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

Percentage Accurate: 82.5% → 97.8%

Time: 10.3s

Alternatives: 14

Speedup: 0.9×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.8% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. 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. associate-/r* N/A

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

   5. frac-times N/A

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

   6. metadata-eval N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. unpow-prod-down N/A

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

   10. associate-/l* N/A

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

   11. *-lft-identity N/A

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

   12. inv-pow N/A

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

   13. clear-num N/A

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

   14. /-lowering-/.f64 N/A

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

   15. /-lowering-/.f64 N/A

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

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

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

   17. /-lowering-/.f64 N/A

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

   18. +-lowering-+.f64 96.8

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

4. Applied egg-rr 96.8%

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

5. Final simplification 96.8%

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

Alternative 2: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-248}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+45}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -50000.0)
       t_0
       (if (<= t_1 1e-248)
         (* (/ x_m z) (/ y_m z))
         (if (<= t_1 1e+45) (* y_m (/ x_m (* z (fma z z z)))) t_0)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_0;
	} else if (t_1 <= 1e-248) {
		tmp = (x_m / z) * (y_m / z);
	} else if (t_1 <= 1e+45) {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -50000.0)
		tmp = t_0;
	elseif (t_1 <= 1e-248)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	elseif (t_1 <= 1e+45)
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -50000.0], t$95$0, If[LessEqual[t$95$1, 1e-248], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+45], 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))) < -5e4 or 9.9999999999999993e44 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 -5e4 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.9999999999999998e-249

   1. Initial program 68.4%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 69.4

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

   5. Simplified 69.4%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. times-frac 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 N/A

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

      6. /-lowering-/.f64 96.0

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

   7. Applied egg-rr 96.0%

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

   if 9.9999999999999998e-249 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.9999999999999993e44

   1. Initial program 92.9%

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

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

      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 98.0

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

   4. Applied egg-rr 98.0%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -50000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{-248}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+45}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -50000:\\ \;\;\;\;\frac{x\_m \cdot \frac{y\_m}{z \cdot z}}{z}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-143}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_0 -50000.0)
       (/ (* x_m (/ y_m (* z z))) z)
       (if (<= t_0 5e-143)
         (/ y_m (* z (/ z x_m)))
         (* x_m (/ y_m (* z (fma z z z))))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -50000.0) {
		tmp = (x_m * (y_m / (z * z))) / z;
	} else if (t_0 <= 5e-143) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = x_m * (y_m / (z * fma(z, z, z)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -50000.0)
		tmp = Float64(Float64(x_m * Float64(y_m / Float64(z * z))) / z);
	elseif (t_0 <= 5e-143)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$0, -50000.0], N[(N[(x$95$m * N[(y$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 5e-143], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$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))) < -5e4

   1. Initial program 84.5%

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

      1. 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. associate-*l/ N/A

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

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

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

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

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

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

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 96.0

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 93.7

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

   7. Simplified 93.7%

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

   if -5e4 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 5.0000000000000002e-143

   1. Initial program 71.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 73.8

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

   5. Simplified 73.8%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. neg-mul-1 N/A

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

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

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

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

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

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

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

      11. distribute-neg-frac2 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-lowering-neg.f64 83.4

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

   7. Applied egg-rr 83.4%

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

   if 5.0000000000000002e-143 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 89.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 91.6

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

   4. Applied egg-rr 91.6%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -50000:\\ \;\;\;\;\frac{x \cdot \frac{y}{z \cdot z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 5 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -50000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (*
     y_s
     (if (<= t_1 -50000.0)
       t_0
       (if (<= t_1 0.4) (* y_m (/ x_m (* z z))) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.4) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    t_1 = (z + 1.0d0) * (z * z)
    if (t_1 <= (-50000.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.4d0) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -50000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.4) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	t_1 = (z + 1.0) * (z * z)
	tmp = 0
	if t_1 <= -50000.0:
		tmp = t_0
	elif t_1 <= 0.4:
		tmp = y_m * (x_m / (z * z))
	else:
		tmp = t_0
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -50000.0)
		tmp = t_0;
	elseif (t_1 <= 0.4)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	t_1 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_1 <= -50000.0)
		tmp = t_0;
	elseif (t_1 <= 0.4)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[t$95$1, -50000.0], t$95$0, If[LessEqual[t$95$1, 0.4], 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))) < -5e4 or 0.40000000000000002 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 86.2%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 87.2

