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

Percentage Accurate: 83.1% → 96.1%

Time: 7.7s

Alternatives: 13

Speedup: 0.9×

• 26.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.1% 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: 96.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

5. Final simplification 96.8%

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

Alternative 2: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{y\_m} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* x_m y_m) (* (+ z 1.0) (* z z))) 2e+25)
     (/ x_m (* (/ (fma z z z) y_m) z))
     (* (/ (/ x_m (fma z z z)) z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e+25) {
		tmp = x_m / ((fma(z, z, z) / y_m) * z);
	} else {
		tmp = ((x_m / fma(z, z, z)) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 2e+25)
		tmp = Float64(x_m / Float64(Float64(fma(z, z, z) / y_m) * z));
	else
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+25], N[(x$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / y$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 2.00000000000000018e25

   1. Initial program 92.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-/l* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 94.0

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

   4. Applied rewrites 94.0%

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

   if 2.00000000000000018e25 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 66.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 89.9

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

   4. Applied rewrites 89.9%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(z, z, z\right)}{y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 10^{-128}:\\ \;\;\;\;\frac{\frac{y\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* x_m y_m) (* (+ z 1.0) (* z z))) 1e-128)
     (* (/ (/ y_m (fma z z z)) z) x_m)
     (* (/ (/ x_m (fma z z z)) z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 1e-128) {
		tmp = ((y_m / fma(z, z, z)) / z) * x_m;
	} else {
		tmp = ((x_m / fma(z, z, z)) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 1e-128)
		tmp = Float64(Float64(Float64(y_m / fma(z, z, z)) / z) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-128], N[(N[(N[(y$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1.00000000000000005e-128

   1. Initial program 92.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. distribute-lft1-in N/A

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

      16. lower-fma.f64 94.1

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

   4. Applied rewrites 94.1%

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

   if 1.00000000000000005e-128 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. lower-fma.f64 90.5

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

   4. Applied rewrites 90.5%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* x_m y_m) (* (+ z 1.0) (* z z))) 5e+305)
     (* (/ x_m (* (fma z z z) z)) y_m)
     (* (/ (/ x_m z) z) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 5e+305) {
		tmp = (x_m / (fma(z, z, z) * z)) * y_m;
	} else {
		tmp = ((x_m / z) / z) * y_m;
	}
	return x_s * (y_s * tmp);
}

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 5e+305)
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y_m);
	else
		tmp = Float64(Float64(Float64(x_m / z) / z) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+305], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 5.00000000000000009e305

   1. Initial program 93.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 97.1

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

   4. Applied rewrites 97.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-/r* N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft-in N/A

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

      13. *-rgt-identity N/A

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. clear-num N/A

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

      17. associate-/r/ N/A

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

      18. associate-*r* N/A

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

      19. div-inv N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 95.9%

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

   if 5.00000000000000009e305 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in z around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 77.9

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

   7. Applied rewrites 77.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y\_m}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{-x\_m}{z} \cdot y\_m\\ \mathbf{else}:\\ \;\;\;\;\left(-x\_m\right) \cdot \frac{y\_m}{z}\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (/ (* x_m y_m) (* (+ z 1.0) (* z z))) 2e+247)
     (* (/ (- x_m) z) y_m)
     (* (- x_m) (/ y_m z))))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e+247) {
		tmp = (-x_m / z) * y_m;
	} else {
		tmp = -x_m * (y_m / z);
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if (((x_m * y_m) / ((z + 1.0d0) * (z * z))) <= 2d+247) then
        tmp = (-x_m / z) * y_m
    else
        tmp = -x_m * (y_m / z)
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e+247) {
		tmp = (-x_m / z) * y_m;
	} else {
		tmp = -x_m * (y_m / z);
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if ((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e+247:
		tmp = (-x_m / z) * y_m
	else:
		tmp = -x_m * (y_m / z)
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y_m) / Float64(Float64(z + 1.0) * Float64(z * z))) <= 2e+247)
		tmp = Float64(Float64(Float64(-x_m) / z) * y_m);
	else
		tmp = Float64(Float64(-x_m) * Float64(y_m / z));
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if (((x_m * y_m) / ((z + 1.0) * (z * z))) <= 2e+247)
		tmp = (-x_m / z) * y_m;
	else
		tmp = -x_m * (y_m / z);
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(N[(x$95$m * y$95$m), $MachinePrecision] / N[(N[(z + 1.0), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+247], N[(N[((-x$95$m) / z), $MachinePrecision] * y$95$m), $MachinePrecision], N[((-x$95$m) * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 1.9999999999999999e247

