Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 98.8%

Time: 13.6s

Alternatives: 17

Speedup: 1.5×

• 54.2% 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: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 2.4 \cdot 10^{-59}:\\ \;\;\;\;\left(\left(3 + 2 \cdot {r\_m}^{-2}\right) - \left(\left(r\_m \cdot \left(v \cdot -0.25 + 0.375\right)\right) \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r\_m}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) + r\_m \cdot \left(w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{v + -1}{r\_m \cdot w}}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 2.4e-59)
   (-
    (-
     (+ 3.0 (* 2.0 (pow r_m -2.0)))
     (* (* (* r_m (+ (* v -0.25) 0.375)) w) (/ w (/ (- 1.0 v) r_m))))
    4.5)
   (-
    (+
     (+ 3.0 (/ 2.0 (* r_m r_m)))
     (* r_m (* w (/ (fma v -0.25 0.375) (/ (+ v -1.0) (* r_m w))))))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2.4e-59) {
		tmp = ((3.0 + (2.0 * pow(r_m, -2.0))) - (((r_m * ((v * -0.25) + 0.375)) * w) * (w / ((1.0 - v) / r_m)))) - 4.5;
	} else {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) + (r_m * (w * (fma(v, -0.25, 0.375) / ((v + -1.0) / (r_m * w)))))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 2.4e-59)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 * (r_m ^ -2.0))) - Float64(Float64(Float64(r_m * Float64(Float64(v * -0.25) + 0.375)) * w) * Float64(w / Float64(Float64(1.0 - v) / r_m)))) - 4.5);
	else
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) + Float64(r_m * Float64(w * Float64(fma(v, -0.25, 0.375) / Float64(Float64(v + -1.0) / Float64(r_m * w)))))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 2.4e-59], N[(N[(N[(3.0 + N[(2.0 * N[Power[r$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(r$95$m * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(w / N[(N[(1.0 - v), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(r$95$m * N[(w * N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.40000000000000015e-59

   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. Step-by-step derivation

      1. associate-/l* 81.9%

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

      2. cancel-sign-sub-inv 81.9%

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

      3. metadata-eval 81.9%

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

      4. +-commutative 81.9%

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

      5. *-commutative 81.9%

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

      6. fma-undefine 81.9%

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

      7. *-commutative 81.9%

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

      8. *-commutative 81.9%

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

      9. associate-/l* 81.9%

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

      10. *-commutative 81.9%

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

      11. associate-*r/ 81.9%

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

      12. associate-*r* 79.1%

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

      13. associate-*l* 93.1%

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

      14. associate-*r* 95.9%

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

   4. Applied egg-rr 96.4%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(\left(v \cdot -0.25 + 0.375\right) \cdot r\right) \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r}}}\right) - 4.5 \]
   5. Step-by-step derivation

      1. *-un-lft-identity 96.4%

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

      2. div-inv 96.4%

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

      3. pow2 96.4%

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

      4. pow-flip 96.6%

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

      5. metadata-eval 96.6%

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

   6. Applied egg-rr 96.6%

      \[\leadsto \left(\left(3 + \color{blue}{1 \cdot \left(2 \cdot {r}^{-2}\right)}\right) - \left(\left(\left(v \cdot -0.25 + 0.375\right) \cdot r\right) \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r}}\right) - 4.5 \]
   7. Step-by-step derivation

      1. *-lft-identity 96.6%

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

   8. Simplified 96.6%

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

   if 2.40000000000000015e-59 < r

   1. Initial program 87.0%

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

      1. associate-/l* 94.0%

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

      2. cancel-sign-sub-inv 94.0%

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

      3. metadata-eval 94.0%

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

      4. +-commutative 94.0%

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

      5. *-commutative 94.0%

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

      6. fma-undefine 94.0%

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

      7. *-commutative 94.0%

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

      8. *-commutative 94.0%

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

      9. associate-/l* 94.0%

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

      10. *-commutative 94.0%

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

      11. associate-*r/ 92.7%

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

      12. associate-*r* 86.5%

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

      13. associate-*l* 87.7%

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

      14. associate-*r* 84.3%

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

   4. Applied egg-rr 84.3%

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

   5. Taylor expanded in r around 0 91.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)} \cdot \frac{w}{\frac{1 - v}{r}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. clear-num 91.6%

