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

Percentage Accurate: 83.4% → 98.4%

Time: 10.2s

Alternatives: 15

Speedup: 0.7×

• 25.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.4% 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: 98.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2.1 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{y\_m}{z}}{\frac{\mathsf{fma}\left(z, z, z\right)}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m \cdot \frac{y\_m}{z + 1}}{z}}{z}\\ \end{array}\right) \end{array} \]

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 2.1e+40)
     (/ (/ y_m z) (/ (fma z z z) x_m))
     (/ (/ (* x_m (/ y_m (+ z 1.0))) 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 <= 2.1e+40) {
		tmp = (y_m / z) / (fma(z, z, z) / x_m);
	} else {
		tmp = ((x_m * (y_m / (z + 1.0))) / z) / z;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2.1e+40)
		tmp = Float64(Float64(y_m / z) / Float64(fma(z, z, z) / x_m));
	else
		tmp = Float64(Float64(Float64(x_m * Float64(y_m / Float64(z + 1.0))) / z) / z);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 2.1e+40], N[(N[(y$95$m / z), $MachinePrecision] / N[(N[(z * z + z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m * N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 2.1000000000000001e40

   1. Initial program 81.1%

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

      1. lift-/.f64 N/A

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

      2. 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. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. associate-/r* N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 91.9

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

   4. Applied rewrites 91.9%

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

   if 2.1000000000000001e40 < y

   1. Initial program 83.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lower-*.f64 N/A

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

      11. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

Alternative 2: 92.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (* x_m (/ y_m (* z (* z z))))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -1000000000.0)
       t_0
       (if (<= t_1 0.0)
         (* (/ y_m z) (/ x_m z))
         (if (<= t_1 2e-5) (* y_m (/ x_m (* z z))) t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (y_m / z) * (x_m / z);
	} else if (t_1 <= 2e-5) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    t_1 = (z + 1.0d0) * (z * z)
    if (t_1 <= (-1000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = (y_m / z) * (x_m / z)
    else if (t_1 <= 2d-5) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = (y_m / z) * (x_m / z);
	} else if (t_1 <= 2e-5) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	t_1 = (z + 1.0) * (z * z)
	tmp = 0
	if t_1 <= -1000000000.0:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = (y_m / z) * (x_m / z)
	elif t_1 <= 2e-5:
		tmp = y_m * (x_m / (z * 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(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (t_1 <= 2e-5)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	t_1 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_1 <= -1000000000.0)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = (y_m / z) * (x_m / z);
	elseif (t_1 <= 2e-5)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -1000000000.0], t$95$0, If[LessEqual[t$95$1, 0.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-5], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e9 or 2.00000000000000016e-5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{3}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -1e9 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 67.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 96.7%

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

   if 0.0 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.00000000000000016e-5

   1. Initial program 91.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-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. lower-/.f64 93.6

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 91.2

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

   7. Applied rewrites 91.2%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -1000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (+ z 1.0) (* z z))) 2e-9)
     (/ x_m (* z (/ (fma z z z) y_m)))
     (/ (* y_m (/ x_m z)) (fma z z z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z + 1.0) * (z * z))) <= 2e-9) {
		tmp = x_m / (z * (fma(z, z, z) / y_m));
	} else {
		tmp = (y_m * (x_m / z)) / fma(z, z, z);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 2e-9)
		tmp = Float64(x_m / Float64(z * Float64(fma(z, z, z) / y_m)));
	else
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / fma(z, z, z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-9], N[(x$95$m / N[(z * N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2.00000000000000012e-9

   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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.2

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 90.2

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

   4. Applied rewrites 90.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 92.7

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-/.f64 92.2

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

   6. Applied rewrites 92.2%

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

   if 2.00000000000000012e-9 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 63.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 85.1

