Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.1%

Time: 14.3s

Alternatives: 15

Speedup: 1.6×

• 52.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Initial Program: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v)))
  4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((3.0d0 + (2.0d0 / (r * r))) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (((w * w) * r) * r)) / (1.0d0 - v))) - 4.5d0
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(Float64(Float64(w * w) * r) * r)) / Float64(1.0 - v))) - 4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r) * r)) / (1.0 - v))) - 4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(w * w), $MachinePrecision] * r), $MachinePrecision] * r), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Alternative 1: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 3.9 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(r\_m \cdot w, \left(r\_m \cdot w\right) \cdot -0.375, -1.5 + \frac{2}{r\_m \cdot r\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{r\_m}, \frac{1}{r\_m}, 3 - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{r\_m}{1 - v}, 4.5\right)\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 3.9e-46)
   (fma (* r_m w) (* (* r_m w) -0.375) (+ -1.5 (/ 2.0 (* r_m r_m))))
   (fma
    (/ 2.0 r_m)
    (/ 1.0 r_m)
    (-
     3.0
     (fma
      (* 0.125 (fma v -2.0 3.0))
      (* (* w (* r_m w)) (/ r_m (- 1.0 v)))
      4.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 3.9e-46) {
		tmp = fma((r_m * w), ((r_m * w) * -0.375), (-1.5 + (2.0 / (r_m * r_m))));
	} else {
		tmp = fma((2.0 / r_m), (1.0 / r_m), (3.0 - fma((0.125 * fma(v, -2.0, 3.0)), ((w * (r_m * w)) * (r_m / (1.0 - v))), 4.5)));
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 3.9e-46)
		tmp = fma(Float64(r_m * w), Float64(Float64(r_m * w) * -0.375), Float64(-1.5 + Float64(2.0 / Float64(r_m * r_m))));
	else
		tmp = fma(Float64(2.0 / r_m), Float64(1.0 / r_m), Float64(3.0 - fma(Float64(0.125 * fma(v, -2.0, 3.0)), Float64(Float64(w * Float64(r_m * w)) * Float64(r_m / Float64(1.0 - v))), 4.5)));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 3.9e-46], N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision] + N[(-1.5 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / r$95$m), $MachinePrecision] * N[(1.0 / r$95$m), $MachinePrecision] + N[(3.0 - N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.9000000000000003e-46

   1. Initial program 87.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.2%

      \[\leadsto \mathsf{fma}\left(r \cdot w, \color{blue}{\left(r \cdot w\right) \cdot -0.375}, -1.5 + \frac{2}{r \cdot r}\right) \]

   if 3.9000000000000003e-46 < r

   1. Initial program 85.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} - \frac{9}{2} \]

      3. associate--l- N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right)} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right) \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + 3\right)} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right) \]

      6. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right) \]

      9. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} + \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right) \]

      10. div-inv N/A

          \[\leadsto \color{blue}{\frac{2}{r} \cdot \frac{1}{r}} + \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{r}, \frac{1}{r}, 3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

      12. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{r}}, \frac{1}{r}, 3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right) \]

      13. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{2}{r}, \color{blue}{\frac{1}{r}}, 3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{r}, \frac{1}{r}, 3 - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 89.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := \left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r\_m \cdot \left(r\_m \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\\ \mathbf{if}\;t\_1 \leq -20000000000000:\\ \;\;\;\;3 - \mathsf{fma}\left(w, \left(w \cdot \left(r\_m \cdot r\_m\right)\right) \cdot 0.25, 4.5\right)\\ \mathbf{elif}\;t\_1 \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(\left(r\_m \cdot w\right) \cdot -0.375, r\_m \cdot w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t\_0\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1
         (+
          (+ 3.0 t_0)
          (/
           (* (* 0.125 (- 3.0 (* 2.0 v))) (* r_m (* r_m (* w w))))
           (+ v -1.0)))))
   (if (<= t_1 -20000000000000.0)
     (- 3.0 (fma w (* (* w (* r_m r_m)) 0.25) 4.5))
     (if (<= t_1 3.1)
       (fma (* (* r_m w) -0.375) (* r_m w) -1.5)
       (+ -1.5 t_0)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = (3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0));
	double tmp;
	if (t_1 <= -20000000000000.0) {
		tmp = 3.0 - fma(w, ((w * (r_m * r_m)) * 0.25), 4.5);
	} else if (t_1 <= 3.1) {
		tmp = fma(((r_m * w) * -0.375), (r_m * w), -1.5);
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	t_1 = Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r_m * Float64(r_m * Float64(w * w)))) / Float64(v + -1.0)))
	tmp = 0.0
	if (t_1 <= -20000000000000.0)
		tmp = Float64(3.0 - fma(w, Float64(Float64(w * Float64(r_m * r_m)) * 0.25), 4.5));
	elseif (t_1 <= 3.1)
		tmp = fma(Float64(Float64(r_m * w) * -0.375), Float64(r_m * w), -1.5);
	else
		tmp = Float64(-1.5 + t_0);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(r$95$m * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000000000.0], N[(3.0 - N[(w * N[(N[(w * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.1], N[(N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision] + -1.5), $MachinePrecision], N[(-1.5 + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2e13

