Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.0%

Time: 9.3s

Alternatives: 14

Speedup: 1.4×

• 53.0% 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: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{r\_m}{1 - v}\\ \mathbf{if}\;r\_m \leq 4.4 \cdot 10^{+209}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - w \cdot \left(\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(t\_0 \cdot \left(w \cdot r\_m\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\_m\right)\right) \cdot t\_0\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ r_m (- 1.0 v))))
   (if (<= r_m 4.4e+209)
     (-
      (-
       (+ 3.0 (/ 2.0 (* r_m r_m)))
       (* w (* (* 0.125 (fma v -2.0 3.0)) (* t_0 (* w r_m)))))
      4.5)
     (- (- 3.0 (* (* (* (fma -0.25 v 0.375) w) (* w r_m)) t_0)) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = r_m / (1.0 - v);
	double tmp;
	if (r_m <= 4.4e+209) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - (w * ((0.125 * fma(v, -2.0, 3.0)) * (t_0 * (w * r_m))))) - 4.5;
	} else {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * w) * (w * r_m)) * t_0)) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(r_m / Float64(1.0 - v))
	tmp = 0.0
	if (r_m <= 4.4e+209)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(w * Float64(Float64(0.125 * fma(v, -2.0, 3.0)) * Float64(t_0 * Float64(w * r_m))))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * w) * Float64(w * r_m)) * t_0)) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 4.4e+209], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(w * N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.3999999999999997e209

   1. Initial program 84.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 90.4%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sub-sign-inv N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

   6. Applied rewrites 97.0%

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

   if 4.3999999999999997e209 < r

   1. Initial program 69.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.7

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

   10. Applied rewrites 99.7%

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

Alternative 2: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ t_1 := \left(3 + t\_0\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\_m\right) \cdot r\_m\right)}{1 - v}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\left(\mathsf{fma}\left(0.25 \cdot w, w, \frac{1.5}{r\_m \cdot r\_m}\right) \cdot r\_m\right) \cdot \left(-r\_m\right)\\ \mathbf{elif}\;t\_1 \leq 2.9936101090905534:\\ \;\;\;\;\left(3 - \left(\left(\left(w \cdot r\_m\right) \cdot r\_m\right) \cdot w\right) \cdot \mathsf{fma}\left(v, 0.125, 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - 1.5\\ \end{array} \end{array} \]

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))) (* (* (* w w) r_m) r_m))
           (- 1.0 v)))))
   (if (<= t_1 -1e+85)
     (* (* (fma (* 0.25 w) w (/ 1.5 (* r_m r_m))) r_m) (- r_m))
     (if (<= t_1 2.9936101090905534)
       (- (- 3.0 (* (* (* (* w r_m) r_m) w) (fma v 0.125 0.375))) 4.5)
       (- t_0 1.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 2.0 / (r_m * r_m);
	double t_1 = (3.0 + t_0) - (((0.125 * (3.0 - (2.0 * v))) * (((w * w) * r_m) * r_m)) / (1.0 - v));
	double tmp;
	if (t_1 <= -1e+85) {
		tmp = (fma((0.25 * w), w, (1.5 / (r_m * r_m))) * r_m) * -r_m;
	} else if (t_1 <= 2.9936101090905534) {
		tmp = (3.0 - ((((w * r_m) * r_m) * w) * fma(v, 0.125, 0.375))) - 4.5;
	} else {
		tmp = t_0 - 1.5;
	}
	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(Float64(Float64(w * w) * r_m) * r_m)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_1 <= -1e+85)
		tmp = Float64(Float64(fma(Float64(0.25 * w), w, Float64(1.5 / Float64(r_m * r_m))) * r_m) * Float64(-r_m));
	elseif (t_1 <= 2.9936101090905534)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r_m) * r_m) * w) * fma(v, 0.125, 0.375))) - 4.5);
	else
		tmp = Float64(t_0 - 1.5);
	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]}, 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[(N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+85], N[(N[(N[(N[(0.25 * w), $MachinePrecision] * w + N[(1.5 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * r$95$m), $MachinePrecision] * (-r$95$m)), $MachinePrecision], If[LessEqual[t$95$1, 2.9936101090905534], N[(N[(3.0 - N[(N[(N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * w), $MachinePrecision] * N[(v * 0.125 + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$0 - 1.5), $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))) < -1e85

   1. Initial program 85.5%

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

   3. 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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 83.7%

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

   if -1e85 < (-.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))) < 2.99361010909055336

   1. Initial program 95.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 84.3%

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

   11. Taylor expanded in v around 0

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

       1. distribute-rgt-out-- N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-out-- N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out-- N/A

