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

Percentage Accurate: 83.2% → 98.0%

Time: 10.6s

Alternatives: 15

Speedup: 0.9×

• 25.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.2% 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: 98.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{\frac{x\_m}{z} \cdot \frac{y\_m}{z}}{z}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (/ (* (/ x_m z) (/ y_m z)) z)) (t_1 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -2e+44)
       t_0
       (if (<= t_1 2e+47) (/ (* y_m (/ x_m (fma z z z))) z) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = ((x_m / z) * (y_m / z)) / z;
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -2e+44) {
		tmp = t_0;
	} else if (t_1 <= 2e+47) {
		tmp = (y_m * (x_m / fma(z, z, z))) / z;
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(Float64(x_m / z) * Float64(y_m / z)) / z)
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2e+44)
		tmp = t_0;
	elseif (t_1 <= 2e+47)
		tmp = Float64(Float64(y_m * Float64(x_m / fma(z, z, z))) / z);
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -2e+44], t$95$0, If[LessEqual[t$95$1, 2e+47], N[(N[(y$95$m * N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2.0000000000000002e44 or 2.0000000000000001e47 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.6%

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

      1. times-frac N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. clear-num N/A

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

      7. inv-pow N/A

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

      8. unpow-prod-down N/A

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

      9. times-frac N/A

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

      10. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 99.2

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

   7. Simplified 99.2%

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

   if -2.0000000000000002e44 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.0000000000000001e47

   1. Initial program 86.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 95.3

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

   4. Applied egg-rr 95.3%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+44}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{+47}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot \frac{y}{z}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;\frac{x\_m}{\left(z \cdot z\right) \cdot \frac{z}{y\_m}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-281}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+49}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z \cdot z}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 -500.0)
       (/ x_m (* (* z z) (/ z y_m)))
       (if (<= t_0 5e-281)
         (/ (* y_m (/ x_m z)) z)
         (if (<= t_0 5e+49)
           (* y_m (/ x_m (* z (fma z z z))))
           (* (/ x_m z) (/ y_m (* z z))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = x_m / ((z * z) * (z / y_m));
	} else if (t_0 <= 5e-281) {
		tmp = (y_m * (x_m / z)) / z;
	} else if (t_0 <= 5e+49) {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	} else {
		tmp = (x_m / z) * (y_m / (z * z));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = Float64(x_m / Float64(Float64(z * z) * Float64(z / y_m)));
	elseif (t_0 <= 5e-281)
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / z);
	elseif (t_0 <= 5e+49)
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	else
		tmp = Float64(Float64(x_m / z) * Float64(y_m / Float64(z * z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -500.0], N[(x$95$m / N[(N[(z * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-281], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e+49], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -500

   1. Initial program 86.5%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 86.2

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

   5. Simplified 86.2%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 93.0

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

   7. Applied egg-rr 93.0%

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

   if -500 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.9999999999999998e-281

   1. Initial program 77.6%

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

      1. times-frac N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. clear-num N/A

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

      7. inv-pow N/A

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

      8. unpow-prod-down N/A

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

      9. times-frac N/A

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

      10. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 97.1%

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

   if 4.9999999999999998e-281 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000004e49

   1. Initial program 96.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 91.4

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

   4. Applied egg-rr 91.4%

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

   if 5.0000000000000004e49 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 74.0%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 74.0

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

   5. Simplified 74.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 91.5

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

   7. Applied egg-rr 91.5%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -500:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot \frac{z}{y}}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-281}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{x\_m}{z} \cdot \frac{y\_m}{z \cdot z}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-281}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+49}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ x_m z) (/ y_m (* z z)))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -500.0)
       t_0
       (if (<= t_1 5e-281)
         (/ (* y_m (/ x_m z)) z)
         (if (<= t_1 5e+49) (* y_m (/ x_m (* z (fma z z z)))) t_0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (x_m / z) * (y_m / (z * z));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 5e-281) {
		tmp = (y_m * (x_m / z)) / z;
	} else if (t_1 <= 5e+49) {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(x_m / z) * Float64(y_m / Float64(z * z)))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 5e-281)
		tmp = Float64(Float64(y_m * Float64(x_m / z)) / z);
	elseif (t_1 <= 5e+49)
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -500.0], t$95$0, If[LessEqual[t$95$1, 5e-281], N[(N[(y$95$m * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+49], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $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))) < -500 or 5.0000000000000004e49 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 80.8%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 80.6