   1. Initial program 85.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} - \frac{9}{2} \]

      3. associate--l- N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v} + \frac{9}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}} + \frac{9}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}, \frac{9}{2}\right)} \]

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) + \frac{9}{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right)} + \frac{9}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right)} \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)} \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{w \cdot \left(\left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right)} + \frac{9}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(w, \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right), \frac{9}{2}\right)} \]

   6. Applied rewrites 92.1%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right)} \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right), \frac{9}{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 87.6%

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right) \]

   10. Taylor expanded in v around inf

       \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \frac{1}{4} \cdot \color{blue}{\left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]

       3. unpow2 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \frac{1}{4} \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot w\right), \frac{9}{2}\right) \]

       4. lower-*.f64 82.9

          \[\leadsto 3 - \mathsf{fma}\left(w, 0.25 \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot w\right), 4.5\right) \]

   12. Applied rewrites 82.9%

       \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{0.25 \cdot \left(\left(r \cdot r\right) \cdot w\right)}, 4.5\right) \]

   if -2e13 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < 3.10000000000000009

   1. Initial program 86.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 54.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.5%

      \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

   9. Taylor expanded in r around inf

      \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 54.1%

       \[\leadsto \mathsf{fma}\left(w \cdot w, \color{blue}{\left(r \cdot r\right) \cdot -0.375}, -1.5\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 93.5%

       \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot \color{blue}{w}, -1.5\right) \]

   if 3.10000000000000009 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 88.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. lower-*.f64 99.8

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -20000000000000:\\ \;\;\;\;3 - \mathsf{fma}\left(w, \left(w \cdot \left(r \cdot r\right)\right) \cdot 0.25, 4.5\right)\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := \left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r\_m \cdot \left(r\_m \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r\_m \cdot r\_m\right)\right)\right)\\ \mathbf{elif}\;t\_1 \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(\left(r\_m \cdot w\right) \cdot -0.375, r\_m \cdot w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t\_0\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1
         (+
          (+ 3.0 t_0)
          (/
           (* (* 0.125 (- 3.0 (* 2.0 v))) (* r_m (* r_m (* w w))))
           (+ v -1.0)))))
   (if (<= t_1 (- INFINITY))
     (* -0.25 (* w (* w (* r_m r_m))))
     (if (<= t_1 3.1)
       (fma (* (* r_m w) -0.375) (* r_m w) -1.5)
       (+ -1.5 t_0)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = (3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = -0.25 * (w * (w * (r_m * r_m)));
	} else if (t_1 <= 3.1) {
		tmp = fma(((r_m * w) * -0.375), (r_m * w), -1.5);
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	t_1 = Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r_m * Float64(r_m * Float64(w * w)))) / Float64(v + -1.0)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(-0.25 * Float64(w * Float64(w * Float64(r_m * r_m))));
	elseif (t_1 <= 3.1)
		tmp = fma(Float64(Float64(r_m * w) * -0.375), Float64(r_m * w), -1.5);
	else
		tmp = Float64(-1.5 + t_0);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(r$95$m * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(-0.25 * N[(w * N[(w * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 3.1], N[(N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision] + -1.5), $MachinePrecision], N[(-1.5 + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -inf.0

   1. Initial program 81.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Applied rewrites 66.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{0.125}{v} - 0.25, -1.5\right)} \]

   6. Taylor expanded in r around inf

      \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \left(\frac{1}{8} \cdot \frac{1}{v} - \frac{1}{4}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.1%

      \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(-0.25 + \frac{0.125}{v}\right)\right)} \]

   9. Taylor expanded in v around inf

      \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot \color{blue}{{w}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 89.2%

       \[\leadsto -0.25 \cdot \left(w \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot w\right)}\right) \]

   if -inf.0 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < 3.10000000000000009

   1. Initial program 90.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 47.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.1%

      \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

   9. Taylor expanded in r around inf

      \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.9%

       \[\leadsto \mathsf{fma}\left(w \cdot w, \color{blue}{\left(r \cdot r\right) \cdot -0.375}, -1.5\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 82.3%

       \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot \color{blue}{w}, -1.5\right) \]

   if 3.10000000000000009 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 88.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. lower-*.f64 99.8