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

       7. metadata-eval N/A

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

       8. *-rgt-identity N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. distribute-lft-out N/A

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

       12. lower-*.f64 N/A

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

   13. Applied rewrites 82.5%

       \[\leadsto \left(3 - \color{blue}{\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot \mathsf{fma}\left(v, 0.125, 0.375\right)}\right) - 4.5 \]

   if 2.99361010909055336 < (-.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 80.0%

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 3: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \left(w \cdot w\right) \cdot r\_m\\ t_1 := \frac{2}{r\_m \cdot r\_m}\\ t_2 := \left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_0 \cdot r\_m\right)}{1 - v}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+85}:\\ \;\;\;\;\left(t\_0 \cdot -0.25\right) \cdot r\_m\\ \mathbf{elif}\;t\_2 \leq 2.9936101090905534:\\ \;\;\;\;\left(3 - \left(\left(\left(w \cdot r\_m\right) \cdot r\_m\right) \cdot w\right) \cdot \mathsf{fma}\left(v, 0.125, 0.375\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (* w w) r_m))
        (t_1 (/ 2.0 (* r_m r_m)))
        (t_2
         (-
          (+ 3.0 t_1)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_0 r_m)) (- 1.0 v)))))
   (if (<= t_2 -1e+85)
     (* (* t_0 -0.25) r_m)
     (if (<= t_2 2.9936101090905534)
       (- (- 3.0 (* (* (* (* w r_m) r_m) w) (fma v 0.125 0.375))) 4.5)
       (- t_1 1.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (w * w) * r_m;
	double t_1 = 2.0 / (r_m * r_m);
	double t_2 = (3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r_m)) / (1.0 - v));
	double tmp;
	if (t_2 <= -1e+85) {
		tmp = (t_0 * -0.25) * r_m;
	} else if (t_2 <= 2.9936101090905534) {
		tmp = (3.0 - ((((w * r_m) * r_m) * w) * fma(v, 0.125, 0.375))) - 4.5;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(w * w) * r_m)
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	t_2 = Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_0 * r_m)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_2 <= -1e+85)
		tmp = Float64(Float64(t_0 * -0.25) * r_m);
	elseif (t_2 <= 2.9936101090905534)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(Float64(w * r_m) * r_m) * w) * fma(v, 0.125, 0.375))) - 4.5);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+85], N[(N[(t$95$0 * -0.25), $MachinePrecision] * r$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.9936101090905534], N[(N[(3.0 - N[(N[(N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * w), $MachinePrecision] * N[(v * 0.125 + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$1 - 1.5), $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))) < -1e85

   1. Initial program 85.5%

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

   3. 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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 87.1

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

   5. Applied rewrites 87.1%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 83.7%

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

   if -1e85 < (-.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))) < 2.99361010909055336

   1. Initial program 95.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in v around 0

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

   10. Applied rewrites 84.3%

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

   11. Taylor expanded in v around 0

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

       1. distribute-rgt-out-- N/A

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

       2. metadata-eval N/A

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

       3. metadata-eval N/A

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

       4. distribute-rgt-out-- N/A

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

       5. associate-*r* N/A

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

       6. distribute-rgt-out-- N/A

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

       7. metadata-eval N/A

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

       8. *-rgt-identity N/A

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

       9. *-commutative N/A

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

       10. *-commutative N/A

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

       11. distribute-lft-out N/A

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

       12. lower-*.f64 N/A

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

   13. Applied rewrites 82.5%

       \[\leadsto \left(3 - \color{blue}{\left(\left(\left(w \cdot r\right) \cdot r\right) \cdot w\right) \cdot \mathsf{fma}\left(v, 0.125, 0.375\right)}\right) - 4.5 \]

   if 2.99361010909055336 < (-.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 80.0%