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

   5. Simplified 87.2%

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

   if -5e4 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.40000000000000002

   1. Initial program 77.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 77.4

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

   5. Simplified 77.4%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. associate-*l/ N/A

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

      9. associate-/r/ N/A

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

      10. clear-num N/A

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

      11. clear-num N/A

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

      12. /-rgt-identity N/A

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

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

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

      14. clear-num N/A

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

      15. associate-/r/ N/A

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

      16. associate-*l/ N/A

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

      17. associate-/r* N/A

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

      18. associate-*l/ N/A

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

      19. *-lft-identity N/A

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

      20. /-lowering-/.f64 N/A

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

      21. *-lowering-*.f64 75.8

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

   7. Applied egg-rr 75.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -50000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0.4:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z \cdot \frac{z}{y\_m}}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (+ z 1.0) (* z z)) 1e+45)
     (/ (* (/ x_m z) y_m) (fma z z z))
     (/ (/ x_m z) (* z (/ z y_m)))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= 1e+45) {
		tmp = ((x_m / z) * y_m) / fma(z, z, z);
	} else {
		tmp = (x_m / z) / (z * (z / y_m));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(z * z)) <= 1e+45)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / fma(z, z, z));
	else
		tmp = Float64(Float64(x_m / z) / Float64(z * Float64(z / y_m)));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], 1e+45], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] / N[(z * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.9999999999999993e44

   1. Initial program 80.9%

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

      1. 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. frac-times N/A

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

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

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

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

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

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

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 95.5

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

   4. Applied egg-rr 95.5%

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

   if 9.9999999999999993e44 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.9%

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

      1. 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. associate-/r* N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. unpow-prod-down N/A

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

      10. associate-/l* N/A

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

      11. *-lft-identity N/A

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

      12. inv-pow N/A

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

      13. clear-num N/A

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

      14. /-lowering-/.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

      17. /-lowering-/.f64 N/A

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

      18. +-lowering-+.f64 97.0

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

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in z around inf

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

      1. /-lowering-/.f64 97.0

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

   7. Simplified 97.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+45}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+45}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* (+ z 1.0) (* z z)) 1e+45)
     (* y_m (/ x_m (* z (fma z z z))))
     (* x_m (/ y_m (* z (* z z))))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= 1e+45) {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	} else {
		tmp = x_m * (y_m / (z * (z * z)));
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(z * z)) <= 1e+45)
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	else
		tmp = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], 1e+45], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y$95$m / N[(z * N[(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))) < 9.9999999999999993e44

   1. Initial program 80.9%

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

      1. *-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 83.6

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

   4. Applied egg-rr 83.6%

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

   if 9.9999999999999993e44 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.9%

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

   3. 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 87.5

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

   5. Simplified 87.5%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+45}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. 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 96.8%

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

5. Final simplification 96.8%

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

Alternative 8: 92.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{y\_m}{z \cdot \frac{z}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (fma z z z))))))
   (*
    x_s
    (*
     y_s
     (if (<= z -2.15e-15)
       t_0
       (if (<= z 5.5e-67) (/ y_m (* z (/ z x_m))) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * fma(z, z, z)));
	double tmp;
	if (z <= -2.15e-15) {
		tmp = t_0;
	} else if (z <= 5.5e-67) {
		tmp = y_m / (z * (z / x_m));
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))))
	tmp = 0.0
	if (z <= -2.15e-15)
		tmp = t_0;
	elseif (z <= 5.5e-67)
		tmp = Float64(y_m / Float64(z * Float64(z / x_m)));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -2.15e-15], t$95$0, If[LessEqual[z, 5.5e-67], N[(y$95$m / N[(z * N[(z / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1499999999999998e-15 or 5.5000000000000003e-67 < z

   1. Initial program 87.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 91.7

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

   4. Applied egg-rr 91.7%

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

   if -2.1499999999999998e-15 < z < 5.5000000000000003e-67

   1. Initial program 73.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 75.6

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

   5. Simplified 75.6%

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

      1. associate-*r/ N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. frac-2neg N/A

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

      5. metadata-eval N/A

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

      6. frac-times N/A

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

      7. neg-mul-1 N/A

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

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

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

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

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

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

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

      11. distribute-neg-frac2 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. neg-lowering-neg.f64 84.6

