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

Percentage Accurate: 82.6% → 98.0%

Time: 11.3s

Alternatives: 19

Speedup: 0.8×

• 25.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 82.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	tmp = 0
	if ((z * z) * (z + 1.0)) <= 0.0:
		tmp = (x_m / z) / (z * ((z + 1.0) / y_m))
	else:
		tmp = (y_m / (z + 1.0)) * ((x_m / z) / z)
	return y_s * (x_s * tmp)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 97.4

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

   4. Applied egg-rr 97.4%

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

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

   1. Initial program 82.6%

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

      1. 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 92.4%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. neg-mul-1 N/A

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

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

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. clear-num N/A

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

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

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

      13. distribute-frac-neg2 N/A

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

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

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

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

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

      21. /-lowering-/.f64 N/A

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

      22. neg-lowering-neg.f64 93.0

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

   6. Applied egg-rr 93.0%

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

4. Final simplification 95.0%

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

Alternative 2: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;x\_m \cdot \frac{\frac{y\_m}{z}}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -1e+14)
       t_0
       (if (<= t_1 0.0)
         (* x_m (/ (/ y_m z) z))
         (if (<= t_1 1e-6) (* y_m (/ x_m (* z z))) t_0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = x_m * ((y_m / z) / z);
	} else if (t_1 <= 1e-6) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    t_1 = (z * z) * (z + 1.0d0)
    if (t_1 <= (-1d+14)) then
        tmp = t_0
    else if (t_1 <= 0.0d0) then
        tmp = x_m * ((y_m / z) / z)
    else if (t_1 <= 1d-6) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_0;
	} else if (t_1 <= 0.0) {
		tmp = x_m * ((y_m / z) / z);
	} else if (t_1 <= 1e-6) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	t_1 = (z * z) * (z + 1.0)
	tmp = 0
	if t_1 <= -1e+14:
		tmp = t_0
	elif t_1 <= 0.0:
		tmp = x_m * ((y_m / z) / z)
	elif t_1 <= 1e-6:
		tmp = y_m * (x_m / (z * z))
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+14)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = Float64(x_m * Float64(Float64(y_m / z) / z));
	elseif (t_1 <= 1e-6)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	t_1 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_1 <= -1e+14)
		tmp = t_0;
	elseif (t_1 <= 0.0)
		tmp = x_m * ((y_m / z) / z);
	elseif (t_1 <= 1e-6)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -1e+14], t$95$0, If[LessEqual[t$95$1, 0.0], N[(x$95$m * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e-6], N[(y$95$m * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e14 or 9.99999999999999955e-7 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 77.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-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 84.9

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

   5. Simplified 84.9%

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

   if -1e14 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 59.4%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 60.3

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

   5. Simplified 60.3%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 77.6

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

   7. Applied egg-rr 77.6%

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

   if 0.0 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.99999999999999955e-7

   1. Initial program 89.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-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 88.6

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

   5. Simplified 88.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/r/ N/A

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

      4. associate-/l/ N/A

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

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

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

      6. associate-/l/ N/A

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

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

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

      8. *-lowering-*.f64 84.5

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

   7. Applied egg-rr 84.5%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 0:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-6}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot \frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 5e-21)
     (/ x_m (* (fma z z z) (/ z y_m)))
     (/ (* (/ x_m z) y_m) (fma z z z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 5e-21) {
		tmp = x_m / (fma(z, z, z) * (z / y_m));
	} else {
		tmp = ((x_m / z) * y_m) / fma(z, z, z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 5e-21)
		tmp = Float64(x_m / Float64(fma(z, z, z) * Float64(z / y_m)));
	else
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / fma(z, z, z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-21], N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 4.99999999999999973e-21

   1. Initial program 86.1%

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

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

      3. un-div-inv N/A

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 89.0

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

   4. Applied egg-rr 89.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 91.3

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

   6. Applied egg-rr 91.3%

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

   if 4.99999999999999973e-21 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 62.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 85.9

