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

Percentage Accurate: 82.6% → 98.3%

Time: 5.7s

Alternatives: 14

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: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 5 \cdot 10^{-312}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y\_m}{1 + z} \cdot x\_m}{z}}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 5e-312)
     (* (/ y_m z) (/ x_m z))
     (/ (/ (* (/ y_m (+ 1.0 z)) x_m) z) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 5e-312) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = (((y_m / (1.0 + z)) * x_m) / z) / z;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y_m * x_m) <= 5d-312) then
        tmp = (y_m / z) * (x_m / z)
    else
        tmp = (((y_m / (1.0d0 + z)) * x_m) / z) / z
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 5e-312) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = (((y_m / (1.0 + z)) * x_m) / z) / z;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if (y_m * x_m) <= 5e-312:
		tmp = (y_m / z) * (x_m / z)
	else:
		tmp = (((y_m / (1.0 + z)) * x_m) / z) / z
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 5e-312)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(Float64(Float64(Float64(y_m / Float64(1.0 + z)) * x_m) / z) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if ((y_m * x_m) <= 5e-312)
		tmp = (y_m / z) * (x_m / z);
	else
		tmp = (((y_m / (1.0 + z)) * x_m) / z) / z;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e-312], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 5.0000000000022e-312

   1. Initial program 82.5%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{2}}} \]
   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 76.5

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

   5. Applied rewrites 76.5%

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

   if 5.0000000000022e-312 < (*.f64 x y)

   1. Initial program 87.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. 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 99.8

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

      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 99.8

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

   4. Applied rewrites 99.8%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 5 \cdot 10^{-312}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{1 + z} \cdot x}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.4% accurate, 0.2× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. inv-pow N/A

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

   14. lower-pow.f64 98.1

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Final simplification 98.1%

   \[\leadsto \left({\left(1 + z\right)}^{-1} \cdot \frac{y}{z}\right) \cdot \frac{x}{z} \]
6. Add Preprocessing

Alternative 3: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{y\_m}{z}}{z} \cdot \frac{x\_m}{z}\\ t_1 := \left(z \cdot z\right) \cdot \left(1 + z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{y\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{x\_m} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 -5e+51)
       t_0
       (if (<= t_1 2e+139) (/ 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 <= -5e+51) {
		tmp = t_0;
	} else if (t_1 <= 2e+139) {
		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 / z) / z) * Float64(x_m / z))
	t_1 = Float64(Float64(z * z) * Float64(1.0 + z))
	tmp = 0.0
	if (t_1 <= -5e+51)
		tmp = t_0;
	elseif (t_1 <= 2e+139)
		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 / z), $MachinePrecision] / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $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, -5e+51], t$95$0, If[LessEqual[t$95$1, 2e+139], 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))) < -5e51 or 2.00000000000000007e139 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 97.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.0

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

   7. Applied rewrites 98.0%

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

   if -5e51 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.00000000000000007e139

   1. Initial program 84.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. times-frac N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. lower-/.f64 N/A

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

      21. remove-double-neg N/A

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

      22. distribute-frac-neg N/A

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

   6. Applied rewrites 93.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq -5 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(1 + z\right) \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(z, z, z\right)}{x} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z}}{z} \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{x\_m}{\left(\frac{z}{y\_m} \cdot z\right) \cdot \left(1 + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{x\_m} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (* z z) (+ 1.0 z))) 5e+115)
     (/ x_m (* (* (/ z y_m) z) (+ 1.0 z)))
     (/ y_m (* (/ (fma z z z) x_m) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z * z) * (1.0 + z))) <= 5e+115) {
		tmp = x_m / (((z / y_m) * z) * (1.0 + z));
	} else {
		tmp = y_m / ((fma(z, z, z) / x_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(z * z) * Float64(1.0 + z))) <= 5e+115)
		tmp = Float64(x_m / Float64(Float64(Float64(z / y_m) * z) * Float64(1.0 + z)));
	else
		tmp = Float64(y_m / Float64(Float64(fma(z, z, z) / x_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+115], N[(x$95$m / N[(N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 5.00000000000000008e115

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

      19. lower-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. associate-*l/ N/A

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

      22. lift-/.f64 N/A

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

      23. clear-num N/A

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

   6. Applied rewrites 93.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lift-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-*.f64 94.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 94.1

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 94.1

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

   8. Applied rewrites 94.1%

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

   if 5.00000000000000008e115 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. times-frac N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. lower-/.f64 N/A