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 4: 88.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \left(w \cdot w\right) \cdot r\_m\\ t_1 := \frac{2}{r\_m \cdot r\_m}\\ t_2 := \left(3 + t\_1\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(t\_0 \cdot r\_m\right)}{1 - v}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+50}:\\ \;\;\;\;\left(t\_0 \cdot -0.25\right) \cdot r\_m\\ \mathbf{elif}\;t\_2 \leq 2.9936101090905534:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot \left(w \cdot w\right), r\_m \cdot r\_m, 3\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 - 1.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (* w w) r_m))
        (t_1 (/ 2.0 (* r_m r_m)))
        (t_2
         (-
          (+ 3.0 t_1)
          (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_0 r_m)) (- 1.0 v)))))
   (if (<= t_2 -2e+50)
     (* (* t_0 -0.25) r_m)
     (if (<= t_2 2.9936101090905534)
       (- (fma (* -0.375 (* w w)) (* r_m r_m) 3.0) 4.5)
       (- t_1 1.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (w * w) * r_m;
	double t_1 = 2.0 / (r_m * r_m);
	double t_2 = (3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r_m)) / (1.0 - v));
	double tmp;
	if (t_2 <= -2e+50) {
		tmp = (t_0 * -0.25) * r_m;
	} else if (t_2 <= 2.9936101090905534) {
		tmp = fma((-0.375 * (w * w)), (r_m * r_m), 3.0) - 4.5;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(w * w) * r_m)
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	t_2 = Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_0 * r_m)) / Float64(1.0 - v)))
	tmp = 0.0
	if (t_2 <= -2e+50)
		tmp = Float64(Float64(t_0 * -0.25) * r_m);
	elseif (t_2 <= 2.9936101090905534)
		tmp = Float64(fma(Float64(-0.375 * Float64(w * w)), Float64(r_m * r_m), 3.0) - 4.5);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+50], N[(N[(t$95$0 * -0.25), $MachinePrecision] * r$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.9936101090905534], N[(N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * r$95$m), $MachinePrecision] + 3.0), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$1 - 1.5), $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))) < -2.0000000000000002e50

   1. Initial program 85.6%

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

   3. 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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 85.9

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 82.7%

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

   if -2.0000000000000002e50 < (-.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))) < 2.99361010909055336

   1. Initial program 98.6%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 78.3%

      \[\leadsto \mathsf{fma}\left(-0.375 \cdot \left(w \cdot w\right), r \cdot r, 3\right) - 4.5 \]

   if 2.99361010909055336 < (-.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 80.0%

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 92.5

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

   5. Applied rewrites 92.5%

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

Alternative 5: 88.7% accurate, 0.8× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* (* w w) r_m)) (t_1 (/ 2.0 (* r_m r_m))))
   (if (<=
        (-
         (+ 3.0 t_1)
         (/ (* (* 0.125 (- 3.0 (* 2.0 v))) (* t_0 r_m)) (- 1.0 v)))
        -10.0)
     (* (* t_0 -0.25) r_m)
     (- t_1 1.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (w * w) * r_m;
	double t_1 = 2.0 / (r_m * r_m);
	double tmp;
	if (((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r_m)) / (1.0 - v))) <= -10.0) {
		tmp = (t_0 * -0.25) * r_m;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (w * w) * r_m
    t_1 = 2.0d0 / (r_m * r_m)
    if (((3.0d0 + t_1) - (((0.125d0 * (3.0d0 - (2.0d0 * v))) * (t_0 * r_m)) / (1.0d0 - v))) <= (-10.0d0)) then
        tmp = (t_0 * (-0.25d0)) * r_m
    else
        tmp = t_1 - 1.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = (w * w) * r_m;
	double t_1 = 2.0 / (r_m * r_m);
	double tmp;
	if (((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r_m)) / (1.0 - v))) <= -10.0) {
		tmp = (t_0 * -0.25) * r_m;
	} else {
		tmp = t_1 - 1.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = (w * w) * r_m
	t_1 = 2.0 / (r_m * r_m)
	tmp = 0
	if ((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r_m)) / (1.0 - v))) <= -10.0:
		tmp = (t_0 * -0.25) * r_m
	else:
		tmp = t_1 - 1.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(w * w) * r_m)
	t_1 = Float64(2.0 / Float64(r_m * r_m))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_1) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(2.0 * v))) * Float64(t_0 * r_m)) / Float64(1.0 - v))) <= -10.0)
		tmp = Float64(Float64(t_0 * -0.25) * r_m);
	else
		tmp = Float64(t_1 - 1.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = (w * w) * r_m;
	t_1 = 2.0 / (r_m * r_m);
	tmp = 0.0;
	if (((3.0 + t_1) - (((0.125 * (3.0 - (2.0 * v))) * (t_0 * r_m)) / (1.0 - v))) <= -10.0)
		tmp = (t_0 * -0.25) * r_m;
	else
		tmp = t_1 - 1.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(w * w), $MachinePrecision] * r$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * r$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -10.0], N[(N[(t$95$0 * -0.25), $MachinePrecision] * r$95$m), $MachinePrecision], N[(t$95$1 - 1.5), $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))) < -10

   1. Initial program 86.2%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 82.6

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

   5. Applied rewrites 82.6%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 79.7%

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

   if -10 < (-.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 80.3%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 91.5

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

   5. Applied rewrites 91.5%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \left(\left(w \cdot 0.125\right) \cdot \left(\frac{r\_m}{1 - v} \cdot \left(w \cdot r\_m\right)\right)\right)\right) - 4.5 \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 91.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 98.3