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

   5. Simplified 80.6%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 92.5

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

   7. Applied egg-rr 92.5%

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

   if -500 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.9999999999999998e-281

   1. Initial program 77.6%

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

      1. times-frac N/A

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

      2. associate-/r* N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. clear-num N/A

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

      7. inv-pow N/A

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

      8. unpow-prod-down N/A

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

      9. times-frac N/A

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

      10. /-lowering-/.f64 N/A

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

   4. Applied egg-rr 97.1%

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

   5. Taylor expanded in z around 0

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

   7. Simplified 97.1%

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

   if 4.9999999999999998e-281 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000004e49

   1. Initial program 96.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 91.4

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

   4. Applied egg-rr 91.4%

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

4. Final simplification 93.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -500:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-281}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{+49}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 10^{-265}:\\ \;\;\;\;\frac{x\_m}{\left(z \cdot z\right) \cdot \frac{z + 1}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* y_m x_m) (* (+ z 1.0) (* z z))) 1e-265)
     (/ x_m (* (* z z) (/ (+ z 1.0) y_m)))
     (/ (* y_m (/ x_m (fma z z z))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z + 1.0) * (z * z))) <= 1e-265) {
		tmp = x_m / ((z * z) * ((z + 1.0) / y_m));
	} else {
		tmp = (y_m * (x_m / fma(z, z, z))) / z;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 1e-265)
		tmp = Float64(x_m / Float64(Float64(z * z) * Float64(Float64(z + 1.0) / y_m)));
	else
		tmp = Float64(Float64(y_m * Float64(x_m / fma(z, z, z))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-265], N[(x$95$m / N[(N[(z * z), $MachinePrecision] * N[(N[(z + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 9.99999999999999985e-266

   1. Initial program 88.6%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. *-lowering-*.f64 92.0

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

   4. Applied egg-rr 92.0%

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

      1. *-lft-identity 92.0

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

   6. Applied egg-rr 92.0%

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

   if 9.99999999999999985e-266 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 73.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 92.0

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

   4. Applied egg-rr 92.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 10^{-265}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot \frac{z + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y\_m \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* y_m x_m) (* (+ z 1.0) (* z z))) 2e+93)
     (* (/ x_m z) (/ y_m (fma z z z)))
     (/ (* y_m (/ x_m (fma z z z))) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((y_m * x_m) / ((z + 1.0) * (z * z))) <= 2e+93) {
		tmp = (x_m / z) * (y_m / fma(z, z, z));
	} else {
		tmp = (y_m * (x_m / fma(z, z, z))) / z;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_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+93)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / fma(z, z, z)));
	else
		tmp = Float64(Float64(y_m * Float64(x_m / fma(z, z, z))) / z);
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$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+93], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$95$m * N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $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)))) < 2.00000000000000009e93

   1. Initial program 89.8%

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

      1. associate-*l* N/A

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

      2. times-frac N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. /-lowering-/.f64 94.9

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

   4. Applied egg-rr 94.9%

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

   if 2.00000000000000009e93 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 64.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*l/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 89.6

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

   4. Applied egg-rr 89.6%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 -500.0)
       (* x_m (/ y_m (* z (* z z))))
       (if (<= t_0 5e-165)
         (/ y_m (* z (/ z x_m)))
         (* y_m (/ x_m (* z (fma z z z))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = x_m * (y_m / (z * (z * z)));
	} else if (t_0 <= 5e-165) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))));
	elseif (t_0 <= 5e-165)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -500.0], N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-165], N[(y$95$m / N[(z * N[(z / x$95$m), $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 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -500

   1. Initial program 86.5%

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

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

      3. /-lowering-/.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. *-lowering-*.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. *-lowering-*.f64 89.0

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

   5. Simplified 89.0%

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

   if -500 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.99999999999999981e-165

   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

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 81.6

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

   5. Simplified 81.6%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 81.2

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

   7. Applied egg-rr 81.2%

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

      1. associate-*l/ N/A

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

      2. frac-times N/A

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

      3. associate-/r/ N/A

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

      4. /-lowering-/.f64 N/A

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

      5. div-inv N/A

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

      6. clear-num N/A

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

      7. *-lowering-*.f64 N/A

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

      8. /-lowering-/.f64 88.3

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

   9. Applied egg-rr 88.3%

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

   if 4.99999999999999981e-165 < (*.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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 88.1