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

   4. Applied rewrites 85.1%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (+ z 1.0) (* z z))) 2e+62)
     (/ x_m (* z (/ (fma z z z) y_m)))
     (/ y_m (* z (/ z x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z + 1.0) * (z * z))) <= 2e+62) {
		tmp = x_m / (z * (fma(z, z, z) / y_m));
	} else {
		tmp = y_m / (z * (z / x_m));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 2e+62)
		tmp = Float64(x_m / Float64(z * Float64(fma(z, z, z) / y_m)));
	else
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+62], N[(x$95$m / N[(z * N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2.00000000000000007e62

   1. Initial program 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.5

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 90.5

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

   4. Applied rewrites 90.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 92.9

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-/.f64 92.5

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

   6. Applied rewrites 92.5%

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

   if 2.00000000000000007e62 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 60.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 73.9%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{z \cdot \frac{\mathsf{fma}\left(z, z, z\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1000000000:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;t\_0 \leq 10^{-214}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{t\_0}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 -1000000000.0)
       (* x_m (/ y_m (* z (fma z z z))))
       (if (<= t_0 1e-214) (* (/ y_m z) (/ x_m z)) (/ (* y_m x_m) t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -1000000000.0) {
		tmp = x_m * (y_m / (z * fma(z, z, z)));
	} else if (t_0 <= 1e-214) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = (y_m * x_m) / t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -1000000000.0)
		tmp = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))));
	elseif (t_0 <= 1e-214)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(Float64(y_m * x_m) / 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[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, -1000000000.0], N[(x$95$m * N[(y$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e-214], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * x$95$m), $MachinePrecision] / t$95$0), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e9

   1. Initial program 80.9%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.5

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 86.5

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

   4. Applied rewrites 86.5%

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

   if -1e9 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.99999999999999913e-215

   1. Initial program 72.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   7. Applied rewrites 96.6%

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

   if 9.99999999999999913e-215 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 90.4%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -1000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{-214}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, z, z\right)\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{t\_0}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* z (fma z z z))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -1000000000.0)
       (* x_m (/ y_m t_0))
       (if (<= t_1 0.0) (* (/ y_m z) (/ x_m z)) (* y_m (/ x_m t_0))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = z * fma(z, z, z);
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = x_m * (y_m / t_0);
	} else if (t_1 <= 0.0) {
		tmp = (y_m / z) * (x_m / z);
	} else {
		tmp = y_m * (x_m / t_0);
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(z * fma(z, z, z))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = Float64(x_m * Float64(y_m / t_0));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	else
		tmp = Float64(y_m * Float64(x_m / 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[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -1000000000.0], N[(x$95$m * N[(y$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e9

   1. Initial program 80.9%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.5

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 86.5

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

   4. Applied rewrites 86.5%

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

   if -1e9 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 67.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.5

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

   5. Applied rewrites 68.5%

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

   7. Applied rewrites 96.7%

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

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

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 89.5

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 89.5

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

   4. Applied rewrites 89.5%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -1000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

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 (* x_m (/ y_m (* z (* z z))))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -1000000000.0)
       t_0
       (if (<= t_1 2e-5) (* y_m (/ x_m (* z z))) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-5) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    t_1 = (z + 1.0d0) * (z * z)
    if (t_1 <= (-1000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= 2d-5) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -1000000000.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-5) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	t_1 = (z + 1.0) * (z * z)
	tmp = 0
	if t_1 <= -1000000000.0:
		tmp = t_0
	elif t_1 <= 2e-5:
		tmp = y_m * (x_m / (z * 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(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -1000000000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-5)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	t_1 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_1 <= -1000000000.0)
		tmp = t_0;
	elseif (t_1 <= 2e-5)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -1000000000.0], t$95$0, If[LessEqual[t$95$1, 2e-5], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e9 or 2.00000000000000016e-5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{3}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -1e9 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.00000000000000016e-5

   1. Initial program 80.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 81.6

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 81.6

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

   4. Applied rewrites 81.6%

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

   5. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 80.3

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

   7. Applied rewrites 80.3%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -1000000000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+62}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* y_m x_m) (* (+ z 1.0) (* z z))) 2e+62)
     (* x_m (/ y_m (* z (fma z z z))))
     (/ y_m (* z (/ z x_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z + 1.0) * (z * z))) <= 2e+62) {
		tmp = x_m * (y_m / (z * fma(z, z, z)));
	} else {
		tmp = y_m / (z * (z / x_m));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 2e+62)
		tmp = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))));
	else
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+62], N[(x$95$m * N[(y$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2.00000000000000007e62