   1. Initial program 92.9%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right)}}{{z}^{2}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x \cdot y\right)}\right)\right)}{{z}^{2}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - -1 \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - -1 \cdot \left(x \cdot y\right)}{{z}^{2}}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right) - x \cdot y\right)}}{{z}^{2}} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(x \cdot \left(y \cdot z\right) - x \cdot y\right)\right)}}{{z}^{2}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)}\right)}{{z}^{2}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{\left(x \cdot y\right) \cdot z} + \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right)}{{z}^{2}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{-1 \cdot \left(x \cdot y\right)}\right)\right)}{{z}^{2}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{\left(x \cdot y\right) \cdot -1}\right)\right)}{{z}^{2}} \]

      11. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot y\right) \cdot \left(z + -1\right)}\right)}{{z}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(z + -1\right) \cdot \left(x \cdot y\right)}\right)}{{z}^{2}} \]

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

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z + -1\right)\right)\right) \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z + -1\right)\right)\right) \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

      15. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(-1 + z\right)}\right)\right) \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      16. distribute-neg-in N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(z\right)\right)\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\left(\color{blue}{1} + \left(\mathsf{neg}\left(z\right)\right)\right) \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      18. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\left(1 - z\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - z\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      20. *-commutative N/A

          \[\leadsto \frac{\left(1 - z\right) \cdot \color{blue}{\left(y \cdot x\right)}}{{z}^{2}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\left(1 - z\right) \cdot \color{blue}{\left(y \cdot x\right)}}{{z}^{2}} \]

      22. unpow2 N/A

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

      23. lower-*.f64 67.5

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

   5. Applied rewrites 67.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 38.1%

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

   if 1.9999999999999999e247 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 62.6%

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

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + x \cdot y}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. remove-double-neg N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right)}}{{z}^{2}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x \cdot y\right)}\right)\right)}{{z}^{2}} \]

      3. sub-neg N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - -1 \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - -1 \cdot \left(x \cdot y\right)}{{z}^{2}}} \]

      5. distribute-lft-out-- N/A

         \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right) - x \cdot y\right)}}{{z}^{2}} \]

      6. mul-1-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(x \cdot \left(y \cdot z\right) - x \cdot y\right)\right)}}{{z}^{2}} \]

      7. sub-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)}\right)}{{z}^{2}} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{\left(x \cdot y\right) \cdot z} + \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right)}{{z}^{2}} \]

      9. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{-1 \cdot \left(x \cdot y\right)}\right)\right)}{{z}^{2}} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{\left(x \cdot y\right) \cdot -1}\right)\right)}{{z}^{2}} \]

      11. distribute-lft-out N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot y\right) \cdot \left(z + -1\right)}\right)}{{z}^{2}} \]

      12. *-commutative N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(z + -1\right) \cdot \left(x \cdot y\right)}\right)}{{z}^{2}} \]

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

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z + -1\right)\right)\right) \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z + -1\right)\right)\right) \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

      15. +-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(-1 + z\right)}\right)\right) \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      16. distribute-neg-in N/A

          \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(z\right)\right)\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      17. metadata-eval N/A

          \[\leadsto \frac{\left(\color{blue}{1} + \left(\mathsf{neg}\left(z\right)\right)\right) \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      18. sub-neg N/A

          \[\leadsto \frac{\color{blue}{\left(1 - z\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      19. lower--.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(1 - z\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

      20. *-commutative N/A

          \[\leadsto \frac{\left(1 - z\right) \cdot \color{blue}{\left(y \cdot x\right)}}{{z}^{2}} \]

      21. lower-*.f64 N/A

          \[\leadsto \frac{\left(1 - z\right) \cdot \color{blue}{\left(y \cdot x\right)}}{{z}^{2}} \]

      22. unpow2 N/A

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

      23. lower-*.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 21.8%

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

   10. Applied rewrites 20.1%

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

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z + 1\right) \cdot \left(z \cdot z\right)} \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(-x\right) \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. lift-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 96.8