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

   4. Applied egg-rr 85.9%

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

Alternative 4: 96.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot \frac{z}{y\_m}}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{\frac{x\_m}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (/ (* x_m y_m) (* (* z z) (+ z 1.0))) 5e-21)
     (/ x_m (* (fma z z z) (/ z y_m)))
     (* y_m (/ (/ x_m z) (fma z z z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z * z) * (z + 1.0))) <= 5e-21) {
		tmp = x_m / (fma(z, z, z) * (z / y_m));
	} else {
		tmp = y_m * ((x_m / z) / fma(z, z, z));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 5e-21)
		tmp = Float64(x_m / Float64(fma(z, z, z) * Float64(z / y_m)));
	else
		tmp = Float64(y_m * Float64(Float64(x_m / z) / fma(z, z, z)));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-21], N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(N[(x$95$m / z), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 4.99999999999999973e-21

   1. Initial program 86.1%

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

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

      3. un-div-inv N/A

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 89.0

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

   4. Applied egg-rr 89.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 91.3

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

   6. Applied egg-rr 91.3%

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

   if 4.99999999999999973e-21 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 62.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 64.2

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

   4. Applied egg-rr 64.2%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. accelerator-lowering-fma.f64 76.0

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

   6. Applied egg-rr 76.0%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{-21}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot \frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{x}{z}}{\mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x\_m}{\left(z \cdot z\right) \cdot \frac{z}{y\_m}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-321}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_0 -1e+14)
       (/ x_m (* (* z z) (/ z y_m)))
       (if (<= t_0 5e-321)
         (/ (* (/ x_m z) y_m) z)
         (* y_m (/ x_m (* z (fma z z z))))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (z * z) * (z + 1.0);
	double tmp;
	if (t_0 <= -1e+14) {
		tmp = x_m / ((z * z) * (z / y_m));
	} else if (t_0 <= 5e-321) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_0 <= -1e+14)
		tmp = Float64(x_m / Float64(Float64(z * z) * Float64(z / y_m)));
	elseif (t_0 <= 5e-321)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$0, -1e+14], N[(x$95$m / N[(N[(z * z), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e-321], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e14

   1. Initial program 81.7%

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

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

      3. un-div-inv N/A

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 87.7

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

   4. Applied egg-rr 87.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 93.5

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

   6. Applied egg-rr 93.5%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 92.6

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

   9. Simplified 92.6%

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

   if -1e14 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.99994e-321

   1. Initial program 59.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-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 60.7

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

   5. Simplified 60.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z}} \]
   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. *-commutative N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. /-lowering-/.f64 98.0

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

   7. Applied egg-rr 98.0%

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

   if 4.99994e-321 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.8%

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

      1. *-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 86.9

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

   4. Applied egg-rr 86.9%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{\left(z \cdot z\right) \cdot \frac{z}{y}}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 5 \cdot 10^{-321}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, z, z\right)\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x\_m \cdot \frac{y\_m}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-321}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{t\_0}\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* z (fma z z z))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -1e+14)
       (* x_m (/ y_m t_0))
       (if (<= t_1 5e-321) (/ (* (/ x_m z) y_m) z) (* y_m (/ x_m t_0))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = z * fma(z, z, z);
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = x_m * (y_m / t_0);
	} else if (t_1 <= 5e-321) {
		tmp = ((x_m / z) * y_m) / z;
	} else {
		tmp = y_m * (x_m / t_0);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(z * fma(z, z, z))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+14)
		tmp = Float64(x_m * Float64(y_m / t_0));
	elseif (t_1 <= 5e-321)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / z);
	else
		tmp = Float64(y_m * Float64(x_m / t_0));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -1e+14], N[(x$95$m * N[(y$95$m / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-321], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / z), $MachinePrecision], N[(y$95$m * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1e14

   1. Initial program 81.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 88.9

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

   4. Applied egg-rr 88.9%

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

   if -1e14 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.99994e-321

   1. Initial program 59.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-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 60.7

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

   5. Simplified 60.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z}} \]
   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. *-commutative N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