   1. Initial program 91.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.8

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 89.8

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

   4. Applied rewrites 89.8%

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

   if 2.00000000000000007e62 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 60.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 54.6

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

   5. Applied rewrites 54.6%

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

   7. Applied rewrites 73.9%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+62}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 10^{-282}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{\frac{y\_m \cdot x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(\left(z + 1\right) \cdot \frac{z}{y\_m}\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* y_m x_m) 1e-282)
     (* (/ y_m z) (/ x_m z))
     (if (<= (* y_m x_m) 2e+227)
       (/ (/ (* y_m x_m) (fma z z z)) z)
       (/ x_m (* z (* (+ z 1.0) (/ 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) <= 1e-282) {
		tmp = (y_m / z) * (x_m / z);
	} else if ((y_m * x_m) <= 2e+227) {
		tmp = ((y_m * x_m) / fma(z, z, z)) / z;
	} else {
		tmp = x_m / (z * ((z + 1.0) * (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) <= 1e-282)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / z));
	elseif (Float64(y_m * x_m) <= 2e+227)
		tmp = Float64(Float64(Float64(y_m * x_m) / fma(z, z, z)) / z);
	else
		tmp = Float64(x_m / Float64(z * Float64(Float64(z + 1.0) * Float64(z / y_m))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e-282], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 2e+227], N[(N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(z * N[(N[(z + 1.0), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 1e-282

   1. Initial program 78.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 68.4

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

   5. Applied rewrites 68.4%

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

   7. Applied rewrites 75.7%

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

   if 1e-282 < (*.f64 x y) < 2.0000000000000002e227

   1. Initial program 88.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-lft1-in N/A

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

      12. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

   if 2.0000000000000002e227 < (*.f64 x y)

   1. Initial program 74.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 81.5

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 81.4

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

   4. Applied rewrites 81.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 87.2

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-/.f64 81.4

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

   6. Applied rewrites 81.4%

      \[\leadsto \frac{x}{\color{blue}{\frac{\mathsf{fma}\left(z, z, z\right)}{y} \cdot z}} \]
   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. associate-/l* N/A

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

      5. remove-double-neg N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-in N/A

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

      8. +-commutative N/A

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

      9. sub-neg N/A

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

      10. lift--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 87.2

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

   8. Applied rewrites 87.2%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-282}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{elif}\;y \cdot x \leq 2 \cdot 10^{+227}:\\ \;\;\;\;\frac{\frac{y \cdot x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(\left(z + 1\right) \cdot \frac{z}{y}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-160}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))))
   (*
    x_s
    (*
     y_s
     (if (<= z -2e+16)
       t_0
       (if (<= z -3e-160)
         (* y_m (/ x_m (* z (fma z z z))))
         (if (<= z 1.0) (* (/ y_m 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 = x_m * (y_m / (z * (z * z)));
	double tmp;
	if (z <= -2e+16) {
		tmp = t_0;
	} else if (z <= -3e-160) {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	} else if (z <= 1.0) {
		tmp = (y_m / 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(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	tmp = 0.0
	if (z <= -2e+16)
		tmp = t_0;
	elseif (z <= -3e-160)
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	elseif (z <= 1.0)
		tmp = Float64(Float64(y_m / z) * Float64(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[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -2e+16], t$95$0, If[LessEqual[z, -3e-160], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -2e16 or 1 < z

   1. Initial program 82.7%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{{z}^{3}}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{{z}^{3}}} \]

      3. lower-/.f64 N/A

         \[\leadsto x \cdot \color{blue}{\frac{y}{{z}^{3}}} \]

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -2e16 < z < -2.99999999999999997e-160