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

6. Applied rewrites 96.8%

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

Alternative 7: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} y\_m = \left|y\right| \\ y\_s = \mathsf{copysign}\left(1, y\right) \\ x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y_m, z] = \mathsf{sort}([x_m, y_m, z])\\ \\ x\_s \cdot \left(y\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y\_m \leq 5 \cdot 10^{-131}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\_m\\ \end{array}\right) \end{array} \]

FPCore

y\_m = (fabs.f64 y)
y\_s = (copysign.f64 #s(literal 1 binary64) y)
x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
(FPCore (x_s y_s x_m y_m z)
 :precision binary64
 (*
  x_s
  (*
   y_s
   (if (<= (* x_m y_m) 5e-131)
     (* (/ x_m z) (/ y_m z))
     (* (/ x_m (* z z)) y_m)))))

C

y\_m = fabs(y);
y\_s = copysign(1.0, y);
x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y_m && y_m < z);
double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 5e-131) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / (z * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Fortran

y\_m = abs(y)
y\_s = copysign(1.0d0, y)
x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, y_s, x_m, y_m, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: y_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x_m * y_m) <= 5d-131) then
        tmp = (x_m / z) * (y_m / z)
    else
        tmp = (x_m / (z * z)) * y_m
    end if
    code = x_s * (y_s * tmp)
end function

Java

y\_m = Math.abs(y);
y\_s = Math.copySign(1.0, y);
x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y_m && y_m < z;
public static double code(double x_s, double y_s, double x_m, double y_m, double z) {
	double tmp;
	if ((x_m * y_m) <= 5e-131) {
		tmp = (x_m / z) * (y_m / z);
	} else {
		tmp = (x_m / (z * z)) * y_m;
	}
	return x_s * (y_s * tmp);
}

Python

y\_m = math.fabs(y)
y\_s = math.copysign(1.0, y)
x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y_m, z] = sort([x_m, y_m, z])
def code(x_s, y_s, x_m, y_m, z):
	tmp = 0
	if (x_m * y_m) <= 5e-131:
		tmp = (x_m / z) * (y_m / z)
	else:
		tmp = (x_m / (z * z)) * y_m
	return x_s * (y_s * tmp)

Julia

y\_m = abs(y)
y\_s = copysign(1.0, y)
x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y_m, z = sort([x_m, y_m, z])
function code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0
	if (Float64(x_m * y_m) <= 5e-131)
		tmp = Float64(Float64(x_m / z) * Float64(y_m / z));
	else
		tmp = Float64(Float64(x_m / Float64(z * z)) * y_m);
	end
	return Float64(x_s * Float64(y_s * tmp))
end

MATLAB

y\_m = abs(y);
y\_s = sign(y) * abs(1.0);
x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y_m, z = num2cell(sort([x_m, y_m, z])){:}
function tmp_2 = code(x_s, y_s, x_m, y_m, z)
	tmp = 0.0;
	if ((x_m * y_m) <= 5e-131)
		tmp = (x_m / z) * (y_m / z);
	else
		tmp = (x_m / (z * z)) * y_m;
	end
	tmp_2 = x_s * (y_s * tmp);
end

Wolfram

y\_m = N[Abs[y], $MachinePrecision]
y\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[y]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y_m, and z should be sorted in increasing order before calling this function.
code[x$95$s_, y$95$s_, x$95$m_, y$95$m_, z_] := N[(x$95$s * N[(y$95$s * If[LessEqual[N[(x$95$m * y$95$m), $MachinePrecision], 5e-131], N[(N[(x$95$m / z), $MachinePrecision] * N[(y$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 5.0000000000000004e-131

   1. Initial program 83.7%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. unpow2 N/A

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

      2. times-frac N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 82.2

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

   5. Applied rewrites 82.2%

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

   if 5.0000000000000004e-131 < (*.f64 x y)

   1. Initial program 90.0%

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

   3. Taylor expanded in z around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 78.1

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

   7. Applied rewrites 78.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y \leq 5 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. lift-fma.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/r/ N/A

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

   12. clear-num N/A

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

   13. lift-/.f64 N/A

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

   14. associate-/r* N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. associate-/r/ N/A

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

   18. clear-num N/A

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

   19. lift-*.f64 N/A

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

   20. *-commutative N/A

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

   21. associate-/r* N/A

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

   22. lower-/.f64 N/A

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

6. Applied rewrites 96.0%

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

7. Final simplification 96.0%

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

Alternative 9: 92.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. *-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. distribute-lft1-in N/A