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

6. Applied rewrites 98.3%

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

Alternative 7: 93.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{+244}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(r\_m, \left(w \cdot r\_m\right) \cdot \left(0.25 \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375 \cdot w, w \cdot \left(r\_m \cdot r\_m\right), t\_0 + 3\right) - 4.5\\ \end{array} \end{array} \]

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) 5e+244)
     (- t_0 (fma r_m (* (* w r_m) (* 0.25 w)) 1.5))
     (- (fma (* -0.375 w) (* w (* r_m r_m)) (+ t_0 3.0)) 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 ((w * w) <= 5e+244) {
		tmp = t_0 - fma(r_m, ((w * r_m) * (0.25 * w)), 1.5);
	} else {
		tmp = fma((-0.375 * w), (w * (r_m * r_m)), (t_0 + 3.0)) - 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 (Float64(w * w) <= 5e+244)
		tmp = Float64(t_0 - fma(r_m, Float64(Float64(w * r_m) * Float64(0.25 * w)), 1.5));
	else
		tmp = Float64(fma(Float64(-0.375 * w), Float64(w * Float64(r_m * r_m)), Float64(t_0 + 3.0)) - 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[N[(w * w), $MachinePrecision], 5e+244], N[(t$95$0 - N[(r$95$m * N[(N[(w * r$95$m), $MachinePrecision] * N[(0.25 * w), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-0.375 * w), $MachinePrecision] * N[(w * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + 3.0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 5.00000000000000022e244

   1. Initial program 89.8%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 85.9%

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

   9. Applied rewrites 89.4%

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

   if 5.00000000000000022e244 < (*.f64 w w)

   1. Initial program 69.6%

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

   3. Taylor expanded in v around 0

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

      1. fp-cancel-sub-sign-inv N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375 \cdot \left(w \cdot w\right), r \cdot r, 3 + \frac{2}{r \cdot r}\right)} - 4.5 \]
   6. Step-by-step derivation

   7. Applied rewrites 99.0%

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

Alternative 8: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 32500000:\\ \;\;\;\;\frac{2}{r\_m \cdot r\_m} - \mathsf{fma}\left(\left(\left(w \cdot r\_m\right) \cdot r\_m\right) \cdot 0.25, w, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(\left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot w\right) \cdot \left(w \cdot r\_m\right)\right) \cdot \frac{r\_m}{1 - v}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 32500000.0)
   (- (/ 2.0 (* r_m r_m)) (fma (* (* (* w r_m) r_m) 0.25) w 1.5))
   (-
    (- 3.0 (* (* (* (fma -0.25 v 0.375) w) (* w r_m)) (/ 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 <= 32500000.0) {
		tmp = (2.0 / (r_m * r_m)) - fma((((w * r_m) * r_m) * 0.25), w, 1.5);
	} else {
		tmp = (3.0 - (((fma(-0.25, v, 0.375) * w) * (w * r_m)) * (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 <= 32500000.0)
		tmp = Float64(Float64(2.0 / Float64(r_m * r_m)) - fma(Float64(Float64(Float64(w * r_m) * r_m) * 0.25), w, 1.5));
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(fma(-0.25, v, 0.375) * w) * Float64(w * r_m)) * 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, 32500000.0], N[(N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(w * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * 0.25), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 - N[(N[(N[(N[(-0.25 * v + 0.375), $MachinePrecision] * w), $MachinePrecision] * N[(w * r$95$m), $MachinePrecision]), $MachinePrecision] * N[(r$95$m / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.25e7

   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. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 86.1

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

   5. Applied rewrites 86.1%

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

   7. Applied rewrites 90.2%

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

   if 3.25e7 < r

   1. Initial program 86.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in r around inf

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in v around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 99.8

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

   10. Applied rewrites 99.8%

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

Alternative 9: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{+244}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(r\_m, \left(w \cdot r\_m\right) \cdot \left(0.25 \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(0.375 \cdot \left(r\_m \cdot r\_m\right)\right) \cdot w, w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= (* w w) 5e+244)
     (- t_0 (fma r_m (* (* w r_m) (* 0.25 w)) 1.5))
     (- t_0 (fma (* (* 0.375 (* r_m r_m)) w) w 1.5)))))