      7. /-lowering-/.f64 98.0

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

   7. Applied egg-rr 98.0%

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

   if 4.99994e-321 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.8%

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

      1. *-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 86.9

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

   4. Applied egg-rr 86.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 5 \cdot 10^{-321}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := x\_m \cdot \frac{y\_m}{z \cdot \left(z \cdot z\right)}\\ t_1 := \left(z \cdot z\right) \cdot \left(z + 1\right)\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* x_m (/ y_m (* z (* z z))))) (t_1 (* (* z z) (+ z 1.0))))
   (*
    y_s
    (*
     x_s
     (if (<= t_1 -1e+14)
       t_0
       (if (<= t_1 1e-6) (* y_m (/ x_m (* z z))) t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_0;
	} else if (t_1 <= 1e-6) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
y\_m = abs(y)
y\_s = copysign(1.0d0, y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(y_s, x_s, x_m, y_m, z)
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x_m * (y_m / (z * (z * z)))
    t_1 = (z * z) * (z + 1.0d0)
    if (t_1 <= (-1d+14)) then
        tmp = t_0
    else if (t_1 <= 1d-6) then
        tmp = y_m * (x_m / (z * z))
    else
        tmp = t_0
    end if
    code = y_s * (x_s * tmp)
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
assert x_m < y_m && y_m < z;
public static double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = x_m * (y_m / (z * (z * z)));
	double t_1 = (z * z) * (z + 1.0);
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_0;
	} else if (t_1 <= 1e-6) {
		tmp = y_m * (x_m / (z * z));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(y_s, x_s, x_m, y_m, z):
	t_0 = x_m * (y_m / (z * (z * z)))
	t_1 = (z * z) * (z + 1.0)
	tmp = 0
	if t_1 <= -1e+14:
		tmp = t_0
	elif t_1 <= 1e-6:
		tmp = y_m * (x_m / (z * z))
	else:
		tmp = t_0
	return y_s * (x_s * tmp)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(x_m * Float64(y_m / Float64(z * Float64(z * z))))
	t_1 = Float64(Float64(z * z) * Float64(z + 1.0))
	tmp = 0.0
	if (t_1 <= -1e+14)
		tmp = t_0;
	elseif (t_1 <= 1e-6)
		tmp = Float64(y_m * Float64(x_m / Float64(z * z)));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(y_s, x_s, x_m, y_m, z)
	t_0 = x_m * (y_m / (z * (z * z)));
	t_1 = (z * z) * (z + 1.0);
	tmp = 0.0;
	if (t_1 <= -1e+14)
		tmp = t_0;
	elseif (t_1 <= 1e-6)
		tmp = y_m * (x_m / (z * z));
	else
		tmp = t_0;
	end
	tmp_2 = y_s * (x_s * tmp);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(x$95$m * N[(y$95$m / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[t$95$1, -1e+14], t$95$0, If[LessEqual[t$95$1, 1e-6], 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))) < -1e14 or 9.99999999999999955e-7 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 77.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. *-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 84.9

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

   5. Simplified 84.9%

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

   if -1e14 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 9.99999999999999955e-7

   1. Initial program 78.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-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 77.6

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

   5. Simplified 77.6%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

      3. associate-/r/ N/A

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

      4. associate-/l/ N/A

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

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

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

      6. associate-/l/ N/A

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

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

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

      8. *-lowering-*.f64 75.1

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

   7. Applied egg-rr 75.1%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -1 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 10^{-6}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z \cdot \frac{z}{y\_m}}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* (* z z) (+ z 1.0)) 2e+32)
     (/ (* (/ x_m z) y_m) (fma z z z))
     (/ (/ x_m z) (* z (/ z y_m)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z * z) * (z + 1.0)) <= 2e+32) {
		tmp = ((x_m / z) * y_m) / fma(z, z, z);
	} else {
		tmp = (x_m / z) / (z * (z / y_m));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z * z) * Float64(z + 1.0)) <= 2e+32)
		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(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], 2e+32], 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))) < 2.00000000000000011e32

   1. Initial program 79.5%

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

      1. times-frac N/A

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

      2. associate-/r* N/A

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

      3. 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 93.5

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

   4. Applied egg-rr 93.5%

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

   if 2.00000000000000011e32 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 98.4

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 98.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 2 \cdot 10^{+32}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z \cdot \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 95.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (((z * z) * (z + 1.0)) <= 1e+62) {
		tmp = ((x_m / z) * y_m) / fma(z, z, z);
	} else {
		tmp = (y_m / z) * ((x_m / z) / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(z * z) * Float64(z + 1.0)) <= 1e+62)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / fma(z, z, z));
	else
		tmp = Float64(Float64(y_m / z) * Float64(Float64(x_m / z) / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.00000000000000004e62