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -\infty:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq 3.1:\\ \;\;\;\;\mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot w, -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r\_m \cdot \left(r\_m \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000000000:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r\_m \cdot r\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t\_0\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<=
        (+
         (+ 3.0 t_0)
         (/
          (* (* 0.125 (- 3.0 (* 2.0 v))) (* r_m (* r_m (* w w))))
          (+ v -1.0)))
        -2000000000000.0)
     (* -0.25 (* w (* w (* r_m r_m))))
     (+ -1.5 t_0))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0))) <= -2000000000000.0) {
		tmp = -0.25 * (w * (w * (r_m * r_m)));
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r_m * r_m)
    if (((3.0d0 + t_0) + (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (r_m * (r_m * (w * w)))) / (v + (-1.0d0)))) <= (-2000000000000.0d0)) then
        tmp = (-0.25d0) * (w * (w * (r_m * r_m)))
    else
        tmp = (-1.5d0) + t_0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0))) <= -2000000000000.0) {
		tmp = -0.25 * (w * (w * (r_m * r_m)));
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = 2.0 / (r_m * r_m)
	tmp = 0
	if ((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0))) <= -2000000000000.0:
		tmp = -0.25 * (w * (w * (r_m * r_m)))
	else:
		tmp = -1.5 + t_0
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) + Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(r_m * Float64(r_m * Float64(w * w)))) / Float64(v + -1.0))) <= -2000000000000.0)
		tmp = Float64(-0.25 * Float64(w * Float64(w * Float64(r_m * r_m))));
	else
		tmp = Float64(-1.5 + t_0);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = 2.0 / (r_m * r_m);
	tmp = 0.0;
	if (((3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0))) <= -2000000000000.0)
		tmp = -0.25 * (w * (w * (r_m * r_m)));
	else
		tmp = -1.5 + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(r$95$m * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2000000000000.0], N[(-0.25 * N[(w * N[(w * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v))) < -2e12

   1. Initial program 85.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \frac{0.125}{v} - 0.25, -1.5\right)} \]

   6. Taylor expanded in r around inf

      \[\leadsto {r}^{2} \cdot \color{blue}{\left({w}^{2} \cdot \left(\frac{1}{8} \cdot \frac{1}{v} - \frac{1}{4}\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.5%

      \[\leadsto \left(r \cdot r\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot \left(-0.25 + \frac{0.125}{v}\right)\right)} \]

   9. Taylor expanded in v around inf

      \[\leadsto \frac{-1}{4} \cdot \left({r}^{2} \cdot \color{blue}{{w}^{2}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 82.2%

       \[\leadsto -0.25 \cdot \left(w \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot w\right)}\right) \]

   if -2e12 < (-.f64 (+.f64 #s(literal 3 binary64) (/.f64 #s(literal 2 binary64) (*.f64 r r))) (/.f64 (*.f64 (*.f64 #s(literal 1/8 binary64) (-.f64 #s(literal 3 binary64) (*.f64 #s(literal 2 binary64) v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 #s(literal 1 binary64) v)))

   1. Initial program 87.8%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      4. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

      8. unpow2 N/A

         \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

      9. lower-*.f64 94.8

         \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 94.8%

      \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2000000000000:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 3.9 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(r\_m \cdot w, \left(r\_m \cdot w\right) \cdot -0.375, -1.5 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_0\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{r\_m}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 3.9e-46)
     (fma (* r_m w) (* (* r_m w) -0.375) (+ -1.5 t_0))
     (-
      (+ 3.0 t_0)
      (fma
       (* 0.125 (fma v -2.0 3.0))
       (* (* w (* r_m w)) (/ r_m (- 1.0 v)))
       4.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 3.9e-46) {
		tmp = fma((r_m * w), ((r_m * w) * -0.375), (-1.5 + t_0));
	} else {
		tmp = (3.0 + t_0) - fma((0.125 * fma(v, -2.0, 3.0)), ((w * (r_m * w)) * (r_m / (1.0 - v))), 4.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 3.9e-46)
		tmp = fma(Float64(r_m * w), Float64(Float64(r_m * w) * -0.375), Float64(-1.5 + t_0));
	else
		tmp = Float64(Float64(3.0 + t_0) - fma(Float64(0.125 * fma(v, -2.0, 3.0)), Float64(Float64(w * Float64(r_m * w)) * Float64(r_m / Float64(1.0 - v))), 4.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 3.9e-46], N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision] + N[(-1.5 + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 3.9000000000000003e-46