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 ((w * w) <= 5e+244) {
		tmp = t_0 - fma(r_m, ((w * r_m) * (0.25 * w)), 1.5);
	} else {
		tmp = t_0 - fma(((0.375 * (r_m * r_m)) * w), w, 1.5);
	}
	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 (Float64(w * w) <= 5e+244)
		tmp = Float64(t_0 - fma(r_m, Float64(Float64(w * r_m) * Float64(0.25 * w)), 1.5));
	else
		tmp = Float64(t_0 - fma(Float64(Float64(0.375 * Float64(r_m * r_m)) * w), w, 1.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 5e+244], N[(t$95$0 - N[(r$95$m * N[(N[(w * r$95$m), $MachinePrecision] * N[(0.25 * w), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(N[(0.375 * N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 5.00000000000000022e244

   1. Initial program 89.8%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 85.9%

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

   9. Applied rewrites 89.4%

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

   if 5.00000000000000022e244 < (*.f64 w w)

   1. Initial program 69.6%

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

   3. 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. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 99.0

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

   5. Applied rewrites 99.0%

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

Alternative 10: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;w \cdot w \leq 5 \cdot 10^{+244}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(r\_m, \left(w \cdot r\_m\right) \cdot \left(0.25 \cdot w\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(0.375 \cdot r\_m\right) \cdot r\_m\right) \cdot w, w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= (* w w) 5e+244)
     (- t_0 (fma r_m (* (* w r_m) (* 0.25 w)) 1.5))
     (- t_0 (fma (* (* (* 0.375 r_m) r_m) w) w 1.5)))))

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 ((w * w) <= 5e+244) {
		tmp = t_0 - fma(r_m, ((w * r_m) * (0.25 * w)), 1.5);
	} else {
		tmp = t_0 - fma((((0.375 * r_m) * r_m) * w), w, 1.5);
	}
	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 (Float64(w * w) <= 5e+244)
		tmp = Float64(t_0 - fma(r_m, Float64(Float64(w * r_m) * Float64(0.25 * w)), 1.5));
	else
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(0.375 * r_m) * r_m) * w), w, 1.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 5e+244], N[(t$95$0 - N[(r$95$m * N[(N[(w * r$95$m), $MachinePrecision] * N[(0.25 * w), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(N[(N[(0.375 * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 5.00000000000000022e244

   1. Initial program 89.8%

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

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

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-*r* N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 85.9%

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

   9. Applied rewrites 89.4%

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

   if 5.00000000000000022e244 < (*.f64 w w)

   1. Initial program 69.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   6. 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. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 99.0

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

   7. Applied rewrites 99.0%

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

Alternative 11: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{-310}:\\ \;\;\;\;t\_0 - 1.5\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \mathsf{fma}\left(\left(\left(0.375 \cdot r\_m\right) \cdot r\_m\right) \cdot w, w, 1.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r_m r_m))))
   (if (<= (* w w) 4e-310)
     (- t_0 1.5)
     (- t_0 (fma (* (* (* 0.375 r_m) r_m) w) w 1.5)))))

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 ((w * w) <= 4e-310) {
		tmp = t_0 - 1.5;
	} else {
		tmp = t_0 - fma((((0.375 * r_m) * r_m) * w), w, 1.5);
	}
	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 (Float64(w * w) <= 4e-310)
		tmp = Float64(t_0 - 1.5);
	else
		tmp = Float64(t_0 - fma(Float64(Float64(Float64(0.375 * r_m) * r_m) * w), w, 1.5));
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 4e-310], N[(t$95$0 - 1.5), $MachinePrecision], N[(t$95$0 - N[(N[(N[(N[(0.375 * r$95$m), $MachinePrecision] * r$95$m), $MachinePrecision] * w), $MachinePrecision] * w + 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 3.999999999999988e-310

   1. Initial program 82.4%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.8

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

   5. Applied rewrites 79.8%

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

   if 3.999999999999988e-310 < (*.f64 w w)

   1. Initial program 83.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in v around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \left(\frac{3}{2} + \frac{3}{8} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]
   6. 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. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-*.f64 91.3

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

   7. Applied rewrites 91.3%

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

Alternative 12: 56.9% accurate, 3.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 81.8%

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

   3. Taylor expanded in 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 49.2

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

   5. Applied rewrites 49.2%

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

   if 1.1499999999999999 < r

   1. Initial program 87.5%

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

   3. Taylor expanded in w around 0

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 22.7

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

   5. Applied rewrites 22.7%

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

   6. Taylor expanded in r around inf

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

   8. Applied rewrites 21.1%

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

Alternative 13: 57.4% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 50.7

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

5. Applied rewrites 50.7%

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

Alternative 14: 14.2% accurate, 73.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

3. Taylor expanded in w around 0

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-/.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 50.7

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

5. Applied rewrites 50.7%

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

6. Taylor expanded in r around inf

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

8. Applied rewrites 13.4%

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

Reproduce

herbie shell --seed 2024326 
(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))