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

   17. lower-fma.f64 94.5

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

4. Applied rewrites 94.5%

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

Alternative 10: 94.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. distribute-lft1-in N/A

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

   9. lift-fma.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 96.8

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

6. Applied rewrites 96.8%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lift-/.f64 N/A

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

   5. clear-num N/A

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

   6. lift-/.f64 N/A

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

   7. lift-fma.f64 N/A

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

   8. *-rgt-identity N/A

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

   9. distribute-lft-in N/A

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

   10. lift-+.f64 N/A

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

   11. times-frac N/A

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

   12. *-commutative N/A

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

   13. times-frac N/A

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

   14. lower-*.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lift-+.f64 N/A

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

   17. distribute-lft-in N/A

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

   18. *-rgt-identity N/A

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

   19. lift-fma.f64 N/A

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

   20. lower-/.f64 96.2

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

8. Applied rewrites 96.2%

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

Alternative 11: 80.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. clear-num N/A

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

   7. un-div-inv N/A

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

   8. lower-/.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 96.8

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

4. Applied rewrites 96.8%

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

5. Taylor expanded in z around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 78.6

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

7. Applied rewrites 78.6%

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

Alternative 12: 75.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 73.9

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

5. Applied rewrites 73.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 77.5

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

7. Applied rewrites 77.5%

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

8. Final simplification 77.5%

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

Alternative 13: 28.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 0

   \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + x \cdot y}{{z}^{2}}} \]
4. Step-by-step derivation

   1. remove-double-neg N/A

      \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right)}}{{z}^{2}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) + \left(\mathsf{neg}\left(\color{blue}{-1 \cdot \left(x \cdot y\right)}\right)\right)}{{z}^{2}} \]

   3. sub-neg N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - -1 \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \left(y \cdot z\right)\right) - -1 \cdot \left(x \cdot y\right)}{{z}^{2}}} \]

   5. distribute-lft-out-- N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot \left(y \cdot z\right) - x \cdot y\right)}}{{z}^{2}} \]

   6. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(x \cdot \left(y \cdot z\right) - x \cdot y\right)\right)}}{{z}^{2}} \]

   7. sub-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot \left(y \cdot z\right) + \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)}\right)}{{z}^{2}} \]

   8. associate-*r* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\color{blue}{\left(x \cdot y\right) \cdot z} + \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)\right)}{{z}^{2}} \]

   9. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{-1 \cdot \left(x \cdot y\right)}\right)\right)}{{z}^{2}} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(\left(\left(x \cdot y\right) \cdot z + \color{blue}{\left(x \cdot y\right) \cdot -1}\right)\right)}{{z}^{2}} \]

   11. distribute-lft-out N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(x \cdot y\right) \cdot \left(z + -1\right)}\right)}{{z}^{2}} \]

   12. *-commutative N/A

       \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(z + -1\right) \cdot \left(x \cdot y\right)}\right)}{{z}^{2}} \]

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

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z + -1\right)\right)\right) \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\left(z + -1\right)\right)\right) \cdot \left(x \cdot y\right)}}{{z}^{2}} \]

   15. +-commutative N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{\left(-1 + z\right)}\right)\right) \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   16. distribute-neg-in N/A

       \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(z\right)\right)\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   17. metadata-eval N/A

       \[\leadsto \frac{\left(\color{blue}{1} + \left(\mathsf{neg}\left(z\right)\right)\right) \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   18. sub-neg N/A

       \[\leadsto \frac{\color{blue}{\left(1 - z\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   19. lower--.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(1 - z\right)} \cdot \left(x \cdot y\right)}{{z}^{2}} \]

   20. *-commutative N/A

       \[\leadsto \frac{\left(1 - z\right) \cdot \color{blue}{\left(y \cdot x\right)}}{{z}^{2}} \]

   21. lower-*.f64 N/A

       \[\leadsto \frac{\left(1 - z\right) \cdot \color{blue}{\left(y \cdot x\right)}}{{z}^{2}} \]

   22. unpow2 N/A

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

   23. lower-*.f64 66.1

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

5. Applied rewrites 66.1%

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

6. Taylor expanded in z around inf

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

8. Applied rewrites 34.2%

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

10. Applied rewrites 34.1%

    \[\leadsto \left(-x\right) \cdot \frac{y}{\color{blue}{z}} \]
11. Add Preprocessing

Developer Target 1: 97.2% 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 2024296 
(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))))