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

Percentage Accurate: 83.2% → 97.6%

Time: 10.6s

Alternatives: 16

Speedup: 0.9×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. times-frac N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*l/ N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. clear-num N/A

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

   10. inv-pow N/A

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

   11. unpow-prod-down N/A

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

   12. times-frac N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.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. Add Preprocessing

Alternative 2: 95.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;x\_m \cdot \frac{\frac{y}{z}}{z \cdot z}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{+143}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot \frac{z \cdot z}{y}}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (if (<= t_0 -2000.0)
      (* x_m (/ (/ y z) (* z z)))
      (if (<= t_0 2e-307)
        (/ (* y (/ x_m z)) z)
        (if (<= t_0 1e+143)
          (* y (/ x_m (* z (fma z z z))))
          (/ x_m (* z (/ (* z z) y)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = x_m * ((y / z) / (z * z));
	} else if (t_0 <= 2e-307) {
		tmp = (y * (x_m / z)) / z;
	} else if (t_0 <= 1e+143) {
		tmp = y * (x_m / (z * fma(z, z, z)));
	} else {
		tmp = x_m / (z * ((z * z) / y));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = Float64(x_m * Float64(Float64(y / z) / Float64(z * z)));
	elseif (t_0 <= 2e-307)
		tmp = Float64(Float64(y * Float64(x_m / z)) / z);
	elseif (t_0 <= 1e+143)
		tmp = Float64(y * Float64(x_m / Float64(z * fma(z, z, z))));
	else
		tmp = Float64(x_m / Float64(z * Float64(Float64(z * z) / y)));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2000.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-307], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 1e+143], N[(y * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(z * N[(N[(z * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3

   1. Initial program 87.7%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.6

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

   5. Simplified 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.6

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

   7. Applied egg-rr 88.6%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 91.5

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

   9. Applied egg-rr 91.5%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999982e-307

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.0

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

   5. Simplified 77.0%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 96.2

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

   9. Applied egg-rr 96.2%

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

   if 1.99999999999999982e-307 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1e143

   1. Initial program 91.5%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 89.1

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 89.1

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

   4. Applied egg-rr 89.1%

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

   if 1e143 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 81.3%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 81.3

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

   5. Simplified 81.3%

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

      1. lift-*.f64 N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. lift-/.f64 N/A

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

      5. un-div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/l/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 95.4

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

   7. Applied egg-rr 95.4%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z \cdot z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+143}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \frac{z \cdot z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;x\_m \cdot \frac{\frac{y}{z}}{z \cdot z}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{+302}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (if (<= t_0 -2000.0)
      (* x_m (/ (/ y z) (* z z)))
      (if (<= t_0 2e-307)
        (/ (* y (/ x_m z)) z)
        (if (<= t_0 1e+302)
          (* y (/ x_m (* z (fma z z z))))
          (* (/ x_m z) (/ y (* z z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = x_m * ((y / z) / (z * z));
	} else if (t_0 <= 2e-307) {
		tmp = (y * (x_m / z)) / z;
	} else if (t_0 <= 1e+302) {
		tmp = y * (x_m / (z * fma(z, z, z)));
	} else {
		tmp = (x_m / z) * (y / (z * z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = Float64(x_m * Float64(Float64(y / z) / Float64(z * z)));
	elseif (t_0 <= 2e-307)
		tmp = Float64(Float64(y * Float64(x_m / z)) / z);
	elseif (t_0 <= 1e+302)
		tmp = Float64(y * Float64(x_m / Float64(z * fma(z, z, z))));
	else
		tmp = Float64(Float64(x_m / z) * Float64(y / Float64(z * z)));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2000.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-307], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 1e+302], N[(y * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3

   1. Initial program 87.7%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 86.6

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

   5. Simplified 86.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.6

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

   7. Applied egg-rr 88.6%

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

      1. lift-*.f64 N/A

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

      2. associate-/r* N/A

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

      3. lift-/.f64 N/A

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

      4. lower-/.f64 91.5

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

   9. Applied egg-rr 91.5%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999982e-307

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.0

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

   5. Simplified 77.0%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 96.2

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

   9. Applied egg-rr 96.2%

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

   if 1.99999999999999982e-307 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.0000000000000001e302

   1. Initial program 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 90.6

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 90.6

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

   4. Applied egg-rr 90.6%

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

   if 1.0000000000000001e302 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 82.2%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.2