   1. Initial program 79.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 93.5

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

   4. Applied egg-rr 93.5%

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

   if 1.00000000000000004e62 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 71.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 96.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. neg-mul-1 N/A

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

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

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. clear-num N/A

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

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

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

      13. distribute-frac-neg2 N/A

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

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

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

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

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

      21. /-lowering-/.f64 N/A

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

      22. neg-lowering-neg.f64 99.8

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

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{y}{-1 - z} \cdot \frac{\frac{x}{z}}{-z}} \]
   7. Step-by-step derivation

      1. frac-times N/A

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

      2. frac-2neg N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. times-frac N/A

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

      7. frac-2neg N/A

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

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

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

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

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \frac{\frac{x}{z}}{\color{blue}{0 - \left(-1 - z\right)}} \cdot \frac{y}{z} \]

      12. associate--r- N/A

          \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(0 - -1\right) + z}} \cdot \frac{y}{z} \]

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 99.8

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in z around inf

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

   11. Simplified 99.8%

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

4. Final simplification 94.9%

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

Alternative 10: 94.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ \begin{array}{l} t_0 := \frac{y\_m}{z} \cdot \frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}\\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 2 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x\_m \cdot y\_m \leq 4 \cdot 10^{+171}:\\ \;\;\;\;\frac{x\_m \cdot y\_m}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array}\right) \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (let* ((t_0 (* (/ y_m z) (/ x_m (fma z z z)))))
   (*
    y_s
    (*
     x_s
     (if (<= (* x_m y_m) 2e-43)
       t_0
       (if (<= (* x_m y_m) 4e+171)
         (/ (* x_m y_m) (* (* z z) (+ z 1.0)))
         t_0))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double t_0 = (y_m / z) * (x_m / fma(z, z, z));
	double tmp;
	if ((x_m * y_m) <= 2e-43) {
		tmp = t_0;
	} else if ((x_m * y_m) <= 4e+171) {
		tmp = (x_m * y_m) / ((z * z) * (z + 1.0));
	} else {
		tmp = t_0;
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	t_0 = Float64(Float64(y_m / z) * Float64(x_m / fma(z, z, z)))
	tmp = 0.0
	if (Float64(x_m * y_m) <= 2e-43)
		tmp = t_0;
	elseif (Float64(x_m * y_m) <= 4e+171)
		tmp = Float64(Float64(x_m * y_m) / Float64(Float64(z * z) * Float64(z + 1.0)));
	else
		tmp = t_0;
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := Block[{t$95$0 = N[(N[(y$95$m / z), $MachinePrecision] * N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(y$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 2e-43], t$95$0, If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 4e+171], N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 2.00000000000000015e-43 or 3.99999999999999982e171 < (*.f64 x y)

   1. Initial program 74.1%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 94.1

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

   4. Applied egg-rr 94.1%

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

   if 2.00000000000000015e-43 < (*.f64 x y) < 3.99999999999999982e171

   1. Initial program 99.6%

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

Alternative 11: 94.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.0%

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

      1. associate-*l* N/A

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

      2. 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 96.5

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

   4. Applied egg-rr 96.5%

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

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

   1. Initial program 82.6%

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

      1. *-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 86.4

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

   4. Applied egg-rr 86.4%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. accelerator-lowering-fma.f64 87.7

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

   6. Applied egg-rr 87.7%

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

4. Final simplification 91.7%

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

Alternative 12: 93.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 72.0%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      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 95.6

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

   4. Applied egg-rr 95.6%

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

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

   1. Initial program 82.6%

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

      1. *-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 86.4

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

   4. Applied egg-rr 86.4%

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

      1. associate-/r* N/A

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

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

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

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

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

      4. accelerator-lowering-fma.f64 87.7

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

   6. Applied egg-rr 87.7%

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

4. Final simplification 91.2%

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

Alternative 13: 98.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.55 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z + 1} \cdot \frac{y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= x_m 1.55e-22)
     (/ (* (/ x_m z) y_m) (fma z z z))
     (* (/ (/ x_m z) (+ z 1.0)) (/ y_m z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 1.55e-22) {
		tmp = ((x_m / z) * y_m) / fma(z, z, z);
	} else {
		tmp = ((x_m / z) / (z + 1.0)) * (y_m / z);
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 1.55e-22)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / fma(z, z, z));
	else
		tmp = Float64(Float64(Float64(x_m / z) / Float64(z + 1.0)) * Float64(y_m / z));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 1.55e-22], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / N[(z + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000006e-22

