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

Percentage Accurate: 83.2% → 96.0%

Time: 8.8s

Alternatives: 15

Speedup: 0.9×

• 25.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 83.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -1 \cdot 10^{+191}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{z} \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (* z z) (+ z 1.0)) -1e+191)
    (/ (* (/ (/ y z) z) x_m) z)
    (/ (* (/ x_m z) y) (fma z z z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((z * z) * (z + 1.0)) <= -1e+191) {
		tmp = (((y / z) / z) * x_m) / z;
	} else {
		tmp = ((x_m / z) * y) / fma(z, z, z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(z * z) * Float64(z + 1.0)) <= -1e+191)
		tmp = Float64(Float64(Float64(Float64(y / z) / z) * x_m) / z);
	else
		tmp = Float64(Float64(Float64(x_m / z) * y) / fma(z, z, z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], -1e+191], N[(N[(N[(N[(y / z), $MachinePrecision] / z), $MachinePrecision] * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -1.00000000000000007e191

   1. Initial program 82.4%

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

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 98.2

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. lift-fma.f64 N/A

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

      19. *-rgt-identity N/A

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

      20. distribute-lft-in N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. associate-/l/ N/A

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

      24. lift-/.f64 N/A

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

      25. lower-*.f64 N/A

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

   6. Applied rewrites 92.8%

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

   7. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.3

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

   9. Applied rewrites 99.3%

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

   if -1.00000000000000007e191 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 87.0%

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

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 97.4

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

   4. Applied rewrites 97.4%

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

Alternative 2: 93.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{-42}:\\ \;\;\;\;\frac{x\_m}{\frac{\mathsf{fma}\left(z, z, z\right)}{y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m y) (* (* z z) (+ z 1.0))) 1e-42)
    (/ x_m (* (/ (fma z z z) y) z))
    (* (/ (/ x_m (fma z z z)) z) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * y) / ((z * z) * (z + 1.0))) <= 1e-42) {
		tmp = x_m / ((fma(z, z, z) / y) * z);
	} else {
		tmp = ((x_m / fma(z, z, z)) / z) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 1e-42)
		tmp = Float64(x_m / Float64(Float64(fma(z, z, z) / y) * z));
	else
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-42], N[(x$95$m / N[(N[(N[(z * z + z), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y), $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)))) < 1.00000000000000004e-42

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      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 94.8

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

   4. Applied rewrites 94.8%

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

   if 1.00000000000000004e-42 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 74.6%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 84.5

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

   4. Applied rewrites 84.5%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 10^{-42}:\\ \;\;\;\;\frac{x}{\frac{\mathsf{fma}\left(z, z, z\right)}{y} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x\_m} \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m y) (* (* z z) (+ z 1.0))) 5e+109)
    (* (/ y (* (fma z z z) z)) x_m)
    (/ y (* (/ z x_m) z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * y) / ((z * z) * (z + 1.0))) <= 5e+109) {
		tmp = (y / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = y / ((z / x_m) * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 5e+109)
		tmp = Float64(Float64(y / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(y / Float64(Float64(z / x_m) * z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+109], N[(N[(y / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(y / N[(N[(z / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))) < 5.0000000000000001e109

   1. Initial program 91.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 97.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 97.8

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

   4. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. lift-fma.f64 N/A

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

      19. *-rgt-identity N/A

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

      20. distribute-lft-in N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. associate-/l/ N/A

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

      24. lift-/.f64 N/A

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

      25. lower-*.f64 N/A

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

   6. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. associate-*l* N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 91.6

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

   8. Applied rewrites 91.6%

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

   if 5.0000000000000001e109 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 71.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 87.7

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

   4. Applied rewrites 87.7%

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

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l* N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-lft1-in N/A