   1. Initial program 87.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 94.2%

      \[\leadsto \mathsf{fma}\left(r \cdot w, \color{blue}{\left(r \cdot w\right) \cdot -0.375}, -1.5 + \frac{2}{r \cdot r}\right) \]

   if 3.9000000000000003e-46 < r

   1. Initial program 85.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} - \frac{9}{2} \]

      3. associate--l- N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v} + \frac{9}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}} + \frac{9}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}, \frac{9}{2}\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 195:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r\_m \cdot r\_m\right) \cdot -0.25\right), w, \frac{2}{r\_m \cdot r\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{r\_m}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 195.0)
   (+ -1.5 (fma (* w (* (* r_m r_m) -0.25)) w (/ 2.0 (* r_m r_m))))
   (-
    3.0
    (fma
     (* 0.125 (fma v -2.0 3.0))
     (* (* w (* r_m w)) (/ r_m (- 1.0 v)))
     4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 195.0) {
		tmp = -1.5 + fma((w * ((r_m * r_m) * -0.25)), w, (2.0 / (r_m * r_m)));
	} else {
		tmp = 3.0 - fma((0.125 * fma(v, -2.0, 3.0)), ((w * (r_m * w)) * (r_m / (1.0 - v))), 4.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 195.0)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r_m * r_m) * -0.25)), w, Float64(2.0 / Float64(r_m * r_m))));
	else
		tmp = Float64(3.0 - fma(Float64(0.125 * fma(v, -2.0, 3.0)), Float64(Float64(w * Float64(r_m * w)) * Float64(r_m / Float64(1.0 - v))), 4.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 195.0], N[(-1.5 + N[(N[(w * N[(N[(r$95$m * r$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 195

   1. Initial program 86.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

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

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   if 195 < r

   1. Initial program 87.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} - \frac{9}{2} \]

      3. associate--l- N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v} + \frac{9}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}} + \frac{9}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}, \frac{9}{2}\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)} \]

   5. Taylor expanded in r around inf

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, \frac{9}{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 195:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 3.9 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{2}{r\_m}}{r\_m}\\ \mathbf{elif}\;r\_m \leq 0.43:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} + \left(r\_m \cdot r\_m\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;r\_m \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;3 - \mathsf{fma}\left(w, \left(w \cdot \left(r\_m \cdot r\_m\right)\right) \cdot 0.25, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(r\_m \cdot w\right) \cdot -0.375, r\_m \cdot w, -1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 3.9e-130)
   (/ (/ 2.0 r_m) r_m)
   (if (<= r_m 0.43)
     (+ (/ 2.0 (* r_m r_m)) (* (* r_m r_m) (* -0.375 (* w w))))
     (if (<= r_m 2.25e+148)
       (- 3.0 (fma w (* (* w (* r_m r_m)) 0.25) 4.5))
       (fma (* (* r_m w) -0.375) (* r_m w) -1.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 3.9e-130) {
		tmp = (2.0 / r_m) / r_m;
	} else if (r_m <= 0.43) {
		tmp = (2.0 / (r_m * r_m)) + ((r_m * r_m) * (-0.375 * (w * w)));
	} else if (r_m <= 2.25e+148) {
		tmp = 3.0 - fma(w, ((w * (r_m * r_m)) * 0.25), 4.5);
	} else {
		tmp = fma(((r_m * w) * -0.375), (r_m * w), -1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 3.9e-130)
		tmp = Float64(Float64(2.0 / r_m) / r_m);
	elseif (r_m <= 0.43)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(Float64(r_m * r_m) * Float64(-0.375 * Float64(w * w))));
	elseif (r_m <= 2.25e+148)
		tmp = Float64(3.0 - fma(w, Float64(Float64(w * Float64(r_m * r_m)) * 0.25), 4.5));
	else
		tmp = fma(Float64(Float64(r_m * w) * -0.375), Float64(r_m * w), -1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 3.9e-130], N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision], If[LessEqual[r$95$m, 0.43], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r$95$m, 2.25e+148], N[(3.0 - N[(w * N[(N[(w * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision] + -1.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if r < 3.9000000000000001e-130

   1. Initial program 86.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

      3. lower-*.f64 54.2

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.2%

      \[\leadsto \frac{\frac{2}{r}}{\color{blue}{r}} \]

   if 3.9000000000000001e-130 < r < 0.429999999999999993

   1. Initial program 90.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around inf

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{8} \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 93.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot -0.375\right)} \]

   if 0.429999999999999993 < r < 2.24999999999999997e148

   1. Initial program 87.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} - \frac{9}{2} \]

      3. associate--l- N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v} + \frac{9}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}} + \frac{9}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}, \frac{9}{2}\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) + \frac{9}{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right)} + \frac{9}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right)} \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)} \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{w \cdot \left(\left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right)} + \frac{9}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(w, \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right), \frac{9}{2}\right)} \]

   6. Applied rewrites 99.9%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right)} \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right), \frac{9}{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.9%

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right) \]

   10. Taylor expanded in v around inf

       \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \frac{1}{4} \cdot \color{blue}{\left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]