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

   5. Simplified 82.2%

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

      1. lift-*.f64 N/A

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

      2. times-frac N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.3

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

   7. Applied egg-rr 94.3%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;x \cdot \frac{\frac{y}{z}}{z \cdot z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+302}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ t_1 := \frac{x\_m}{z} \cdot \frac{y}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \frac{x\_m}{z}}{z}\\ \mathbf{elif}\;t\_0 \leq 10^{+302}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))) (t_1 (* (/ x_m z) (/ y (* z z)))))
   (*
    x_s
    (if (<= t_0 -2000.0)
      t_1
      (if (<= t_0 2e-307)
        (/ (* y (/ x_m z)) z)
        (if (<= t_0 1e+302) (* y (/ x_m (* z (fma z z z)))) t_1))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double t_1 = (x_m / z) * (y / (z * z));
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = t_1;
	} else if (t_0 <= 2e-307) {
		tmp = (y * (x_m / z)) / z;
	} else if (t_0 <= 1e+302) {
		tmp = y * (x_m / (z * fma(z, z, z)));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	t_1 = Float64(Float64(x_m / z) * Float64(y / Float64(z * z)))
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = t_1;
	elseif (t_0 <= 2e-307)
		tmp = Float64(Float64(y * Float64(x_m / z)) / z);
	elseif (t_0 <= 1e+302)
		tmp = Float64(y * Float64(x_m / Float64(z * fma(z, z, z))));
	else
		tmp = t_1;
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m / z), $MachinePrecision] * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2000.0], t$95$1, If[LessEqual[t$95$0, 2e-307], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$0, 1e+302], N[(y * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3 or 1.0000000000000001e302 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.4%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 84.7

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

   5. Simplified 84.7%

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

      1. lift-*.f64 N/A

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

      2. times-frac N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 91.9

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

   7. Applied egg-rr 91.9%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999982e-307

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.0

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

   5. Simplified 77.0%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 96.2

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

   9. Applied egg-rr 96.2%

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

   if 1.99999999999999982e-307 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.0000000000000001e302

   1. Initial program 89.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 90.6

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 90.6

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

   4. Applied egg-rr 90.6%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 10^{+302}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, z, z\right)\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;x\_m \cdot \frac{y}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (fma z z z))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (if (<= t_1 -2000.0)
      (* x_m (/ y t_0))
      (if (<= t_1 2e-307) (/ (* y (/ x_m z)) z) (* y (/ x_m t_0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * fma(z, z, z);
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -2000.0) {
		tmp = x_m * (y / t_0);
	} else if (t_1 <= 2e-307) {
		tmp = (y * (x_m / z)) / z;
	} else {
		tmp = y * (x_m / t_0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(z * fma(z, z, z))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2000.0)
		tmp = Float64(x_m * Float64(y / t_0));
	elseif (t_1 <= 2e-307)
		tmp = Float64(Float64(y * Float64(x_m / z)) / z);
	else
		tmp = Float64(y * Float64(x_m / t_0));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -2000.0], N[(x$95$m * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-307], N[(N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3

   1. Initial program 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. times-frac N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. inv-pow N/A

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

      11. unpow-prod-down N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.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 98.6%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. *-lft-identity N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lift-fma.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied egg-rr 89.7%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 1.99999999999999982e-307

   1. Initial program 76.6%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 77.0

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

   5. Simplified 77.0%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

      1. associate-*r/ N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 96.2

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

   9. Applied egg-rr 96.2%

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

   if 1.99999999999999982e-307 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.1%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 89.2

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 89.2

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

   4. Applied egg-rr 89.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 2 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, z, z\right)\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;x\_m \cdot \frac{y}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x\_m \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (fma z z z))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (if (<= t_1 -2000.0)
      (* x_m (/ y t_0))
      (if (<= t_1 0.0) (/ (* x_m (/ y z)) z) (* y (/ x_m t_0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * fma(z, z, z);
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -2000.0) {
		tmp = x_m * (y / t_0);
	} else if (t_1 <= 0.0) {
		tmp = (x_m * (y / z)) / z;
	} else {
		tmp = y * (x_m / t_0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(z * fma(z, z, z))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2000.0)
		tmp = Float64(x_m * Float64(y / t_0));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x_m * Float64(y / z)) / z);
	else
		tmp = Float64(y * Float64(x_m / t_0));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -2000.0], N[(x$95$m * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3

   1. Initial program 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. times-frac N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. inv-pow N/A

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

      11. unpow-prod-down N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.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 98.6%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. *-lft-identity N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lift-fma.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied egg-rr 89.7%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 76.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.8

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

   5. Simplified 76.8%

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

      1. associate-/r* N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l/ N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.2