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

   7. Applied egg-rr 84.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.15 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{y}{z \cdot \frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (fma z z z))))))
   (*
    x_s
    (*
     y_s
     (if (<= z -1.05e-15)
       t_0
       (if (<= z 2.6e-61) (/ (* (/ x_m z) y_m) z) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * fma(z, z, z)));
	double tmp;
	if (z <= -1.05e-15) {
		tmp = t_0;
	} else if (z <= 2.6e-61) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))))
	tmp = 0.0
	if (z <= -1.05e-15)
		tmp = t_0;
	elseif (z <= 2.6e-61)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -1.05e-15], t$95$0, If[LessEqual[z, 2.6e-61], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e-15 or 2.6000000000000001e-61 < z

   1. Initial program 87.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 91.7

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

   4. Applied egg-rr 91.7%

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

   if -1.0499999999999999e-15 < z < 2.6000000000000001e-61

   1. Initial program 73.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 75.6

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

   5. Simplified 75.6%

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

      1. associate-*r/ N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. *-lowering-*.f64 87.4

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

   7. Applied egg-rr 87.4%

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

      1. associate-*l/ N/A

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

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

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

      3. /-lowering-/.f64 95.4

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

   9. Applied egg-rr 95.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-15}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (fma z z z))))))
   (*
    x_s
    (*
     y_s
     (if (<= z -2.4e-16)
       t_0
       (if (<= z 1.8e-61) (* (/ x_m z) (/ y_m z)) t_0))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * fma(z, z, z)));
	double tmp;
	if (z <= -2.4e-16) {
		tmp = t_0;
	} else if (z <= 1.8e-61) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = t_0;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * fma(z, z, z))))
	tmp = 0.0
	if (z <= -2.4e-16)
		tmp = t_0;
	elseif (z <= 1.8e-61)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = t_0;
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * N[(y$95$s * If[LessEqual[z, -2.4e-16], t$95$0, If[LessEqual[z, 1.8e-61], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.40000000000000005e-16 or 1.80000000000000007e-61 < z

   1. Initial program 87.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. accelerator-lowering-fma.f64 91.7

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

   4. Applied egg-rr 91.7%

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

   if -2.40000000000000005e-16 < z < 1.80000000000000007e-61

   1. Initial program 73.9%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 75.6

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

   5. Simplified 75.6%

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. times-frac 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 N/A

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

      6. /-lowering-/.f64 96.0

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

   7. Applied egg-rr 96.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 95.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

   1. clear-num N/A

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

   2. associate-*l* N/A

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

   3. times-frac N/A

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

   4. associate-/r* N/A

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

   5. clear-num N/A

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

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

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

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

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

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

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

   9. distribute-lft-in N/A

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

   10. *-rgt-identity N/A

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

   11. accelerator-lowering-fma.f64 95.7

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

4. Applied egg-rr 95.7%

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

Alternative 12: 93.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

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

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

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

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

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

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

   7. distribute-lft-in N/A

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

   8. *-rgt-identity N/A

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

   9. accelerator-lowering-fma.f64 95.8

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

4. Applied egg-rr 95.8%

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

Alternative 13: 74.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in z around 0

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

   1. associate-/l* N/A

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 66.5

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

5. Simplified 66.5%

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

   1. clear-num N/A

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

   2. associate-*r/ N/A

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

   3. div-inv N/A

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

   4. times-frac N/A

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

   5. *-lft-identity N/A

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

   6. associate-*l/ N/A

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

   7. associate-/r* N/A

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

   8. associate-*l/ N/A

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

   9. associate-/r/ N/A

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

   10. clear-num N/A

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

   11. clear-num N/A

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

   12. /-rgt-identity N/A

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

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

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

   14. clear-num N/A

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

   15. associate-/r/ N/A

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

   16. associate-*l/ N/A

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

   17. associate-/r* N/A

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

   18. associate-*l/ N/A

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

   19. *-lft-identity N/A

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

   20. /-lowering-/.f64 N/A

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

   21. *-lowering-*.f64 65.8

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

7. Applied egg-rr 65.8%

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

8. Final simplification 65.8%

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

Alternative 14: 69.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.2%

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

3. Taylor expanded in z around 0

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

   1. associate-/l* N/A

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

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

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

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

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 66.5

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

5. Simplified 66.5%

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

Developer Target 1: 96.9% 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 2024204 
(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))))