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

      13. lift-fma.f64 N/A

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

      14. clear-num N/A

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

      15. lift-/.f64 N/A

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

      16. associate-*l/ N/A

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

      17. *-lft-identity N/A

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

      18. lower-/.f64 N/A

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

   6. Applied rewrites 82.3%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 76.4

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

   9. Applied rewrites 76.4%

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

Alternative 4: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \leq 5 \cdot 10^{+109}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (/ (* x_m y) (* (* z z) (+ z 1.0))) 5e+109)
    (* (/ y (* (fma z z z) z)) x_m)
    (* (/ (/ x_m z) z) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((x_m * y) / ((z * z) * (z + 1.0))) <= 5e+109) {
		tmp = (y / (fma(z, z, z) * z)) * x_m;
	} else {
		tmp = ((x_m / z) / z) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(x_m * y) / Float64(Float64(z * z) * Float64(z + 1.0))) <= 5e+109)
		tmp = Float64(Float64(y / Float64(fma(z, z, z) * z)) * x_m);
	else
		tmp = Float64(Float64(Float64(x_m / z) / z) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(x$95$m * y), $MachinePrecision] / N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+109], N[(N[(y / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision] * y), $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)))) < 5.0000000000000001e109

   1. Initial program 91.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 97.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 97.8

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

   4. Applied rewrites 97.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. lift-fma.f64 N/A

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

      19. *-rgt-identity N/A

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

      20. distribute-lft-in N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. associate-/l/ N/A

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

      24. lift-/.f64 N/A

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

      25. lower-*.f64 N/A

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

   6. Applied rewrites 97.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. associate-*l* N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 91.6

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

   8. Applied rewrites 91.6%

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

   if 5.0000000000000001e109 < (/.f64 (*.f64 x y) (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))))

   1. Initial program 71.8%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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 82.3

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

   4. Applied rewrites 82.3%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 76.3

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

   7. Applied rewrites 76.3%

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

Alternative 5: 95.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq -5 \cdot 10^{+93}:\\ \;\;\;\;\frac{\frac{y}{z}}{\frac{z}{x\_m} \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{z} \cdot y}{\mathsf{fma}\left(z, z, z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (* z z) (+ z 1.0)) -5e+93)
    (/ (/ y z) (* (/ z x_m) z))
    (/ (* (/ x_m z) y) (fma z z z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((z * z) * (z + 1.0)) <= -5e+93) {
		tmp = (y / z) / ((z / x_m) * z);
	} else {
		tmp = ((x_m / z) * y) / fma(z, z, z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(z * z) * Float64(z + 1.0)) <= -5e+93)
		tmp = Float64(Float64(y / z) / Float64(Float64(z / x_m) * z));
	else
		tmp = Float64(Float64(Float64(x_m / z) * y) / fma(z, z, z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], -5e+93], N[(N[(y / z), $MachinePrecision] / N[(N[(z / x$95$m), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision] / N[(z * z + z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < -5.0000000000000001e93

   1. Initial program 83.3%

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

      1. lift-/.f64 N/A

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

      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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -5.0000000000000001e93 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 86.8%

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

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. distribute-lft1-in N/A

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

      14. lower-fma.f64 97.4

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

   4. Applied rewrites 97.4%

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

4. Final simplification 98.0%

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

Alternative 6: 32.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. Initial program 90.4%

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

   3. Taylor expanded in z around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv 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 69.8

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

   5. Applied rewrites 69.8%

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

   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 39.2%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 37.0%

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

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

   1. Initial program 78.1%

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

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

      2. metadata-eval N/A

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

      3. cancel-sign-sub-inv 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 62.5

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

   5. Applied rewrites 62.5%

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

   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 15.1%

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

Alternative 7: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \leq 5 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{x\_m}{\mathsf{fma}\left(z, z, z\right)}}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot x\_m}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* (* z z) (+ z 1.0)) 5e+30)
    (* (/ (/ x_m (fma z z z)) z) y)
    (/ (* (/ y z) x_m) (* z z)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (((z * z) * (z + 1.0)) <= 5e+30) {
		tmp = ((x_m / fma(z, z, z)) / z) * y;
	} else {
		tmp = ((y / z) * x_m) / (z * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(Float64(z * z) * Float64(z + 1.0)) <= 5e+30)
		tmp = Float64(Float64(Float64(x_m / fma(z, z, z)) / z) * y);
	else
		tmp = Float64(Float64(Float64(y / z) * x_m) / Float64(z * z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(N[(z * z), $MachinePrecision] * N[(z + 1.0), $MachinePrecision]), $MachinePrecision], 5e+30], N[(N[(N[(x$95$m / N[(z * z + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64))) < 4.9999999999999998e30

   1. Initial program 86.0%

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
   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. *-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 88.8

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

   4. Applied rewrites 88.8%

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

   if 4.9999999999999998e30 < (*.f64 (*.f64 z z) (+.f64 z #s(literal 1 binary64)))