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 88.2

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

   4. Applied egg-rr 88.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0:\\ \;\;\;\;\frac{x \cdot \frac{y}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := z \cdot \mathsf{fma}\left(z, z, z\right)\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;x\_m \cdot \frac{y}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (fma z z z))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (if (<= t_1 -2000.0)
      (* x_m (/ y t_0))
      (if (<= t_1 0.0) (* (/ x_m z) (/ y z)) (* y (/ x_m t_0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * fma(z, z, z);
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -2000.0) {
		tmp = x_m * (y / t_0);
	} else if (t_1 <= 0.0) {
		tmp = (x_m / z) * (y / z);
	} else {
		tmp = y * (x_m / t_0);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(z * fma(z, z, z))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2000.0)
		tmp = Float64(x_m * Float64(y / t_0));
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x_m / z) * Float64(y / z));
	else
		tmp = Float64(y * Float64(x_m / t_0));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -2000.0], N[(x$95$m * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3

   1. Initial program 87.7%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. times-frac N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. associate-*l/ N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

      10. inv-pow N/A

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

      11. unpow-prod-down N/A

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

      12. times-frac N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.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 98.6%

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

      1. *-lft-identity N/A

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

      2. lift-+.f64 N/A

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

      3. *-lft-identity N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lift-fma.f64 N/A

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

      15. associate-/l/ N/A

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

      16. lift-*.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-*r/ N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

   6. Applied egg-rr 89.7%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 76.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.8

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

   5. Simplified 76.8%

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

      1. lift-*.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.2

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 88.2

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

   4. Applied egg-rr 88.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;x\_m \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (if (<= t_0 -2000.0)
      (* x_m (/ y (* z (* z z))))
      (if (<= t_0 0.0)
        (* (/ x_m z) (/ y z))
        (* y (/ x_m (* z (fma z z z)))))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = x_m * (y / (z * (z * z)));
	} else if (t_0 <= 0.0) {
		tmp = (x_m / z) * (y / z);
	} else {
		tmp = y * (x_m / (z * fma(z, z, z)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = Float64(x_m * Float64(y / Float64(z * Float64(z * z))));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(x_m / z) * Float64(y / z));
	else
		tmp = Float64(y * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2000.0], N[(x$95$m * N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(x$95$m / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / N[(z * N[(z * z + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3

   1. Initial program 87.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.6

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

   5. Simplified 88.6%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.0

   1. Initial program 76.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 76.8

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

   5. Simplified 76.8%

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

      1. lift-*.f64 N/A

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

      2. associate-*r/ N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. times-frac N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

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

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.2

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 88.2

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

   4. Applied egg-rr 88.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := z \cdot \left(z \cdot z\right)\\ t_1 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;x\_m \cdot \frac{y}{t\_0}\\ \mathbf{elif}\;t\_1 \leq 0.2:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{t\_0}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* z (* z z))) (t_1 (* (+ z 1.0) (* z z))))
   (*
    x_s
    (if (<= t_1 -2000.0)
      (* x_m (/ y t_0))
      (if (<= t_1 0.2) (* y (/ x_m (* z z))) (* y (/ x_m t_0)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * (z * z);
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -2000.0) {
		tmp = x_m * (y / t_0);
	} else if (t_1 <= 0.2) {
		tmp = y * (x_m / (z * z));
	} else {
		tmp = y * (x_m / t_0);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = z * (z * z)
    t_1 = (z + 1.0d0) * (z * z)
    if (t_1 <= (-2000.0d0)) then
        tmp = x_m * (y / t_0)
    else if (t_1 <= 0.2d0) then
        tmp = y * (x_m / (z * z))
    else
        tmp = y * (x_m / t_0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = z * (z * z);
	double t_1 = (z + 1.0) * (z * z);
	double tmp;
	if (t_1 <= -2000.0) {
		tmp = x_m * (y / t_0);
	} else if (t_1 <= 0.2) {
		tmp = y * (x_m / (z * z));
	} else {
		tmp = y * (x_m / t_0);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = z * (z * z)
	t_1 = (z + 1.0) * (z * z)
	tmp = 0
	if t_1 <= -2000.0:
		tmp = x_m * (y / t_0)
	elif t_1 <= 0.2:
		tmp = y * (x_m / (z * z))
	else:
		tmp = y * (x_m / t_0)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(z * Float64(z * z))
	t_1 = Float64(Float64(z + 1.0) * Float64(z * z))
	tmp = 0.0
	if (t_1 <= -2000.0)
		tmp = Float64(x_m * Float64(y / t_0));
	elseif (t_1 <= 0.2)
		tmp = Float64(y * Float64(x_m / Float64(z * z)));
	else
		tmp = Float64(y * Float64(x_m / t_0));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = z * (z * z);
	t_1 = (z + 1.0) * (z * z);
	tmp = 0.0;
	if (t_1 <= -2000.0)
		tmp = x_m * (y / t_0);
	elseif (t_1 <= 0.2)
		tmp = y * (x_m / (z * z));
	else
		tmp = y * (x_m / t_0);
	end
	tmp_2 = 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -2000.0], N[(x$95$m * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.2], N[(y * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / t$95$0), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3