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

   4. Applied egg-rr 88.1%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -500:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-165}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -500:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 -500.0)
       (* x_m (/ y_m (* z (* z z))))
       (if (<= t_0 0.0)
         (* (/ x_m z) (/ y_m z))
         (* y_m (/ x_m (* z (fma z z z))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -500.0) {
		tmp = x_m * (y_m / (z * (z * z)));
	} else if (t_0 <= 0.0) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -500.0)
		tmp = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -500.0], N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $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 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -500

   1. Initial program 86.5%

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

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

      3. /-lowering-/.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. *-lowering-*.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. *-lowering-*.f64 89.0

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

   5. Simplified 89.0%

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

   if -500 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 75.6%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 75.6

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

   5. Simplified 75.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 99.8

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

   7. Applied egg-rr 99.8%

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

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

   1. Initial program 85.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 87.4

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

   4. Applied egg-rr 87.4%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -500:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_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))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -500.0)
       t_0
       (if (<= t_1 2e-5) (* y_m (/ x_m (* z z))) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_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 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-5) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_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 <= (-500.0d0)) 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 = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_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 <= -500.0) {
		tmp = t_0;
	} else if (t_1 <= 2e-5) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_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 <= -500.0:
		tmp = t_0
	elif t_1 <= 2e-5:
		tmp = y_m * (x_m / (z * z))
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_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 <= -500.0)
		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(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_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 <= -500.0)
		tmp = t_0;
	elseif (t_1 <= 2e-5)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$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[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -500.0], 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))) < -500 or 2.00000000000000016e-5 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 81.3%

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

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

      3. /-lowering-/.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. *-lowering-*.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. *-lowering-*.f64 86.3

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

   5. Simplified 86.3%

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

   if -500 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 2.00000000000000016e-5

   1. Initial program 85.9%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 83.2

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

   5. Simplified 83.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 81.1

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

   7. Applied egg-rr 81.1%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -500:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-5}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 10^{-252}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 10^{+255}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \left(x\_m \cdot \frac{1}{z \cdot \mathsf{fma}\left(z, z, z\right)}\right)\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m x_m) 1e-252)
     (* (/ x_m z) (/ y_m z))
     (if (<= (* y_m x_m) 1e+255)
       (/ (* y_m x_m) (* (+ z 1.0) (* z z)))
       (* y_m (* x_m (/ 1.0 (* z (fma z z z))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 1e-252) {
		tmp = (x_m / z) * (y_m / z);
	} else if ((y_m * x_m) <= 1e+255) {
		tmp = (y_m * x_m) / ((z + 1.0) * (z * z));
	} else {
		tmp = y_m * (x_m * (1.0 / (z * fma(z, z, z))));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 1e-252)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (Float64(y_m * x_m) <= 1e+255)
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z)));
	else
		tmp = Float64(y_m * Float64(x_m * Float64(1.0 / Float64(z * fma(z, z, z)))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e-252], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e+255], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m * N[(1.0 / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x y) < 9.99999999999999943e-253

   1. Initial program 81.0%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 73.1

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

   7. Applied egg-rr 73.1%

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

   if 9.99999999999999943e-253 < (*.f64 x y) < 9.99999999999999988e254

   1. Initial program 92.0%

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

   if 9.99999999999999988e254 < (*.f64 x y)

   1. Initial program 65.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 81.0

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

   4. Applied egg-rr 81.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. accelerator-lowering-fma.f64 81.0

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

   6. Applied egg-rr 81.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-252}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;y \cdot x \leq 10^{+255}:\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot \frac{1}{z \cdot \mathsf{fma}\left(z, z, z\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 10^{-252}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* y_m x_m) 1e-252)
     (* (/ x_m z) (/ y_m z))
     (if (<= (* y_m x_m) 5e+272)
       (/ (* y_m x_m) (* (+ z 1.0) (* z z)))
       (* y_m (/ x_m (* z (fma z z z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((y_m * x_m) <= 1e-252) {
		tmp = (x_m / z) * (y_m / z);
	} else if ((y_m * x_m) <= 5e+272) {
		tmp = (y_m * x_m) / ((z + 1.0) * (z * z));
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(y_m * x_m) <= 1e-252)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (Float64(y_m * x_m) <= 5e+272)
		tmp = Float64(Float64(y_m * x_m) / Float64(Float64(z + 1.0) * Float64(z * z)));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e-252], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 5e+272], N[(N[(y$95$m * x$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(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 3 regimes