   1. Initial program 86.1%

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

      1. 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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.9

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

   7. Applied rewrites 97.9%

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

Alternative 8: 91.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot x\_m}{z \cdot z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= z -5e-16)
    (* (/ y (* (fma z z z) z)) x_m)
    (if (<= z 3.2e-8) (* (/ (/ x_m z) z) y) (/ (* (/ y z) x_m) (* z z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -5e-16) {
		tmp = (y / (fma(z, z, z) * z)) * x_m;
	} else if (z <= 3.2e-8) {
		tmp = ((x_m / z) / z) * y;
	} else {
		tmp = ((y / z) * x_m) / (z * z);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -5e-16)
		tmp = Float64(Float64(y / Float64(fma(z, z, z) * z)) * x_m);
	elseif (z <= 3.2e-8)
		tmp = Float64(Float64(Float64(x_m / z) / z) * y);
	else
		tmp = Float64(Float64(Float64(y / z) * x_m) / Float64(z * z));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -5e-16], N[(N[(y / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[z, 3.2e-8], N[(N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5.0000000000000004e-16

   1. Initial program 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 98.4

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. lift-fma.f64 N/A

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

      19. *-rgt-identity N/A

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

      20. distribute-lft-in N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. associate-/l/ N/A

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

      24. lift-/.f64 N/A

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

      25. lower-*.f64 N/A

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

   6. Applied rewrites 93.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. associate-*l* N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 91.0

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

   8. Applied rewrites 91.0%

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

   if -5.0000000000000004e-16 < z < 3.2000000000000002e-8

   1. Initial program 86.1%

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

      1. 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 87.4

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

   4. Applied rewrites 87.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 87.2

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

   7. Applied rewrites 87.2%

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

   if 3.2000000000000002e-8 < z

   1. Initial program 86.9%

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

      1. lift-/.f64 N/A

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

      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. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. distribute-lft1-in N/A

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

      15. lower-fma.f64 98.0

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

   4. Applied rewrites 98.0%

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

   5. Taylor expanded in z around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.5

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

   7. Applied rewrites 97.5%

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

Alternative 9: 95.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

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

4. Applied rewrites 93.5%

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

5. Final simplification 93.5%

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

Alternative 10: 91.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ \begin{array}{l} t_0 := \frac{y}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (* (/ y (* (fma z z z) z)) x_m)))
   (*
    x_s
    (if (<= z -3.4e-34) t_0 (if (<= z 1.2e-14) (* (/ x_m z) (/ y z)) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (y / (fma(z, z, z) * z)) * x_m;
	double tmp;
	if (z <= -3.4e-34) {
		tmp = t_0;
	} else if (z <= 1.2e-14) {
		tmp = (x_m / z) * (y / z);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(y / Float64(fma(z, z, z) * z)) * x_m)
	tmp = 0.0
	if (z <= -3.4e-34)
		tmp = t_0;
	elseif (z <= 1.2e-14)
		tmp = Float64(Float64(x_m / z) * Float64(y / z));
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(y / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -3.4e-34], t$95$0, If[LessEqual[z, 1.2e-14], N[(N[(x$95$m / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4000000000000001e-34 or 1.2e-14 < z

   1. Initial program 86.4%

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

      1. 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

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. distribute-lft1-in N/A

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

      9. lift-fma.f64 N/A

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

      10. div-inv N/A

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

      11. associate-*r* N/A

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

      12. div-inv N/A

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

      13. clear-num N/A

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

      14. lift-/.f64 N/A

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

      15. div-inv N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/r/ N/A

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

      18. lift-fma.f64 N/A

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

      19. *-rgt-identity N/A

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

      20. distribute-lft-in N/A

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

      21. +-commutative N/A

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

      22. lift-+.f64 N/A

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

      23. associate-/l/ N/A

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

      24. lift-/.f64 N/A

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

      25. lower-*.f64 N/A

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

   6. Applied rewrites 94.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-lft-in N/A

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

      9. *-lft-identity N/A

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

      10. distribute-rgt-in N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. associate-*l* N/A