       3. unpow2 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \frac{1}{4} \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot w\right), \frac{9}{2}\right) \]

       4. lower-*.f64 97.5

          \[\leadsto 3 - \mathsf{fma}\left(w, 0.25 \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot w\right), 4.5\right) \]

   12. Applied rewrites 97.5%

       \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{0.25 \cdot \left(\left(r \cdot r\right) \cdot w\right)}, 4.5\right) \]

   if 2.24999999999999997e148 < r

   1. Initial program 85.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.6%

      \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

   9. Taylor expanded in r around inf

      \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.5%

       \[\leadsto \mathsf{fma}\left(w \cdot w, \color{blue}{\left(r \cdot r\right) \cdot -0.375}, -1.5\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 90.2%

       \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot \color{blue}{w}, -1.5\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.9 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 0.43:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;r \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;3 - \mathsf{fma}\left(w, \left(w \cdot \left(r \cdot r\right)\right) \cdot 0.25, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot w, -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 195:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r\_m \cdot r\_m\right) \cdot -0.25\right), w, \frac{2}{r\_m \cdot r\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(w, \left(r\_m \cdot \left(w \cdot \frac{r\_m}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 195.0)
   (+ -1.5 (fma (* w (* (* r_m r_m) -0.25)) w (/ 2.0 (* r_m r_m))))
   (-
    3.0
    (fma w (* (* r_m (* w (/ r_m (- 1.0 v)))) (fma v -0.25 0.375)) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 195.0) {
		tmp = -1.5 + fma((w * ((r_m * r_m) * -0.25)), w, (2.0 / (r_m * r_m)));
	} else {
		tmp = 3.0 - fma(w, ((r_m * (w * (r_m / (1.0 - v)))) * fma(v, -0.25, 0.375)), 4.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 195.0)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r_m * r_m) * -0.25)), w, Float64(2.0 / Float64(r_m * r_m))));
	else
		tmp = Float64(3.0 - fma(w, Float64(Float64(r_m * Float64(w * Float64(r_m / Float64(1.0 - v)))) * fma(v, -0.25, 0.375)), 4.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 195.0], N[(-1.5 + N[(N[(w * N[(N[(r$95$m * r$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 - N[(w * N[(N[(r$95$m * N[(w * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 195

   1. Initial program 86.5%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

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

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Applied rewrites 90.7%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   if 195 < r

   1. Initial program 87.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} - \frac{9}{2} \]

      3. associate--l- N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v} + \frac{9}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}} + \frac{9}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}, \frac{9}{2}\right)} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) + \frac{9}{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right)} + \frac{9}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right)} \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)} \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{w \cdot \left(\left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right)} + \frac{9}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(w, \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right), \frac{9}{2}\right)} \]

   6. Applied rewrites 93.6%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right)} \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right), \frac{9}{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 93.6%

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 195:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r\_m \cdot r\_m\right) \cdot -0.25\right), w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r\_m \cdot \left(w \cdot \left(r\_m \cdot w\right)\right), -0.375, -1.5 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 2.25e+148)
     (+ -1.5 (fma (* w (* (* r_m r_m) -0.25)) w t_0))
     (fma (* r_m (* w (* r_m w))) -0.375 (+ -1.5 t_0)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 2.25e+148) {
		tmp = -1.5 + fma((w * ((r_m * r_m) * -0.25)), w, t_0);
	} else {
		tmp = fma((r_m * (w * (r_m * w))), -0.375, (-1.5 + t_0));
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 2.25e+148)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r_m * r_m) * -0.25)), w, t_0));
	else
		tmp = fma(Float64(r_m * Float64(w * Float64(r_m * w))), -0.375, Float64(-1.5 + t_0));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 2.25e+148], N[(-1.5 + N[(N[(w * N[(N[(r$95$m * r$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(r$95$m * N[(w * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.375 + N[(-1.5 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.24999999999999997e148

   1. Initial program 87.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

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

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   if 2.24999999999999997e148 < r

   1. Initial program 85.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 85.5%

      \[\leadsto \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), \color{blue}{-0.375}, -1.5 + \frac{2}{r \cdot r}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 90.2%

      \[\leadsto \mathsf{fma}\left(r \cdot \left(\left(r \cdot w\right) \cdot w\right), -0.375, -1.5 + \frac{2}{r \cdot r}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot \left(w \cdot \left(r \cdot w\right)\right), -0.375, -1.5 + \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r\_m \cdot r\_m\right) \cdot -0.25\right), w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r\_m \cdot w, \left(r\_m \cdot w\right) \cdot -0.375, -1.5 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= r_m 2.25e+148)
     (+ -1.5 (fma (* w (* (* r_m r_m) -0.25)) w t_0))
     (fma (* r_m w) (* (* r_m w) -0.375) (+ -1.5 t_0)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double tmp;
	if (r_m <= 2.25e+148) {
		tmp = -1.5 + fma((w * ((r_m * r_m) * -0.25)), w, t_0);
	} else {
		tmp = fma((r_m * w), ((r_m * w) * -0.375), (-1.5 + t_0));
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 2.25e+148)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r_m * r_m) * -0.25)), w, t_0));
	else
		tmp = fma(Float64(r_m * w), Float64(Float64(r_m * w) * -0.375), Float64(-1.5 + t_0));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 2.25e+148], N[(-1.5 + N[(N[(w * N[(N[(r$95$m * r$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(r$95$m * w), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision] + N[(-1.5 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.24999999999999997e148

