Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 98.5%

Time: 13.8s

Alternatives: 12

Speedup: 1.8×

• 53.4% 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: 84.5% 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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 5 \cdot 10^{-127}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r\_m \cdot r\_m\right)\right), w, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_0\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{r\_m}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

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 5e-127)
     (+ -1.5 (fma (* w (* -0.25 (* r_m r_m))) w 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 <= 5e-127) {
		tmp = -1.5 + fma((w * (-0.25 * (r_m * r_m))), w, 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 <= 5e-127)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r_m * r_m))), w, 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, 5e-127], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + 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 < 4.9999999999999997e-127

   1. Initial program 82.1%

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

   3. Taylor expanded in v around 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 90.9%

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

   if 4.9999999999999997e-127 < r

   1. Initial program 87.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-127}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := -1.5 + t\_0\\ t_2 := \left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r\_m \cdot \left(r\_m \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(w \cdot \left(r\_m \cdot -0.25\right), r\_m \cdot w, t\_1\right)\\ \mathbf{elif}\;t\_2 \leq 3:\\ \;\;\;\;3 - \mathsf{fma}\left(r\_m, \frac{w}{1 - v} \cdot \left(r\_m \cdot \left(w \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right)\right), 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (t_1 (+ -1.5 t_0))
        (t_2
         (+
          (+ 3.0 t_0)
          (/
           (* (* 0.125 (- 3.0 (* 2.0 v))) (* r_m (* r_m (* w w))))
           (+ v -1.0)))))
   (if (<= t_2 (- INFINITY))
     (fma (* w (* r_m -0.25)) (* r_m w) t_1)
     (if (<= t_2 3.0)
       (-
        3.0
        (fma r_m (* (/ w (- 1.0 v)) (* r_m (* w (fma v -0.25 0.375)))) 4.5))
       t_1))))

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 = -1.5 + t_0;
	double t_2 = (3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = fma((w * (r_m * -0.25)), (r_m * w), t_1);
	} else if (t_2 <= 3.0) {
		tmp = 3.0 - fma(r_m, ((w / (1.0 - v)) * (r_m * (w * fma(v, -0.25, 0.375)))), 4.5);
	} else {
		tmp = t_1;
	}
	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(-1.5 + t_0)
	t_2 = 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_2 <= Float64(-Inf))
		tmp = fma(Float64(w * Float64(r_m * -0.25)), Float64(r_m * w), t_1);
	elseif (t_2 <= 3.0)
		tmp = Float64(3.0 - fma(r_m, Float64(Float64(w / Float64(1.0 - v)) * Float64(r_m * Float64(w * fma(v, -0.25, 0.375)))), 4.5));
	else
		tmp = t_1;
	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[(-1.5 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = 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$2, (-Infinity)], N[(N[(w * N[(r$95$m * -0.25), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$2, 3.0], N[(3.0 - N[(r$95$m * N[(N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(w * N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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 82.1%

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

   3. Taylor expanded in 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 7.4

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in v around inf

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

      1. associate--r+ N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 93.0

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

   8. Applied rewrites 93.0%

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

   10. Applied rewrites 98.9%

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

   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

   1. Initial program 86.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 82.0

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 82.0

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 93.6

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

   4. Applied rewrites 93.6%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 93.6%

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

   9. Applied rewrites 98.1%

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

   if 3 < (-.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 83.7%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.1%

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

Alternative 3: 92.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := \left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r\_m \cdot \left(r\_m \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r\_m \cdot r\_m\right)\right), w, t\_0\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_0 - r\_m \cdot \left(r\_m \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t\_0\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (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))
     (+ -1.5 (fma (* w (* -0.25 (* r_m r_m))) w t_0))
     (if (<= t_1 -2e+20)
       (- t_0 (* r_m (* r_m (* 0.375 (* w w)))))
       (+ -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 = -1.5 + fma((w * (-0.25 * (r_m * r_m))), w, t_0);
	} else if (t_1 <= -2e+20) {
		tmp = t_0 - (r_m * (r_m * (0.375 * (w * w))));
	} 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(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r_m * r_m))), w, t_0));
	elseif (t_1 <= -2e+20)
		tmp = Float64(t_0 - Float64(r_m * Float64(r_m * Float64(0.375 * Float64(w * w)))));
	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[(-1.5 + N[(N[(w * N[(-0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+20], N[(t$95$0 - N[(r$95$m * N[(r$95$m * N[(0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 82.1%