   1. Initial program 87.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.6

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

   5. Simplified 88.6%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 84.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.6

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

   5. Simplified 85.6%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 80.5

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

   7. Applied egg-rr 80.5%

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

   if 0.20000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 83.8%

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

   3. Taylor expanded in z around inf

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

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 82.3

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

   5. Simplified 82.3%

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

      1. lift-*.f64 N/A

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

      2. times-frac N/A

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

      3. clear-num N/A

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

      4. clear-num N/A

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

      5. frac-times N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. associate-/r/ N/A

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

      9. associate-*l/ N/A

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

      10. lift-*.f64 N/A

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

      11. clear-num N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 89.2

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

   7. Applied egg-rr 89.2%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0.2:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 84.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \left(z + 1\right) \cdot \left(z \cdot z\right)\\ t_1 := x\_m \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -2000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.2:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (+ z 1.0) (* z z))) (t_1 (* x_m (/ y (* z (* z z))))))
   (*
    x_s
    (if (<= t_0 -2000.0) t_1 (if (<= t_0 0.2) (* y (/ x_m (* z z))) t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double t_1 = x_m * (y / (z * (z * z)));
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.2) {
		tmp = y * (x_m / (z * z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (z + 1.0d0) * (z * z)
    t_1 = x_m * (y / (z * (z * z)))
    if (t_0 <= (-2000.0d0)) then
        tmp = t_1
    else if (t_0 <= 0.2d0) then
        tmp = y * (x_m / (z * z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (z + 1.0) * (z * z);
	double t_1 = x_m * (y / (z * (z * z)));
	double tmp;
	if (t_0 <= -2000.0) {
		tmp = t_1;
	} else if (t_0 <= 0.2) {
		tmp = y * (x_m / (z * z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	t_0 = (z + 1.0) * (z * z)
	t_1 = x_m * (y / (z * (z * z)))
	tmp = 0
	if t_0 <= -2000.0:
		tmp = t_1
	elif t_0 <= 0.2:
		tmp = y * (x_m / (z * z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(z + 1.0) * Float64(z * z))
	t_1 = Float64(x_m * Float64(y / Float64(z * Float64(z * z))))
	tmp = 0.0
	if (t_0 <= -2000.0)
		tmp = t_1;
	elseif (t_0 <= 0.2)
		tmp = Float64(y * Float64(x_m / Float64(z * z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (z + 1.0) * (z * z);
	t_1 = x_m * (y / (z * (z * z)));
	tmp = 0.0;
	if (t_0 <= -2000.0)
		tmp = t_1;
	elseif (t_0 <= 0.2)
		tmp = y * (x_m / (z * z));
	else
		tmp = t_1;
	end
	tmp_2 = 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -2000.0], t$95$1, If[LessEqual[t$95$0, 0.2], N[(y * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -2e3 or 0.20000000000000001 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 85.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 88.9

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

   5. Simplified 88.9%

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

   if -2e3 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 0.20000000000000001

   1. Initial program 84.7%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 85.6

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

   5. Simplified 85.6%

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

      1. lift-*.f64 N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. associate-/r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 80.5

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

   7. Applied egg-rr 80.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2000:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{elif}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq 0.2:\\ \;\;\;\;y \cdot \frac{x}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+79}:\\ \;\;\;\;x\_m \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (+ z 1.0) (* z z)) -2e+79)
    (* x_m (/ y (* z (* z z))))
    (* y (/ x_m (* z (fma z z z)))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((z + 1.0) * (z * z)) <= -2e+79) {
		tmp = x_m * (y / (z * (z * z)));
	} else {
		tmp = y * (x_m / (z * fma(z, z, z)));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(z + 1.0) * Float64(z * z)) <= -2e+79)
		tmp = Float64(x_m * Float64(y / Float64(z * Float64(z * z))));
	else
		tmp = Float64(y * Float64(x_m / Float64(z * fma(z, z, z))));
	end
	return 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision], -2e+79], N[(x$95$m * N[(y / N[(z * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x$95$m / N[(z * 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))) < -1.99999999999999993e79

   1. Initial program 87.1%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. cube-mult N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 89.2

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

   5. Simplified 89.2%

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

   if -1.99999999999999993e79 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 85.7