   1. Initial program 77.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 94.5

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

   4. Applied egg-rr 94.5%

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

   if 1.55000000000000006e-22 < x

   1. Initial program 77.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 92.6%

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

      1. *-commutative N/A

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

      2. clear-num N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. frac-2neg N/A

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

      6. neg-mul-1 N/A

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

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

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

      8. times-frac N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. clear-num N/A

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

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

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

      13. distribute-frac-neg2 N/A

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

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

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

      15. +-commutative N/A

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

      16. distribute-neg-in N/A

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

      17. metadata-eval N/A

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

      18. unsub-neg N/A

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

      19. --lowering--.f64 N/A

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

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

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

      21. /-lowering-/.f64 N/A

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

      22. neg-lowering-neg.f64 93.7

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

   6. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\frac{y}{-1 - z} \cdot \frac{\frac{x}{z}}{-z}} \]
   7. Step-by-step derivation

      1. frac-times N/A

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

      2. frac-2neg N/A

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

      3. *-commutative N/A

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

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

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

      5. distribute-lft-neg-in N/A

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

      6. times-frac N/A

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

      7. frac-2neg N/A

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

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

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

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

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \frac{\frac{x}{z}}{\color{blue}{0 - \left(-1 - z\right)}} \cdot \frac{y}{z} \]

      12. associate--r- N/A

          \[\leadsto \frac{\frac{x}{z}}{\color{blue}{\left(0 - -1\right) + z}} \cdot \frac{y}{z} \]

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 94.9

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

   8. Applied egg-rr 94.9%

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

Alternative 14: 97.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y\_m}{\mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot \left(\left(z + 1\right) \cdot \frac{z}{y\_m}\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= x_m 5e-25)
     (/ (* (/ x_m z) y_m) (fma z z z))
     (/ x_m (* z (* (+ z 1.0) (/ z y_m))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if (x_m <= 5e-25) {
		tmp = ((x_m / z) * y_m) / fma(z, z, z);
	} else {
		tmp = x_m / (z * ((z + 1.0) * (z / y_m)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (x_m <= 5e-25)
		tmp = Float64(Float64(Float64(x_m / z) * y_m) / fma(z, z, z));
	else
		tmp = Float64(x_m / Float64(z * Float64(Float64(z + 1.0) * Float64(z / y_m))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[x$95$m, 5e-25], N[(N[(N[(x$95$m / z), $MachinePrecision] * y$95$m), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z * N[(N[(z + 1.0), $MachinePrecision] * N[(z / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999962e-25

   1. Initial program 77.5%

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

      1. times-frac N/A

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

      2. associate-/r* N/A

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

      3. 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 94.4

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

   4. Applied egg-rr 94.4%

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

   if 4.99999999999999962e-25 < x

   1. Initial program 78.7%

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

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

      3. un-div-inv N/A

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 87.3

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

   4. Applied egg-rr 87.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. /-lowering-/.f64 90.4

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

   6. Applied egg-rr 90.4%

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

      1. distribute-lft1-in N/A

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

      2. remove-double-neg N/A

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

      3. metadata-eval N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. neg-sub0 N/A

          \[\leadsto \frac{x}{z \cdot \left(\color{blue}{\left(0 - \left(-1 - z\right)\right)} \cdot \frac{z}{y}\right)} \]

      12. associate--r- N/A

          \[\leadsto \frac{x}{z \cdot \left(\color{blue}{\left(\left(0 - -1\right) + z\right)} \cdot \frac{z}{y}\right)} \]

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. metadata-eval N/A

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

      18. /-lowering-/.f64 90.4

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

   8. Applied egg-rr 90.4%

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

Alternative 15: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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. 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 95.0%