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

      21. remove-double-neg N/A

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

      22. distribute-frac-neg N/A

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

   6. Applied rewrites 85.9%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{x}{\left(\frac{z}{y} \cdot z\right) \cdot \left(1 + z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(z, z, z\right)}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{x\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{y\_m} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{x\_m} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (* z z) (+ 1.0 z))) 5e+115)
     (/ x_m (* (/ (fma z z z) y_m) z))
     (/ y_m (* (/ (fma z z z) x_m) z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z * z) * (1.0 + z))) <= 5e+115) {
		tmp = x_m / ((fma(z, z, z) / y_m) * z);
	} else {
		tmp = y_m / ((fma(z, z, z) / x_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(z * z) * Float64(1.0 + z))) <= 5e+115)
		tmp = Float64(x_m / Float64(Float64(fma(z, z, z) / y_m) * z));
	else
		tmp = Float64(y_m / Float64(Float64(fma(z, z, z) / x_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+115], N[(x$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 5.00000000000000008e115

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

      19. lower-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. associate-*l/ N/A

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

      22. lift-/.f64 N/A

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

      23. clear-num N/A

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

   6. Applied rewrites 93.7%

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

   if 5.00000000000000008e115 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. times-frac N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. lower-/.f64 N/A

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

      21. remove-double-neg N/A

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

      22. distribute-frac-neg N/A

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

   6. Applied rewrites 85.9%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 5 \cdot 10^{+115}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(z, z, z\right)}{y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{\mathsf{fma}\left(z, z, z\right)}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 50:\\ \;\;\;\;\frac{x\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{y\_m} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (* z z) (+ 1.0 z))) 50.0)
     (/ x_m (* (/ (fma z z z) y_m) z))
     (* (/ (/ x_m (fma z z z)) z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z * z) * (1.0 + z))) <= 50.0) {
		tmp = x_m / ((fma(z, z, z) / y_m) * z);
	} else {
		tmp = ((x_m / fma(z, z, z)) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(z * z) * Float64(1.0 + z))) <= 50.0)
		tmp = Float64(x_m / Float64(Float64(fma(z, z, z) / y_m) * z));
	else
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 50.0], N[(x$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 50

   1. Initial program 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

      19. lower-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. associate-*l/ N/A

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

      22. lift-/.f64 N/A

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

      23. clear-num N/A

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

   6. Applied rewrites 93.4%

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

   if 50 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 86.6

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

   4. Applied rewrites 86.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 50:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(z, z, z\right)}{y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 0:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 10^{+171}:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)} \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{1 + z}}{\frac{z}{y\_m} \cdot z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 0.0)
     (* (/ y_m z) (/ x_m z))
     (if (<= (* y_m x_m) 1e+171)
       (/ (* (/ y_m (fma z z z)) x_m) z)
       (/ (/ x_m (+ 1.0 z)) (* (/ z y_m) z)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 0.0) {
		tmp = (y_m / z) * (x_m / z);
	} else if ((y_m * x_m) <= 1e+171) {
		tmp = ((y_m / fma(z, z, z)) * x_m) / z;
	} else {
		tmp = (x_m / (1.0 + z)) / ((z / y_m) * z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 0.0)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (Float64(y_m * x_m) <= 1e+171)
		tmp = Float64(Float64(Float64(y_m / fma(z, z, z)) * x_m) / z);
	else
		tmp = Float64(Float64(x_m / Float64(1.0 + z)) / Float64(Float64(z / y_m) * z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 0.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e+171], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] / N[(N[(z / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 0.0 < (*.f64 x y) < 9.99999999999999954e170

   1. Initial program 89.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 99.3

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

   4. Applied rewrites 99.3%

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

   if 9.99999999999999954e170 < (*.f64 x y)

   1. Initial program 79.8%

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. times-frac N/A

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

      7. associate-/r* N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 96.8

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

   4. Applied rewrites 96.8%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;y \cdot x \leq 10^{+171}:\\ \;\;\;\;\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)} \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + z}}{\frac{z}{y} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 10^{+80}:\\ \;\;\;\;\frac{y\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{\frac{z}{x\_m} \cdot z}\\ \end{array}\right) \end{array} \]

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 (<= (/ (* y_m x_m) (* (* z z) (+ 1.0 z))) 1e+80)
     (* (/ 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 (((y_m * x_m) / ((z * z) * (1.0 + z))) <= 1e+80) {
		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(y_m * x_m) / Float64(Float64(z * z) * Float64(1.0 + z))) <= 1e+80)
		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[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+80], 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)))) < 1e80

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.0

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.0

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

   4. Applied rewrites 98.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

      19. lower-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. associate-*l/ N/A

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

      22. lift-/.f64 N/A

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

      23. clear-num N/A

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

   6. Applied rewrites 93.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 94.1

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 94.1

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

   8. Applied rewrites 94.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   10. Applied rewrites 91.5%

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

   if 1e80 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. times-frac N/A

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

      16. lift-*.f64 N/A

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

      17. associate-/l* N/A

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

      18. clear-num N/A

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

      19. un-div-inv N/A

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

      20. lower-/.f64 N/A

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

      21. remove-double-neg N/A

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

      22. distribute-frac-neg N/A

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

   6. Applied rewrites 85.9%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 80.6

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

   9. Applied rewrites 80.6%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z \cdot z\right) \cdot \left(1 + z\right)} \leq 10^{+80}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x} \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.1% accurate, 0.7× speedup