   1. Initial program 87.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

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

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   if 2.24999999999999997e148 < r

   1. Initial program 85.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.2%

      \[\leadsto \mathsf{fma}\left(r \cdot w, \color{blue}{\left(r \cdot w\right) \cdot -0.375}, -1.5 + \frac{2}{r \cdot r}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(r \cdot w, \left(r \cdot w\right) \cdot -0.375, -1.5 + \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r\_m \cdot r\_m\right) \cdot -0.25\right), w, \frac{2}{r\_m \cdot r\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(r\_m \cdot w\right) \cdot -0.375, r\_m \cdot w, -1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 2.25e+148)
   (+ -1.5 (fma (* w (* (* r_m r_m) -0.25)) w (/ 2.0 (* r_m r_m))))
   (fma (* (* r_m w) -0.375) (* r_m w) -1.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2.25e+148) {
		tmp = -1.5 + fma((w * ((r_m * r_m) * -0.25)), w, (2.0 / (r_m * r_m)));
	} else {
		tmp = fma(((r_m * w) * -0.375), (r_m * w), -1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 2.25e+148)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(Float64(r_m * r_m) * -0.25)), w, Float64(2.0 / Float64(r_m * r_m))));
	else
		tmp = fma(Float64(Float64(r_m * w) * -0.375), Float64(r_m * w), -1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 2.25e+148], N[(-1.5 + N[(N[(w * N[(N[(r$95$m * r$95$m), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.24999999999999997e148

   1. Initial program 87.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}}} \]

      3. distribute-neg-in N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{2}\right)\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} + 2 \cdot \frac{1}{{r}^{2}} \]

      4. metadata-eval N/A

         \[\leadsto \left(\color{blue}{\frac{-3}{2}} + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

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

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      6. metadata-eval N/A

         \[\leadsto \left(\frac{-3}{2} + \color{blue}{\frac{-1}{4}} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right) + 2 \cdot \frac{1}{{r}^{2}} \]

      7. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \color{blue}{\frac{-3}{2} + \left(\frac{-1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right) + 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot {w}^{2}} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      10. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot \color{blue}{\left(w \cdot w\right)} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      11. associate-*r* N/A

          \[\leadsto \frac{-3}{2} + \left(\color{blue}{\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w\right) \cdot w} + 2 \cdot \frac{1}{{r}^{2}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{-3}{2} + \color{blue}{\mathsf{fma}\left(\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right) \cdot w}, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{4} \cdot {r}^{2}\right)} \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      15. unpow2 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \color{blue}{\left(r \cdot r\right)}\right) \cdot w, w, 2 \cdot \frac{1}{{r}^{2}}\right) \]

      17. associate-*r/ N/A

          \[\leadsto \frac{-3}{2} + \mathsf{fma}\left(\left(\frac{-1}{4} \cdot \left(r \cdot r\right)\right) \cdot w, w, \color{blue}{\frac{2 \cdot 1}{{r}^{2}}}\right) \]

   5. Applied rewrites 91.6%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]

   if 2.24999999999999997e148 < r

   1. Initial program 85.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.6%

      \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

   9. Taylor expanded in r around inf

      \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.5%

       \[\leadsto \mathsf{fma}\left(w \cdot w, \color{blue}{\left(r \cdot r\right) \cdot -0.375}, -1.5\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 90.2%