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

   3. Taylor expanded in v around 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 97.9%

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

   if -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))) < -2e20

   1. Initial program 98.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 4.9%

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

   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 65.6%

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

   if -2e20 < (-.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 83.0%

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

   3. Taylor expanded in 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 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -\infty:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{r \cdot r} - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := \left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r\_m \cdot \left(r\_m \cdot \left(w \cdot w\right)\right)\right)}{v + -1}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(r\_m \cdot r\_m\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;t\_1 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_0 - r\_m \cdot \left(r\_m \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t\_0\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m)))
        (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))
     (* (* r_m r_m) (* -0.25 (* w w)))
     (if (<= t_1 -2e+20)
       (- t_0 (* r_m (* r_m (* 0.375 (* w w)))))
       (+ -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 = (r_m * r_m) * (-0.25 * (w * w));
	} else if (t_1 <= -2e+20) {
		tmp = t_0 - (r_m * (r_m * (0.375 * (w * w))));
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

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 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.POSITIVE_INFINITY) {
		tmp = (r_m * r_m) * (-0.25 * (w * w));
	} else if (t_1 <= -2e+20) {
		tmp = t_0 - (r_m * (r_m * (0.375 * (w * w))));
	} 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)
	t_1 = (3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (r_m * r_m) * (-0.25 * (w * w))
	elif t_1 <= -2e+20:
		tmp = t_0 - (r_m * (r_m * (0.375 * (w * w))))
	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(Float64(r_m * r_m) * Float64(-0.25 * Float64(w * w)));
	elseif (t_1 <= -2e+20)
		tmp = Float64(t_0 - Float64(r_m * Float64(r_m * Float64(0.375 * Float64(w * w)))));
	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);
	t_1 = (3.0 + t_0) + (((0.125 * (3.0 - (2.0 * v))) * (r_m * (r_m * (w * w)))) / (v + -1.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (r_m * r_m) * (-0.25 * (w * w));
	elseif (t_1 <= -2e+20)
		tmp = t_0 - (r_m * (r_m * (0.375 * (w * w))));
	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]}, 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[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -2e+20], N[(t$95$0 - N[(r$95$m * N[(r$95$m * N[(0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 82.1%

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

   3. Taylor expanded in 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 7.4

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

   5. Applied rewrites 7.4%

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

   6. Taylor expanded in v around inf

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

      1. associate--r+ N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 93.0

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

   8. Applied rewrites 93.0%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 90.8%

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

   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))) < -2e20

   1. Initial program 98.3%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 4.9%

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

   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 65.6%

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

   if -2e20 < (-.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 83.0%

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

   3. Taylor expanded in 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 94.0

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

   5. Applied rewrites 94.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -\infty:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\frac{2}{r \cdot r} - r \cdot \left(r \cdot \left(0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;\left(3 + t\_0\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r\_m \cdot \left(r\_m \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -1 \cdot 10^{+29}:\\ \;\;\;\;\left(r\_m \cdot r\_m\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t\_0\\ \end{array} \end{array} \]

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)))
        -1e+29)
     (* (* r_m r_m) (* -0.25 (* w w)))
     (+ -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))) <= -1e+29) {
		tmp = (r_m * r_m) * (-0.25 * (w * w));
	} 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)))) <= (-1d+29)) then
        tmp = (r_m * r_m) * ((-0.25d0) * (w * w))
    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))) <= -1e+29) {
		tmp = (r_m * r_m) * (-0.25 * (w * w));
	} 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))) <= -1e+29:
		tmp = (r_m * r_m) * (-0.25 * (w * w))
	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))) <= -1e+29)
		tmp = Float64(Float64(r_m * r_m) * Float64(-0.25 * Float64(w * w)));
	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))) <= -1e+29)
		tmp = (r_m * r_m) * (-0.25 * (w * w));
	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], -1e+29], N[(N[(r$95$m * r$95$m), $MachinePrecision] * N[(-0.25 * N[(w * w), $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))) < -9.99999999999999914e28