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

      17. distribute-lft-in N/A

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

      18. *-rgt-identity N/A

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

      19. lift-fma.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

   if -3.4000000000000001e-34 < z < 1.2e-14

   1. Initial program 85.6%

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

   3. Taylor expanded in z around 0

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

      1. 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 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 11: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y \leq 2 \cdot 10^{-144}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\mathsf{fma}\left(z, z, z\right) \cdot z} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m y) 2e-144)
    (* (/ x_m z) (/ y z))
    (* (/ x_m (* (fma z z z) z)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((x_m * y) <= 2e-144) {
		tmp = (x_m / z) * (y / z);
	} else {
		tmp = (x_m / (fma(z, z, z) * z)) * y;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(x_m * y) <= 2e-144)
		tmp = Float64(Float64(x_m / z) * Float64(y / z));
	else
		tmp = Float64(Float64(x_m / Float64(fma(z, z, z) * z)) * y);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(x$95$m * y), $MachinePrecision], 2e-144], N[(N[(x$95$m / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(N[(z * z + z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.9999999999999999e-144

   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 79.9

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

   5. Applied rewrites 79.9%

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

   if 1.9999999999999999e-144 < (*.f64 x y)

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      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 92.6

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

   4. Applied rewrites 92.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. associate-/l/ N/A

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

      5. lift-fma.f64 N/A

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

      6. *-lft-identity N/A

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

      7. distribute-rgt-in N/A

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

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

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

      9. associate-*l* N/A

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

      10. frac-2neg N/A

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

      11. lower-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. distribute-lft-in N/A

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

      14. *-rgt-identity N/A

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

      15. lift-*.f64 N/A

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

      16. distribute-rgt-in N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 87.7

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

   6. Applied rewrites 87.7%

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

Alternative 12: 77.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z] = \mathsf{sort}([x_m, y, z])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \cdot y \leq 2 \cdot 10^{-144}:\\ \;\;\;\;\frac{x\_m}{z} \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z \cdot z} \cdot y\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= (* x_m y) 2e-144) (* (/ x_m z) (/ y z)) (* (/ x_m (* z z)) y))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
assert(x_m < y && y < z);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((x_m * y) <= 2e-144) {
		tmp = (x_m / z) * (y / z);
	} else {
		tmp = (x_m / (z * z)) * y;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x_m * y) <= 2d-144) then
        tmp = (x_m / z) * (y / z)
    else
        tmp = (x_m / (z * z)) * y
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
assert x_m < y && y < z;
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if ((x_m * y) <= 2e-144) {
		tmp = (x_m / z) * (y / z);
	} else {
		tmp = (x_m / (z * z)) * y;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
[x_m, y, z] = sort([x_m, y, z])
def code(x_s, x_m, y, z):
	tmp = 0
	if (x_m * y) <= 2e-144:
		tmp = (x_m / z) * (y / z)
	else:
		tmp = (x_m / (z * z)) * y
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
x_m, y, z = sort([x_m, y, z])
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (Float64(x_m * y) <= 2e-144)
		tmp = Float64(Float64(x_m / z) * Float64(y / z));
	else
		tmp = Float64(Float64(x_m / Float64(z * z)) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
x_m, y, z = num2cell(sort([x_m, y, z])){:}
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if ((x_m * y) <= 2e-144)
		tmp = (x_m / z) * (y / z);
	else
		tmp = (x_m / (z * z)) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, and z should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[N[(x$95$m * y), $MachinePrecision], 2e-144], N[(N[(x$95$m / z), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < 1.9999999999999999e-144

   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 79.9

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

   5. Applied rewrites 79.9%

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

   if 1.9999999999999999e-144 < (*.f64 x y)

   1. Initial program 91.1%

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

      1. lift-/.f64 N/A

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

      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 92.6

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

   4. Applied rewrites 92.6%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 74.6

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

   7. Applied rewrites 74.6%

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

Alternative 13: 94.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

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

   6. associate-/r* N/A

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

   7. frac-times N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. distribute-lft1-in N/A

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

   14. lower-fma.f64 96.0

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

4. Applied rewrites 96.0%

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

Alternative 14: 73.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

   \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)} \]
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. *-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 89.8

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

4. Applied rewrites 89.8%

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

5. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f64 73.1

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

7. Applied rewrites 73.1%

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

Alternative 15: 30.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.0%

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

3. Taylor expanded in z around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv 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 67.2

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

5. Applied rewrites 67.2%

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

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 30.5%

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

9. Taylor expanded in z around inf

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

11. Applied rewrites 29.4%

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

Developer Target 1: 96.1% 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 2024255 
(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))))