)}\\ \end{array}\right) \end{array} \]

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 3.8e+253)
     (/ (* (/ x_m z) y_m) (fma z z z))
     (/ y_m (* (* (/ z x_m) z) (+ 1.0 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 <= 3.8e+253) {
		tmp = ((x_m / z) * y_m) / fma(z, z, z);
	} else {
		tmp = y_m / (((z / x_m) * z) * (1.0 + 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 (y_m <= 3.8e+253)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / fma(z, z, z));
	else
		tmp = Float64(y_m / Float64(Float64(Float64(z / x_m) * z) * Float64(1.0 + 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[y$95$m, 3.8e+253], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(N[(z / x$95$m), $MachinePrecision] * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.79999999999999989e253

   1. Initial program 84.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot z}} \cdot \frac{y}{z + 1} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z}} \cdot \frac{y}{z + 1} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{z \cdot \left(z + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{z}} \cdot y}{z \cdot \left(z + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right) \cdot z}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right)} \cdot z} \]

      13. distribute-lft1-in N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{z \cdot z + z}} \]

      14. lower-fma.f64 94.6

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}} \]

   if 3.79999999999999989e253 < y

   1. Initial program 65.5%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot z}} \cdot \frac{y}{z + 1} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z}} \cdot \frac{y}{z + 1} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{z \cdot \left(z + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{z}} \cdot y}{z \cdot \left(z + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right) \cdot z}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right)} \cdot z} \]

      13. distribute-lft1-in N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{z \cdot z + z}} \]

      14. lower-fma.f64 66.1

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{\mathsf{fma}\left(z, z, z\right)} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right)} \cdot \frac{x}{z}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{\mathsf{fma}\left(z, z, z\right)} \cdot \color{blue}{\frac{x}{z}} \]

      6. clear-num N/A

         \[\leadsto \frac{y}{\mathsf{fma}\left(z, z, z\right)} \cdot \color{blue}{\frac{1}{\frac{z}{x}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{\frac{z}{x}}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{\frac{z}{x}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}}{\frac{z}{x}} \]

      10. lower-/.f64 79.7

          \[\leadsto \frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{\color{blue}{\frac{z}{x}}} \]

   6. Applied rewrites 79.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{\frac{z}{x}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{\frac{z}{x}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}}{\frac{z}{x}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{z}{x} \cdot \mathsf{fma}\left(z, z, z\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\frac{z}{x} \cdot \mathsf{fma}\left(z, z, z\right)}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{z}{x}} \cdot \mathsf{fma}\left(z, z, z\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{z \cdot \mathsf{fma}\left(z, z, z\right)}{x}}} \]

      7. remove-double-neg N/A

         \[\leadsto \frac{y}{\frac{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(z \cdot \mathsf{fma}\left(z, z, z\right)\right)\right)\right)}}{x}} \]

      8. distribute-lft-neg-out N/A

         \[\leadsto \frac{y}{\frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \mathsf{fma}\left(z, z, z\right)}\right)}{x}} \]

      9. lift-fma.f64 N/A

         \[\leadsto \frac{y}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(z \cdot z + z\right)}\right)}{x}} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \left(z \cdot z + z\right)\right)}{x}}} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{y}{\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\mathsf{fma}\left(z, z, z\right)}\right)}{x}} \]

      12. *-commutative N/A

          \[\leadsto \frac{y}{\frac{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(z, z, z\right) \cdot \left(\mathsf{neg}\left(z\right)\right)}\right)}{x}} \]

      13. distribute-rgt-neg-in N/A

          \[\leadsto \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(z, z, z\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right)}}{x}} \]

      14. remove-double-neg N/A

          \[\leadsto \frac{y}{\frac{\mathsf{fma}\left(z, z, z\right) \cdot \color{blue}{z}}{x}} \]

      15. lower-*.f64 72.7

          \[\leadsto \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(z, z, z\right) \cdot z}}{x}} \]

   8. Applied rewrites 72.7%

      \[\leadsto \color{blue}{\frac{y}{\frac{\mathsf{fma}\left(z, z, z\right) \cdot z}{x}}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\frac{\mathsf{fma}\left(z, z, z\right) \cdot z}{x}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{y}{\frac{\color{blue}{\mathsf{fma}\left(z, z, z\right) \cdot z}}{x}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z\right) \cdot \frac{z}{x}}} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z \cdot z + z\right)} \cdot \frac{z}{x}} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(z + 1\right) \cdot z\right)} \cdot \frac{z}{x}} \]

      6. associate-*l* N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z + 1\right) \cdot \left(z \cdot \frac{z}{x}\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(z + 1\right) \cdot \left(z \cdot \frac{z}{x}\right)}} \]

      8. +-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{\left(1 + z\right)} \cdot \left(z \cdot \frac{z}{x}\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(1 + z\right)} \cdot \left(z \cdot \frac{z}{x}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{y}{\left(1 + z\right) \cdot \color{blue}{\left(z \cdot \frac{z}{x}\right)}} \]

      11. lower-/.f64 86.6

          \[\leadsto \frac{y}{\left(1 + z\right) \cdot \left(z \cdot \color{blue}{\frac{z}{x}}\right)} \]