       \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot \color{blue}{w}, -1.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(\left(r \cdot r\right) \cdot -0.25\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot w, -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := w \cdot \left(r\_m \cdot r\_m\right)\\ \mathbf{if}\;r\_m \leq 0.43:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} + -0.375 \cdot \left(w \cdot t\_0\right)\\ \mathbf{elif}\;r\_m \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;3 - \mathsf{fma}\left(w, t\_0 \cdot 0.25, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(r\_m \cdot w\right) \cdot -0.375, r\_m \cdot w, -1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* w (* r_m r_m))))
   (if (<= r_m 0.43)
     (+ (/ 2.0 (* r_m r_m)) (* -0.375 (* w t_0)))
     (if (<= r_m 2.25e+148)
       (- 3.0 (fma w (* t_0 0.25) 4.5))
       (fma (* (* r_m w) -0.375) (* r_m w) -1.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = w * (r_m * r_m);
	double tmp;
	if (r_m <= 0.43) {
		tmp = (2.0 / (r_m * r_m)) + (-0.375 * (w * t_0));
	} else if (r_m <= 2.25e+148) {
		tmp = 3.0 - fma(w, (t_0 * 0.25), 4.5);
	} else {
		tmp = fma(((r_m * w) * -0.375), (r_m * w), -1.5);
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(w * Float64(r_m * r_m))
	tmp = 0.0
	if (r_m <= 0.43)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) + Float64(-0.375 * Float64(w * t_0)));
	elseif (r_m <= 2.25e+148)
		tmp = Float64(3.0 - fma(w, Float64(t_0 * 0.25), 4.5));
	else
		tmp = fma(Float64(Float64(r_m * w) * -0.375), Float64(r_m * w), -1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(w * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 0.43], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(w * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r$95$m, 2.25e+148], N[(3.0 - N[(w * N[(t$95$0 * 0.25), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(r$95$m * w), $MachinePrecision] * -0.375), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision] + -1.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 0.429999999999999993

   1. Initial program 86.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.0%

      \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

   9. Taylor expanded in w around inf

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{8} \cdot \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right) \cdot \color{blue}{-0.375} \]

   if 0.429999999999999993 < r < 2.24999999999999997e148

   1. Initial program 87.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \frac{9}{2}} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} - \frac{9}{2} \]

      3. associate--l- N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}}{1 - v} + \frac{9}{2}\right) \]

      7. associate-/l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}} + \frac{9}{2}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right), \frac{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}{1 - v}, \frac{9}{2}\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) + \frac{9}{2}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right)} + \frac{9}{2}\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}\right)} \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      5. associate-*l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{\left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)} \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right) + \frac{9}{2}\right) \]

      6. associate-*l* N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{w \cdot \left(\left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right)\right)} + \frac{9}{2}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(w, \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right) \cdot \left(\frac{1}{8} \cdot \mathsf{fma}\left(v, -2, 3\right)\right), \frac{9}{2}\right)} \]

   6. Applied rewrites 99.9%

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right)} \]

   7. Taylor expanded in r around inf

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, \frac{-1}{4}, \frac{3}{8}\right), \frac{9}{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 99.9%

      \[\leadsto \color{blue}{3} - \mathsf{fma}\left(w, \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right), 4.5\right) \]

   10. Taylor expanded in v around inf

       \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{\frac{1}{4} \cdot \left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \frac{1}{4} \cdot \color{blue}{\left({r}^{2} \cdot w\right)}, \frac{9}{2}\right) \]

       3. unpow2 N/A

          \[\leadsto 3 - \mathsf{fma}\left(w, \frac{1}{4} \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot w\right), \frac{9}{2}\right) \]

       4. lower-*.f64 97.5

          \[\leadsto 3 - \mathsf{fma}\left(w, 0.25 \cdot \left(\color{blue}{\left(r \cdot r\right)} \cdot w\right), 4.5\right) \]

   12. Applied rewrites 97.5%

       \[\leadsto 3 - \mathsf{fma}\left(w, \color{blue}{0.25 \cdot \left(\left(r \cdot r\right) \cdot w\right)}, 4.5\right) \]

   if 2.24999999999999997e148 < r

   1. Initial program 85.1%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 55.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.6%

      \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

   9. Taylor expanded in r around inf

      \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 55.5%

       \[\leadsto \mathsf{fma}\left(w \cdot w, \color{blue}{\left(r \cdot r\right) \cdot -0.375}, -1.5\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 90.2%

       \[\leadsto \mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot \color{blue}{w}, -1.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.43:\\ \;\;\;\;\frac{2}{r \cdot r} + -0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{elif}\;r \leq 2.25 \cdot 10^{+148}:\\ \;\;\;\;3 - \mathsf{fma}\left(w, \left(w \cdot \left(r \cdot r\right)\right) \cdot 0.25, 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(r \cdot w\right) \cdot -0.375, r \cdot w, -1.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.15:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.15) (/ 2.0 (* r_m r_m)) -1.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.15) {
		tmp = 2.0 / (r_m * r_m);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.15d0) then
        tmp = 2.0d0 / (r_m * r_m)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.15) {
		tmp = 2.0 / (r_m * r_m);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.15:
		tmp = 2.0 / (r_m * r_m)
	else:
		tmp = -1.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.15)
		tmp = Float64(2.0 / Float64(r_m * r_m));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.15)
		tmp = 2.0 / (r_m * r_m);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.15], N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 86.9%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