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

5. Final simplification 95.0%

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

Alternative 16: 89.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 1.5 \cdot 10^{-175}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* x_m y_m) 1.5e-175)
     (* (/ x_m z) (/ y_m z))
     (* y_m (/ x_m (* z (fma z z z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 1.5e-175) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 1.5e-175)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 1.5e-175], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.5e-175

   1. Initial program 73.8%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-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 67.0

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

   5. Simplified 67.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z \cdot z}} \]
   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 76.3

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

   7. Applied egg-rr 76.3%

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

   if 1.5e-175 < (*.f64 x y)

   1. Initial program 84.0%

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

      1. *-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 79.7

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

   4. Applied egg-rr 79.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 1.5 \cdot 10^{-175}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ y\_s \cdot \left(x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 5 \cdot 10^{-209}:\\ \;\;\;\;x\_m \cdot \frac{\frac{y\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y\_m \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array}\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (y_s x_s x_m y_m z)
 :precision binary64
 (*
  y_s
  (*
   x_s
   (if (<= (* x_m y_m) 5e-209)
     (* x_m (/ (/ y_m z) z))
     (* y_m (/ x_m (* z (fma z z z))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
y\_m = fabs(y);
y\_s = copysign(1.0, y);
assert(x_m < y_m && y_m < z);
double code(double y_s, double x_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 5e-209) {
		tmp = x_m * ((y_m / z) / z);
	} else {
		tmp = y_m * (x_m / (z * fma(z, z, z)));
	}
	return y_s * (x_s * tmp);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
y\_m = abs(y)
y\_s = copysign(1.0, y)
x_m, y_m, z = sort([x_m, y_m, z])
function code(y_s, x_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 5e-209)
		tmp = Float64(x_m * Float64(Float64(y_m / z) / z));
	else
		tmp = Float64(y_m * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return Float64(y_s * Float64(x_s * tmp))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[y$95$s_, x$95$s_, x$95$m_, y$95$m_, z_] := N[(y$95$s * N[(x$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 5e-209], N[(x$95$m * N[(N[(y$95$m / z), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(y$95$m * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 5.0000000000000005e-209

   1. Initial program 72.9%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 65.9

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

   5. Simplified 65.9%

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

      1. associate-/r* N/A

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

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

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

      3. /-lowering-/.f64 69.2

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

   7. Applied egg-rr 69.2%

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

   if 5.0000000000000005e-209 < (*.f64 x y)

   1. Initial program 84.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. associate-*l* N/A

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

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

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

      8. distribute-lft-in N/A

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

      9. *-rgt-identity N/A

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

      10. accelerator-lowering-fma.f64 80.7

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

   4. Applied egg-rr 80.7%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-209}:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 74.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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 69.9

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

5. Simplified 69.9%

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

   1. clear-num N/A

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

   2. un-div-inv N/A

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

   3. associate-/r/ N/A

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

   4. associate-/l/ N/A

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

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

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

   6. associate-/l/ N/A

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

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

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

   8. *-lowering-*.f64 68.5

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

7. Applied egg-rr 68.5%

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

8. Final simplification 68.5%

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

Alternative 19: 68.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.9%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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 69.9

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

5. Simplified 69.9%

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

Developer Target 1: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z 249.6182814532307)
   (/ (* y (/ x z)) (+ z (* z z)))
   (/ (* (/ (/ y z) (+ 1.0 z)) x) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < 249.6182814532307d0) then
        tmp = (y * (x / z)) / (z + (z * z))
    else
        tmp = (((y / z) / (1.0d0 + z)) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < 249.6182814532307) {
		tmp = (y * (x / z)) / (z + (z * z));
	} else {
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < 249.6182814532307:
		tmp = (y * (x / z)) / (z + (z * z))
	else:
		tmp = (((y / z) / (1.0 + z)) * x) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < 249.6182814532307)
		tmp = Float64(Float64(y * Float64(x / z)) / Float64(z + Float64(z * z)));
	else
		tmp = Float64(Float64(Float64(Float64(y / z) / Float64(1.0 + z)) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < 249.6182814532307)
		tmp = (y * (x / z)) / (z + (z * z));
	else
		tmp = (((y / z) / (1.0 + z)) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, 249.6182814532307], N[(N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision] / N[(z + N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(y / z), $MachinePrecision] / N[(1.0 + z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]

Reproduce

herbie shell --seed 2024198 
(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))))