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

      2. un-div-inv 91.5%

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

      3. associate-*r* 91.5%

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

      4. +-commutative 91.5%

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

      5. fma-define 91.5%

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

   7. Applied egg-rr 91.5%

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

      1. associate-/l* 98.5%

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

      2. associate-*l* 98.6%

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

      3. fma-define 98.6%

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

      4. *-commutative 98.6%

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

      5. fma-undefine 98.6%

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

      6. associate-/l/ 99.8%

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

      7. *-commutative 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.4 \cdot 10^{-59}:\\ \;\;\;\;\left(\left(3 + 2 \cdot {r}^{-2}\right) - \left(\left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right) \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + r \cdot \left(w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{v + -1}{r \cdot w}}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := 3 + \frac{2}{r\_m \cdot r\_m}\\ \mathbf{if}\;r\_m \leq 7.8 \cdot 10^{-82}:\\ \;\;\;\;\left(t\_0 + \left(\left(r\_m \cdot \left(v \cdot -0.25 + 0.375\right)\right) \cdot w\right) \cdot \frac{w}{\frac{v + -1}{r\_m}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + r\_m \cdot \left(w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{v + -1}{r\_m \cdot w}}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (+ 3.0 (/ 2.0 (* r_m r_m)))))
   (if (<= r_m 7.8e-82)
     (-
      (+ t_0 (* (* (* r_m (+ (* v -0.25) 0.375)) w) (/ w (/ (+ v -1.0) r_m))))
      4.5)
     (-
      (+ t_0 (* r_m (* w (/ (fma v -0.25 0.375) (/ (+ v -1.0) (* r_m w))))))
      4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = 3.0 + (2.0 / (r_m * r_m));
	double tmp;
	if (r_m <= 7.8e-82) {
		tmp = (t_0 + (((r_m * ((v * -0.25) + 0.375)) * w) * (w / ((v + -1.0) / r_m)))) - 4.5;
	} else {
		tmp = (t_0 + (r_m * (w * (fma(v, -0.25, 0.375) / ((v + -1.0) / (r_m * w)))))) - 4.5;
	}
	return tmp;
}

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(3.0 + Float64(2.0 / Float64(r_m * r_m)))
	tmp = 0.0
	if (r_m <= 7.8e-82)
		tmp = Float64(Float64(t_0 + Float64(Float64(Float64(r_m * Float64(Float64(v * -0.25) + 0.375)) * w) * Float64(w / Float64(Float64(v + -1.0) / r_m)))) - 4.5);
	else
		tmp = Float64(Float64(t_0 + Float64(r_m * Float64(w * Float64(fma(v, -0.25, 0.375) / Float64(Float64(v + -1.0) / Float64(r_m * w)))))) - 4.5);
	end
	return tmp
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 7.8e-82], N[(N[(t$95$0 + N[(N[(N[(r$95$m * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(w / N[(N[(v + -1.0), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(t$95$0 + N[(r$95$m * N[(w * N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(N[(v + -1.0), $MachinePrecision] / N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 7.79999999999999947e-82

   1. Initial program 79.9%

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

      1. associate-/l* 81.5%

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

      2. cancel-sign-sub-inv 81.5%

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

      3. metadata-eval 81.5%

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

      4. +-commutative 81.5%

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

      5. *-commutative 81.5%

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

      6. fma-undefine 81.5%

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

      7. *-commutative 81.5%

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

      8. *-commutative 81.5%

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

      9. associate-/l* 81.5%

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

      10. *-commutative 81.5%

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

      11. associate-*r/ 81.5%

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

      12. associate-*r* 78.6%

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

      13. associate-*l* 92.9%

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

      14. associate-*r* 95.8%

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

   4. Applied egg-rr 96.4%

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

   if 7.79999999999999947e-82 < r

   1. Initial program 87.6%

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

      1. associate-/l* 94.3%

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

      2. cancel-sign-sub-inv 94.3%

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

      3. metadata-eval 94.3%

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

      4. +-commutative 94.3%

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

      5. *-commutative 94.3%

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

      6. fma-undefine 94.3%

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

      7. *-commutative 94.3%

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

      8. *-commutative 94.3%

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

      9. associate-/l* 94.2%

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

      10. *-commutative 94.2%

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

      11. associate-*r/ 93.1%

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

      12. associate-*r* 87.1%

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

      13. associate-*l* 88.3%

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

      14. associate-*r* 85.0%

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

   4. Applied egg-rr 85.0%

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

   5. Taylor expanded in r around 0 92.0%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(r \cdot \left(w \cdot \left(0.375 + -0.25 \cdot v\right)\right)\right)} \cdot \frac{w}{\frac{1 - v}{r}}\right) - 4.5 \]
   6. Step-by-step derivation