   10. Applied rewrites 86.6%

       \[\leadsto \frac{y}{\color{blue}{\left(1 + z\right) \cdot \left(z \cdot \frac{z}{x}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\left(\frac{z}{x} \cdot z\right) \cdot \left(1 + z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.1 \cdot 10^{-187}:\\ \;\;\;\;\frac{y\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{x\_m} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= x_m 1.1e-187)
     (/ y_m (* (/ (fma z z z) x_m) z))
     (/ (* x_m (/ y_m (fma z z z))) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.1e-187) {
		tmp = y_m / ((fma(z, z, z) / x_m) * z);
	} else {
		tmp = (x_m * (y_m / fma(z, z, z))) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.1e-187)
		tmp = Float64(y_m / Float64(Float64(fma(z, z, z) / x_m) * z));
	else
		tmp = Float64(Float64(x_m * Float64(y_m / fma(z, z, z))) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[x$95$m, 1.1e-187], N[(y$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.10000000000000004e-187

   1. Initial program 82.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(z \cdot z\right) \cdot \left(z + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{z \cdot z}} \cdot \frac{y}{z + 1} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z}} \cdot \frac{y}{z + 1} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{z \cdot \left(z + 1\right)} \]

      10. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{x}{z}} \cdot y}{z \cdot \left(z + 1\right)} \]

      11. *-commutative N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right) \cdot z}} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right)} \cdot z} \]

      13. distribute-lft1-in N/A

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{z \cdot z + z}} \]

      14. lower-fma.f64 93.8

          \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}} \]

   4. Applied rewrites 93.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{\mathsf{fma}\left(z, z, z\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{z}} \cdot y}{\mathsf{fma}\left(z, z, z\right)} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{z}}}{\mathsf{fma}\left(z, z, z\right)} \]

      5. associate-/l* N/A

         \[\leadsto \frac{\color{blue}{x \cdot \frac{y}{z}}}{\mathsf{fma}\left(z, z, z\right)} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(z, z, z\right)} \cdot \frac{y}{z}} \]

      7. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}} \]

      8. clear-num N/A

         \[\leadsto \frac{y}{z} \cdot \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}} \]

      9. lift-/.f64 N/A

         \[\leadsto \frac{y}{z} \cdot \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(z, z, z\right)}{x}}} \]

      10. times-frac N/A

          \[\leadsto \color{blue}{\frac{y \cdot 1}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{x}}} \]

      11. *-rgt-identity N/A

          \[\leadsto \frac{\color{blue}{y}}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{x}} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{x}}} \]

      13. lower-/.f64 91.1

          \[\leadsto \color{blue}{\frac{y}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{x}}} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{x}}} \]

      15. *-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\frac{\mathsf{fma}\left(z, z, z\right)}{x} \cdot z}} \]

      16. lower-*.f64 91.1

          \[\leadsto \frac{y}{\color{blue}{\frac{\mathsf{fma}\left(z, z, z\right)}{x} \cdot z}} \]

   6. Applied rewrites 91.1%

      \[\leadsto \color{blue}{\frac{y}{\frac{\mathsf{fma}\left(z, z, z\right)}{x} \cdot z}} \]

   if 1.10000000000000004e-187 < x

   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. 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 92.8

          \[\leadsto \frac{x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}}}{z} \]

   4. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 95.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \frac{\frac{x\_m}{z} \cdot y\_m}{\mathsf{fma}\left(z, z, z\right)}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (* x_s (* y_s (/ (* (/ x_m z) y_m) (fma z z z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * (((x_m / z) * y_m) / fma(z, z, z)));
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	return Float64(x_s * Float64(y_s * Float64(Float64(Float64(x_m / z) * y_m) / fma(z, z, z))))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.0%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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. lift-*.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{z \cdot z}} \cdot \frac{y}{z + 1} \]

   6. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{z}}{z}} \cdot \frac{y}{z + 1} \]

   7. frac-times N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{z \cdot \left(z + 1\right)}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{z} \cdot y}}{z \cdot \left(z + 1\right)} \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\color{blue}{\frac{x}{z}} \cdot y}{z \cdot \left(z + 1\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right) \cdot z}} \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\left(z + 1\right)} \cdot z} \]

   13. distribute-lft1-in N/A

       \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{z \cdot z + z}} \]

   14. lower-fma.f64 93.1

       \[\leadsto \frac{\frac{x}{z} \cdot y}{\color{blue}{\mathsf{fma}\left(z, z, z\right)}} \]

4. Applied rewrites 93.1%

   \[\leadsto \color{blue}{\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}} \]
5. Add Preprocessing

Alternative 15: 75.2% 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 83.0%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{x \cdot y}{\color{blue}{{z}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

   2. lower-*.f64 68.6

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]

5. Applied rewrites 68.6%

   \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x \cdot y}{z \cdot z}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot y}}{z \cdot z} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot x}}{z \cdot z} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot z}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot z}} \]

   6. lower-/.f64 71.0

      \[\leadsto y \cdot \color{blue}{\frac{x}{z \cdot z}} \]

7. Applied rewrites 71.0%

   \[\leadsto \color{blue}{y \cdot \frac{x}{z \cdot z}} \]

8. Final simplification 71.0%

   \[\leadsto \frac{x}{z \cdot z} \cdot y \]
9. Add Preprocessing

Developer Target 1: 97.4% 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 2024276 
(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))))