2. if (*.f64 x y) < 9.99999999999999943e-253

   1. Initial program 81.0%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 73.1

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

   7. Applied egg-rr 73.1%

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

   if 9.99999999999999943e-253 < (*.f64 x y) < 4.99999999999999973e272

   1. Initial program 92.0%

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

   if 4.99999999999999973e272 < (*.f64 x y)

   1. Initial program 65.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 81.0

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

   4. Applied egg-rr 81.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-252}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;y \cdot x \leq 5 \cdot 10^{+272}:\\ \;\;\;\;\frac{y \cdot x}{\left(z + 1\right) \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, z, z\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;y\_m \cdot x\_m \leq 10^{-252}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;y\_m \cdot x\_m \leq 10^{+255}:\\ \;\;\;\;\frac{y\_m \cdot x\_m}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{t\_0}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* z (fma z z z))))
   (*
    y_s
    (*
     x_s
     (if (<= (* y_m x_m) 1e-252)
       (* (/ x_m z) (/ y_m z))
       (if (<= (* y_m x_m) 1e+255)
         (/ (* y_m x_m) t_0)
         (* y_m (/ x_m t_0))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = z * fma(z, z, z);
	double tmp;
	if ((y_m * x_m) <= 1e-252) {
		tmp = (x_m / z) * (y_m / z);
	} else if ((y_m * x_m) <= 1e+255) {
		tmp = (y_m * x_m) / t_0;
	} else {
		tmp = y_m * (x_m / t_0);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(z * fma(z, z, z))
	tmp = 0.0
	if (Float64(y_m * x_m) <= 1e-252)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (Float64(y_m * x_m) <= 1e+255)
		tmp = Float64(Float64(y_m * x_m) / t_0);
	else
		tmp = Float64(y_m * Float64(x_m / t_0));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e-252], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y$95$m * x$95$m), $MachinePrecision], 1e+255], N[(N[(y$95$m * x$95$m), $MachinePrecision] / t$95$0), $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 x y) < 9.99999999999999943e-253

   1. Initial program 81.0%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 68.6

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

   5. Simplified 68.6%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. /-lowering-/.f64 73.1

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

   7. Applied egg-rr 73.1%

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

   if 9.99999999999999943e-253 < (*.f64 x y) < 9.99999999999999988e254

   1. Initial program 92.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. *-rgt-identity N/A

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

      6. accelerator-lowering-fma.f64 92.0

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

   4. Applied egg-rr 92.0%

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

   if 9.99999999999999988e254 < (*.f64 x y)

   1. Initial program 65.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 81.0

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

   4. Applied egg-rr 81.0%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x \leq 10^{-252}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;y \cdot x \leq 10^{+255}:\\ \;\;\;\;\frac{y \cdot x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+44}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* (+ z 1.0) (* z z)) -2e+44)
     (* x_m (/ y_m (* z (* z z))))
     (* y_m (/ x_m (* z (fma z z z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= -2e+44) {
		tmp = x_m * (y_m / (z * (z * z)));
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(z * z)) <= -2e+44)
		tmp = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], -2e+44], N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $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 (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2.0000000000000002e44

   1. Initial program 86.1%

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

   3. Taylor expanded in z around 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. *-lowering-*.f64 N/A

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

      3. /-lowering-/.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. *-lowering-*.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. *-lowering-*.f64 88.9

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

   5. Simplified 88.9%

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

   if -2.0000000000000002e44 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-lowering-*.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 84.4

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

   4. Applied egg-rr 84.4%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+44}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(N[(y$95$m / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

   1. times-frac N/A

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

   2. associate-/r* N/A

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

   3. associate-*l/ N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. clear-num N/A

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

   7. inv-pow N/A

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

   8. unpow-prod-down N/A

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

   9. times-frac N/A

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

   10. /-lowering-/.f64 N/A

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

4. Applied egg-rr 97.3%

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

Alternative 14: 95.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

   1. associate-*l* N/A

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

   2. times-frac N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. /-lowering-/.f64 93.6

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

4. Applied egg-rr 93.6%

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

5. Final simplification 93.6%

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

Alternative 15: 75.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, 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[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.5%

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

3. Taylor expanded in z around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 70.8

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

5. Simplified 70.8%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lowering-*.f64 71.9

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

7. Applied egg-rr 71.9%

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

8. Final simplification 71.9%

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

Developer Target 1: 96.8% 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 2024196 
(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))))