      3. lower-*.f64 55.4

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r}} \]

   if 1.1499999999999999 < r

   1. Initial program 86.2%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   2. Add Preprocessing

   3. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

      9. distribute-neg-in N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

      10. associate-*r* N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

      16. lower-fma.f64 N/A

          \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

   6. Taylor expanded in w around 0

      \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.4%

      \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

   9. Taylor expanded in r around inf

      \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 64.3%

       \[\leadsto \mathsf{fma}\left(w \cdot w, \color{blue}{\left(r \cdot r\right) \cdot -0.375}, -1.5\right) \]

   12. Taylor expanded in w around 0

       \[\leadsto \frac{-3}{2} \]
   13. Step-by-step derivation

   14. Applied rewrites 33.4%

       \[\leadsto -1.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 58.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ -1.5 + \frac{2}{r\_m \cdot r\_m} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (+ -1.5 (/ 2.0 (* r_m r_m))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return -1.5 + (2.0 / (r_m * r_m));
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = (-1.5d0) + (2.0d0 / (r_m * r_m))
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return -1.5 + (2.0 / (r_m * r_m));
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -1.5 + (2.0 / (r_m * r_m))

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(-1.5 + Float64(2.0 / Float64(r_m * r_m)))
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = -1.5 + (2.0 / (r_m * r_m));
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(-1.5 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.7%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\frac{-3}{2}} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

   4. lower-+.f64 N/A

      \[\leadsto \color{blue}{\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{-3}{2} + \frac{\color{blue}{2}}{{r}^{2}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{\frac{2}{{r}^{2}}} \]

   8. unpow2 N/A

      \[\leadsto \frac{-3}{2} + \frac{2}{\color{blue}{r \cdot r}} \]

   9. lower-*.f64 57.2

      \[\leadsto -1.5 + \frac{2}{\color{blue}{r \cdot r}} \]

5. Applied rewrites 57.2%

   \[\leadsto \color{blue}{-1.5 + \frac{2}{r \cdot r}} \]
6. Add Preprocessing

Alternative 15: 14.9% accurate, 73.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ -1.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 -1.5)

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return -1.5;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = -1.5d0
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return -1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -1.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return -1.5
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = -1.5;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := -1.5

Derivation

1. Initial program 86.7%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Add Preprocessing

3. Taylor expanded in v around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right)} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(\mathsf{neg}\left(\left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \frac{2}{r \cdot r} + \left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right) + \frac{3}{2}\right)}\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right)} \]

   10. associate-*r* N/A

       \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{\left(\frac{3}{8} \cdot {r}^{2}\right) \cdot {w}^{2}}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{r \cdot r} + \left(\left(\mathsf{neg}\left(\color{blue}{{w}^{2} \cdot \left(\frac{3}{8} \cdot {r}^{2}\right)}\right)\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

       \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{{w}^{2} \cdot \left(\mathsf{neg}\left(\frac{3}{8} \cdot {r}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

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

       \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \color{blue}{\left(\left(\mathsf{neg}\left(\frac{3}{8}\right)\right) \cdot {r}^{2}\right)} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\color{blue}{\frac{-3}{8}} \cdot {r}^{2}\right) + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \frac{2}{r \cdot r} + \left({w}^{2} \cdot \left(\frac{-3}{8} \cdot {r}^{2}\right) + \color{blue}{\frac{-3}{2}}\right) \]

   16. lower-fma.f64 N/A

       \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\mathsf{fma}\left({w}^{2}, \frac{-3}{8} \cdot {r}^{2}, \frac{-3}{2}\right)} \]

5. Applied rewrites 75.7%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \mathsf{fma}\left(w \cdot w, \left(r \cdot r\right) \cdot -0.375, -1.5\right)} \]

6. Taylor expanded in w around 0

   \[\leadsto \frac{2}{r \cdot r} + \frac{-3}{2} \]
7. Step-by-step derivation

8. Applied rewrites 57.2%

   \[\leadsto \frac{2}{r \cdot r} + -1.5 \]

9. Taylor expanded in r around inf

   \[\leadsto {r}^{2} \cdot \color{blue}{\left(\frac{-3}{8} \cdot {w}^{2} - \frac{3}{2} \cdot \frac{1}{{r}^{2}}\right)} \]
10. Step-by-step derivation

11. Applied rewrites 42.3%

    \[\leadsto \mathsf{fma}\left(w \cdot w, \color{blue}{\left(r \cdot r\right) \cdot -0.375}, -1.5\right) \]

12. Taylor expanded in w around 0

    \[\leadsto \frac{-3}{2} \]
13. Step-by-step derivation

14. Applied rewrites 18.2%

    \[\leadsto -1.5 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024226 
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3.0 (/ 2.0 (* r r))) (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* (* (* w w) r) r)) (- 1.0 v))) 4.5))