   1. Initial program 85.4%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 6.3

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

   5. Applied rewrites 6.3%

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

   6. Taylor expanded in v around inf

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

      1. associate--r+ N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. sub-neg N/A

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

      13. metadata-eval N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 85.2

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

   8. Applied rewrites 85.2%

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

   9. Taylor expanded in r around inf

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

   11. Applied rewrites 81.3%

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

   if -9.99999999999999914e28 < (-.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 82.9%

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

   3. Taylor expanded in 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 93.4

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

   5. Applied rewrites 93.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{v + -1} \leq -1 \cdot 10^{+29}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 75000000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r\_m \cdot r\_m\right)\right), w, \frac{2}{r\_m \cdot r\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{r\_m}{1 - v}, 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 75000000.0)
   (+ -1.5 (fma (* w (* -0.25 (* r_m r_m))) 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 <= 75000000.0) {
		tmp = -1.5 + fma((w * (-0.25 * (r_m * r_m))), 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 <= 75000000.0)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r_m * r_m))), 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, 75000000.0], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $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 < 7.5e7

   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

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   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 91.2%

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

   if 7.5e7 < r

   1. Initial program 90.6%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 75000000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;3 - \mathsf{fma}\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right), \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}, 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 75000000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r\_m \cdot r\_m\right)\right), w, \frac{2}{r\_m \cdot r\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + r\_m \cdot \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(\left(r\_m \cdot w\right) \cdot \frac{w}{v + -1}\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 75000000.0)
   (+ -1.5 (fma (* w (* -0.25 (* r_m r_m))) w (/ 2.0 (* r_m r_m))))
   (-
    (+ 3.0 (* r_m (* (fma v -0.25 0.375) (* (* r_m w) (/ w (+ v -1.0))))))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 75000000.0) {
		tmp = -1.5 + fma((w * (-0.25 * (r_m * r_m))), w, (2.0 / (r_m * r_m)));
	} else {
		tmp = (3.0 + (r_m * (fma(v, -0.25, 0.375) * ((r_m * w) * (w / (v + -1.0)))))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 75000000.0)
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r_m * r_m))), w, Float64(2.0 / Float64(r_m * r_m))));
	else
		tmp = Float64(Float64(3.0 + Float64(r_m * Float64(fma(v, -0.25, 0.375) * Float64(Float64(r_m * w) * Float64(w / Float64(v + -1.0)))))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 75000000.0], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(r$95$m * N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] * N[(w / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 7.5e7

   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

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   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 91.2%

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

   if 7.5e7 < r

   1. Initial program 90.6%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 88.8

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

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

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

      13. lower-fma.f64 N/A

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

      14. metadata-eval 88.8

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. associate-*l* N/A

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

      18. lower-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.0

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

   4. Applied rewrites 95.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. associate-*l* N/A

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

      13. metadata-eval N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - r \cdot \left(\left(v \cdot \color{blue}{\frac{-1}{4}} + 3 \cdot \frac{1}{8}\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{1 - v}\right)\right) - \frac{9}{2} \]

      14. metadata-eval N/A

          \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - r \cdot \left(\left(v \cdot \frac{-1}{4} + \color{blue}{\frac{3}{8}}\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{1 - v}\right)\right) - \frac{9}{2} \]

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/l* N/A

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

      19. lower-*.f64 N/A

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

      20. lower-/.f64 95.0

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

   6. Applied rewrites 95.0%

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

   7. Taylor expanded in r around inf

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

   9. Applied rewrites 95.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 75000000:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + r \cdot \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{w}{v + -1}\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;w \cdot w \leq 10^{+44}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(r\_m \cdot 0.375\right), r\_m, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r\_m \cdot r\_m\right)\right), w, t\_0\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= (* w w) 1e+44)
     (- t_0 (fma (* (* w w) (* r_m 0.375)) r_m 1.5))
     (+ -1.5 (fma (* w (* -0.25 (* r_m r_m))) w 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 ((w * w) <= 1e+44) {
		tmp = t_0 - fma(((w * w) * (r_m * 0.375)), r_m, 1.5);
	} else {
		tmp = -1.5 + fma((w * (-0.25 * (r_m * r_m))), w, 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(w * w) <= 1e+44)
		tmp = Float64(t_0 - fma(Float64(Float64(w * w) * Float64(r_m * 0.375)), r_m, 1.5));
	else
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r_m * r_m))), w, 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[N[(w * w), $MachinePrecision], 1e+44], N[(t$95$0 - N[(N[(N[(w * w), $MachinePrecision] * N[(r$95$m * 0.375), $MachinePrecision]), $MachinePrecision] * r$95$m + 1.5), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.0000000000000001e44