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-+.f64 N/A

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

      17. distribute-lft1-in N/A

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

      18. lower-fma.f64 85.7

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

   4. Applied egg-rr 85.7%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z + 1\right) \cdot \left(z \cdot z\right) \leq -2 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \frac{y}{z \cdot \left(z \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z \cdot \mathsf{fma}\left(z, z, z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 94.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. times-frac N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. distribute-lft1-in N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-/.f64 95.0

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

4. Applied egg-rr 95.0%

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

5. Final simplification 95.0%

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

Alternative 13: 93.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-+.f64 N/A

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

   14. distribute-lft1-in N/A

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

   15. lower-fma.f64 94.8

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

4. Applied egg-rr 94.8%

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

Alternative 14: 74.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (y * (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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(y * N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in z around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 76.1

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

5. Simplified 76.1%

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

   1. lift-*.f64 N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. associate-/r/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l/ N/A

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

   7. lift-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. lift-*.f64 N/A

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

   12. lower-/.f64 74.1

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

7. Applied egg-rr 74.1%

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

8. Final simplification 74.1%

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

Alternative 15: 70.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * (x_m * (y / (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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(x$95$m * N[(y / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in z around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 76.1

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

5. Simplified 76.1%

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

Alternative 16: 24.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * ((x_m * -y) / 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]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(N[(x$95$m * (-y)), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.3%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. times-frac N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. associate-*l/ N/A

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

   7. clear-num N/A

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

   8. inv-pow N/A

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

   9. clear-num N/A

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

   10. inv-pow N/A

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

   11. unpow-prod-down N/A

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

   12. times-frac N/A

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

   13. lift-*.f64 N/A

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

   14. lower-/.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. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + x \cdot y}{{z}^{2}}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + x \cdot y}{{z}^{2}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{y \cdot x}}{{z}^{2}} \]

   3. cancel-sign-sub N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - \left(\mathsf{neg}\left(y\right)\right) \cdot x}}{{z}^{2}} \]

   4. *-commutative N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\left(y \cdot z\right) \cdot x\right)} - \left(\mathsf{neg}\left(y\right)\right) \cdot x}{{z}^{2}} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot \left(y \cdot z\right)\right) \cdot x} - \left(\mathsf{neg}\left(y\right)\right) \cdot x}{{z}^{2}} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\left(-1 \cdot \left(y \cdot z\right)\right) \cdot x - \color{blue}{\left(-1 \cdot y\right)} \cdot x}{{z}^{2}} \]

   7. distribute-rgt-out-- N/A

      \[\leadsto \frac{\color{blue}{x \cdot \left(-1 \cdot \left(y \cdot z\right) - -1 \cdot y\right)}}{{z}^{2}} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{x \cdot \left(-1 \cdot \left(y \cdot z\right) - -1 \cdot y\right)}}{{z}^{2}} \]

   9. sub-neg N/A

      \[\leadsto \frac{x \cdot \color{blue}{\left(-1 \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(-1 \cdot y\right)\right)\right)}}{{z}^{2}} \]

   10. mul-1-neg N/A

       \[\leadsto \frac{x \cdot \left(-1 \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right)\right)\right)}{{z}^{2}} \]

   11. remove-double-neg N/A

       \[\leadsto \frac{x \cdot \left(-1 \cdot \left(y \cdot z\right) + \color{blue}{y}\right)}{{z}^{2}} \]

   12. +-commutative N/A

       \[\leadsto \frac{x \cdot \color{blue}{\left(y + -1 \cdot \left(y \cdot z\right)\right)}}{{z}^{2}} \]

   13. mul-1-neg N/A

       \[\leadsto \frac{x \cdot \left(y + \color{blue}{\left(\mathsf{neg}\left(y \cdot z\right)\right)}\right)}{{z}^{2}} \]

   14. unsub-neg N/A

       \[\leadsto \frac{x \cdot \color{blue}{\left(y - y \cdot z\right)}}{{z}^{2}} \]

   15. lower--.f64 N/A

       \[\leadsto \frac{x \cdot \color{blue}{\left(y - y \cdot z\right)}}{{z}^{2}} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{x \cdot \left(y - \color{blue}{y \cdot z}\right)}{{z}^{2}} \]

   17. unpow2 N/A

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

   18. lower-*.f64 55.3

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

7. Simplified 55.3%

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

8. Taylor expanded in z around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-*.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 24.5

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

10. Simplified 24.5%

    \[\leadsto \color{blue}{\frac{x \cdot \left(-y\right)}{z}} \]
11. Add Preprocessing

Developer Target 1: 96.3% 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 2024208 
(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))))