   1. Initial program 91.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. lower-/.f64 97.3

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 97.4

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

   4. Applied rewrites 97.4%

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

   if -2.99999999999999997e-160 < z < 1

   1. Initial program 75.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 71.7

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

   5. Applied rewrites 71.7%

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

   7. Applied rewrites 94.0%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+16}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;z \leq -3 \cdot 10^{-160}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{z \cdot z}{y\_m}}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= z -2.4e-11)
     (* x_m (/ y_m (* z (fma z z z))))
     (if (<= z 1.0) (/ y_m (* z (/ z x_m))) (/ x_m (* z (/ (* z z) y_m))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (z <= -2.4e-11) {
		tmp = x_m * (y_m / (z * fma(z, z, z)));
	} else if (z <= 1.0) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = x_m / (z * ((z * z) / y_m));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (z <= -2.4e-11)
		tmp = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))));
	elseif (z <= 1.0)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(x_m / Float64(z * Float64(Float64(z * z) / y_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[z, -2.4e-11], N[(x$95$m * N[(y$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z * N[(N[(z * z), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.4000000000000001e-11

   1. Initial program 81.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.9

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 86.9

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

   4. Applied rewrites 86.9%

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

   if -2.4000000000000001e-11 < z < 1

   1. Initial program 79.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 74.7

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

   5. Applied rewrites 74.7%

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

   7. Applied rewrites 92.5%

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

   if 1 < z

   1. Initial program 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 84.8

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 84.8

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

   4. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. distribute-lft1-in N/A

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

      7. lift-+.f64 N/A

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

      8. associate-*l/ N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 91.1

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. associate-*l/ N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-/.f64 88.0

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

   6. Applied rewrites 88.0%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 88.0

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

   9. Applied rewrites 88.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z \cdot z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	return x_s * (y_s * ((x_m / z) / (z * ((z + 1.0) / 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 * ((z + 1.0d0) / 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 * ((z + 1.0) / 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 * ((z + 1.0) / 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 / z) / Float64(z * Float64(Float64(z + 1.0) / 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 * ((z + 1.0) / y_m))));
end

Wolfram

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

Derivation

1. Initial program 81.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. clear-num N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. clear-num N/A

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

   12. inv-pow N/A

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

   13. unpow-prod-down N/A

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

   14. associate-/l* N/A

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

   15. *-lft-identity N/A

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

   16. inv-pow N/A

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

   17. clear-num N/A

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

   18. lower-/.f64 N/A

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

   19. lower-/.f64 N/A

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

   20. lower-*.f64 N/A

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

   21. lower-/.f64 95.6

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

4. Applied rewrites 95.6%

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

5. Final simplification 95.6%

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

Alternative 13: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \leq 2 \cdot 10^{+226}:\\ \;\;\;\;\frac{y\_m}{z} \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= y_m 2e+226)
     (* (/ y_m z) (/ x_m (fma z z z)))
     (* y_m (/ x_m (* z (fma z z z))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (y_m <= 2e+226) {
		tmp = (y_m / z) * (x_m / fma(z, z, z));
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (y_m <= 2e+226)
		tmp = Float64(Float64(y_m / z) * Float64(x_m / fma(z, z, z)));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[y$95$m, 2e+226], N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999992e226

   1. Initial program 81.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 91.3

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

   4. Applied rewrites 91.3%

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

   if 1.99999999999999992e226 < y

   1. Initial program 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 95.6

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. lower-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 95.6

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

   4. Applied rewrites 95.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+226}:\\ \;\;\;\;\frac{y}{z} \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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. lower-/.f64 83.3

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. lower-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. distribute-lft1-in N/A

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

   15. lower-fma.f64 83.3

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

4. Applied rewrites 83.3%

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

5. Taylor expanded in z around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 69.3

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

7. Applied rewrites 69.3%

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

8. Final simplification 69.3%

   \[\leadsto y \cdot \frac{x}{z \cdot z} \]
9. Add Preprocessing

Alternative 15: 69.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

3. Taylor expanded in z around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 65.4

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

5. Applied rewrites 65.4%

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

Developer Target 1: 96.7% 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 2024234 
(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))))