   1. Initial program 91.8%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 89.6%

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

   if 1.0000000000000001e44 < (*.f64 w w)

   1. Initial program 74.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+44}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\left(w \cdot w\right) \cdot \left(r \cdot 0.375\right), r, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;w \cdot w \leq 10^{+44}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(r\_m \cdot \left(r\_m \cdot \left(w \cdot w\right)\right), 0.375, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r\_m \cdot r\_m\right)\right), w, t\_0\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= (* w w) 1e+44)
     (- t_0 (fma (* r_m (* r_m (* w w))) 0.375 1.5))
     (+ -1.5 (fma (* w (* -0.25 (* r_m r_m))) w 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 ((w * w) <= 1e+44) {
		tmp = t_0 - fma((r_m * (r_m * (w * w))), 0.375, 1.5);
	} else {
		tmp = -1.5 + fma((w * (-0.25 * (r_m * r_m))), w, 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(w * w) <= 1e+44)
		tmp = Float64(t_0 - fma(Float64(r_m * Float64(r_m * Float64(w * w))), 0.375, 1.5));
	else
		tmp = Float64(-1.5 + fma(Float64(w * Float64(-0.25 * Float64(r_m * r_m))), w, 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[N[(w * w), $MachinePrecision], 1e+44], N[(t$95$0 - N[(N[(r$95$m * N[(r$95$m * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375 + 1.5), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(N[(w * N[(-0.25 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * w + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.0000000000000001e44

   1. Initial program 91.8%

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

   3. Taylor expanded in 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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 89.6%

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

   if 1.0000000000000001e44 < (*.f64 w w)

   1. Initial program 74.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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 97.1%

      \[\leadsto \color{blue}{-1.5 + \mathsf{fma}\left(\left(-0.25 \cdot \left(r \cdot r\right)\right) \cdot w, w, \frac{2}{r \cdot r}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+44}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right), 0.375, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \mathsf{fma}\left(w \cdot \left(-0.25 \cdot \left(r \cdot r\right)\right), w, \frac{2}{r \cdot r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 93.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \mathsf{fma}\left(w \cdot \left(r\_m \cdot -0.25\right), r\_m \cdot w, -1.5 + \frac{2}{r\_m \cdot r\_m}\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (fma (* w (* r_m -0.25)) (* r_m w) (+ -1.5 (/ 2.0 (* r_m r_m)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return fma((w * (r_m * -0.25)), (r_m * w), (-1.5 + (2.0 / (r_m * r_m))));
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	return fma(Float64(w * Float64(r_m * -0.25)), Float64(r_m * w), Float64(-1.5 + Float64(2.0 / Float64(r_m * r_m))))
end

Wolfram

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

Derivation

1. Initial program 83.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 43.5

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

5. Applied rewrites 43.5%

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

6. Taylor expanded in v around inf

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

   1. associate--r+ N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. sub-neg N/A

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

   13. metadata-eval N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 83.4

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

8. Applied rewrites 83.4%

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

10. Applied rewrites 92.9%

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

11. Final simplification 92.9%

    \[\leadsto \mathsf{fma}\left(w \cdot \left(r \cdot -0.25\right), r \cdot w, -1.5 + \frac{2}{r \cdot r}\right) \]
12. Add Preprocessing

Alternative 11: 56.5% 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 83.9%

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

3. Taylor expanded in 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 58.1

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

5. Applied rewrites 58.1%

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

Alternative 12: 44.1% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.9%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
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 43.5

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

5. Applied rewrites 43.5%

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

Reproduce

herbie shell --seed 2024216 
(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))