Math

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

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 0.0)
     (* (/ y_m z) (/ x_m z))
     (/ (* (/ y_m (fma z z z)) x_m) z)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 0.0) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = ((y_m / fma(z, z, z)) * x_m) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

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

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 0.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 0.0

   1. Initial program 83.1%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if 0.0 < (*.f64 x y)

   1. Initial program 86.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. 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 95.6

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

   4. Applied rewrites 95.6%

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

4. Final simplification 85.0%

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

Alternative 10: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(z, z, z\right) \cdot z\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x\_m}{t\_0} \cdot y\_m\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{t\_0} \cdot x\_m\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (fma z z z) z)))
   (*
    x_s
    (*
     y_s
     (if (<= z -5e-109)
       (* (/ x_m t_0) y_m)
       (if (<= z 3.4e-16) (* (/ y_m z) (/ x_m z)) (* (/ y_m t_0) x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = fma(z, z, z) * z;
	double tmp;
	if (z <= -5e-109) {
		tmp = (x_m / t_0) * y_m;
	} else if (z <= 3.4e-16) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = (y_m / t_0) * x_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(fma(z, z, z) * z)
	tmp = 0.0
	if (z <= -5e-109)
		tmp = Float64(Float64(x_m / t_0) * y_m);
	elseif (z <= 3.4e-16)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(Float64(y_m / t_0) * x_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -5e-109], N[(N[(x$95$m / t$95$0), $MachinePrecision] * y$95$m), $MachinePrecision], If[LessEqual[z, 3.4e-16], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m / t$95$0), $MachinePrecision] * x$95$m), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -5.0000000000000002e-109

   1. Initial program 85.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 96.3

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 96.3

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

   4. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. times-frac N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l/ N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 87.8

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

   6. Applied rewrites 87.8%

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

   if -5.0000000000000002e-109 < z < 3.4e-16

   1. Initial program 82.3%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if 3.4e-16 < z

   1. Initial program 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 98.2

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. clear-num N/A

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

      18. un-div-inv N/A

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

      19. lower-/.f64 N/A

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

      20. *-lft-identity N/A

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

      21. associate-*l/ N/A

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

      22. lift-/.f64 N/A

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

      23. clear-num N/A

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

   6. Applied rewrites 93.4%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-lft1-in N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l/ N/A

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

      8. frac-2neg N/A

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

      9. lift-neg.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. associate-/r* N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 93.2

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 93.2

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

   8. Applied rewrites 93.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   10. Applied rewrites 89.1%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 4 \cdot 10^{-275}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 4e-275)
     (* (/ y_m z) (/ x_m z))
     (* (/ x_m (* (fma z z z) z)) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 4e-275) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 4e-275)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 4e-275], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 3.99999999999999974e-275

   1. Initial program 82.1%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 77.6

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

   5. Applied rewrites 77.6%

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

   if 3.99999999999999974e-275 < (*.f64 x y)

   1. Initial program 88.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. div-inv N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. inv-pow N/A

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

      14. lower-pow.f64 97.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 97.5

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

   4. Applied rewrites 97.5%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-/.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow-1 N/A

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

      8. un-div-inv N/A

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

      9. associate-/r* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. times-frac N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l/ N/A

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

      18. lower-*.f64 N/A

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

      19. lower-/.f64 87.8

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

   6. Applied rewrites 87.8%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 4 \cdot 10^{-275}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 2e-83)
     (* (/ y_m z) (/ x_m z))
     (* (/ x_m (* z z)) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 2e-83) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = (x_m / (z * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y_m * x_m) <= 2d-83) then
        tmp = (y_m / z) * (x_m / z)
    else
        tmp = (x_m / (z * z)) * y_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 2e-83) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = (x_m / (z * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if (y_m * x_m) <= 2e-83:
		tmp = (y_m / z) * (x_m / z)
	else:
		tmp = (x_m / (z * z)) * y_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 2e-83)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if ((y_m * x_m) <= 2e-83)
		tmp = (y_m / z) * (x_m / z);
	else
		tmp = (x_m / (z * z)) * y_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 2e-83], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.0000000000000001e-83

   1. Initial program 83.5%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if 2.0000000000000001e-83 < (*.f64 x y)

   1. Initial program 87.9%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 70.5

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

   5. Applied rewrites 70.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]
   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 72.1

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

   7. Applied rewrites 72.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 92.1% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. distribute-lft1-in N/A

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

   17. lower-fma.f64 92.2

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

4. Applied rewrites 92.2%

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

Alternative 14: 75.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in z around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 71.5

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

5. Applied rewrites 71.5%

   \[\leadsto \frac{x \cdot y}{\color{blue}{z \cdot z}} \]
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 75.8

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

7. Applied rewrites 75.8%

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

8. Final simplification 75.8%

   \[\leadsto \frac{x}{z \cdot z} \cdot y \]
9. Add Preprocessing

Developer Target 1: 96.5% 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 2024308 
(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))))