      1. clear-num 92.0%

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

      2. un-div-inv 91.9%

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

      3. associate-*r* 91.9%

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

      4. +-commutative 91.9%

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

      5. fma-define 91.9%

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

   7. Applied egg-rr 91.9%

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

      1. associate-/l* 98.6%

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

      2. associate-*l* 98.6%

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

      3. fma-define 98.6%

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

      4. *-commutative 98.6%

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

      5. fma-undefine 98.6%

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

      6. associate-/l/ 99.8%

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

      7. *-commutative 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.8 \cdot 10^{-82}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right) \cdot w\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + r \cdot \left(w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{v + -1}{r \cdot w}}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{w}{\frac{v + -1}{r\_m}}\\ \mathbf{if}\;r\_m \leq 5200000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) + \left(\left(r\_m \cdot \left(v \cdot -0.25 + 0.375\right)\right) \cdot w\right) \cdot t\_0\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(w \cdot \left(r\_m \cdot t\_0\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ w (/ (+ v -1.0) r_m))))
   (if (<= r_m 5200000.0)
     (-
      (+
       (+ 3.0 (/ 2.0 (* r_m r_m)))
       (* (* (* r_m (+ (* v -0.25) 0.375)) w) t_0))
      4.5)
     (+ 3.0 (- (* (* w (* r_m t_0)) (* 0.125 (+ 3.0 (* -2.0 v)))) 4.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = w / ((v + -1.0) / r_m);
	double tmp;
	if (r_m <= 5200000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) + (((r_m * ((v * -0.25) + 0.375)) * w) * t_0)) - 4.5;
	} else {
		tmp = 3.0 + (((w * (r_m * t_0)) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	}
	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) :: tmp
    t_0 = w / ((v + (-1.0d0)) / r_m)
    if (r_m <= 5200000.0d0) then
        tmp = ((3.0d0 + (2.0d0 / (r_m * r_m))) + (((r_m * ((v * (-0.25d0)) + 0.375d0)) * w) * t_0)) - 4.5d0
    else
        tmp = 3.0d0 + (((w * (r_m * t_0)) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))) - 4.5d0)
    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 / ((v + -1.0) / r_m);
	double tmp;
	if (r_m <= 5200000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) + (((r_m * ((v * -0.25) + 0.375)) * w) * t_0)) - 4.5;
	} else {
		tmp = 3.0 + (((w * (r_m * t_0)) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = w / ((v + -1.0) / r_m)
	tmp = 0
	if r_m <= 5200000.0:
		tmp = ((3.0 + (2.0 / (r_m * r_m))) + (((r_m * ((v * -0.25) + 0.375)) * w) * t_0)) - 4.5
	else:
		tmp = 3.0 + (((w * (r_m * t_0)) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5)
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(w / Float64(Float64(v + -1.0) / r_m))
	tmp = 0.0
	if (r_m <= 5200000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) + Float64(Float64(Float64(r_m * Float64(Float64(v * -0.25) + 0.375)) * w) * t_0)) - 4.5);
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(w * Float64(r_m * t_0)) * Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))) - 4.5));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = w / ((v + -1.0) / r_m);
	tmp = 0.0;
	if (r_m <= 5200000.0)
		tmp = ((3.0 + (2.0 / (r_m * r_m))) + (((r_m * ((v * -0.25) + 0.375)) * w) * t_0)) - 4.5;
	else
		tmp = 3.0 + (((w * (r_m * t_0)) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(w / N[(N[(v + -1.0), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 5200000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(r$95$m * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 + N[(N[(N[(w * N[(r$95$m * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 5.2e6

   1. Initial program 80.8%

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

      1. associate-/l* 83.3%

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

      2. cancel-sign-sub-inv 83.3%

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

      3. metadata-eval 83.3%

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

      4. +-commutative 83.3%

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

      5. *-commutative 83.3%

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

      6. fma-undefine 83.3%

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

      7. *-commutative 83.3%

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

      8. *-commutative 83.3%

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

      9. associate-/l* 83.3%

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

      10. *-commutative 83.3%

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

      11. associate-*r/ 82.8%

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

      12. associate-*r* 80.1%

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

      13. associate-*l* 93.1%

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

      14. associate-*r* 94.7%

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

   4. Applied egg-rr 95.2%

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

   if 5.2e6 < r

   1. Initial program 87.0%

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

      1. associate--l- 87.0%

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

      2. associate-*l* 77.3%

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

      3. sqr-neg 77.3%

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

      4. associate-*l* 87.0%

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

      5. associate-/l* 92.7%

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

      6. fma-define 92.7%

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

   3. Simplified 92.7%

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

      1. associate-/l* 92.7%

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

      2. *-commutative 92.7%

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

      3. associate-*r/ 92.7%

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

      4. *-commutative 92.7%

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

      5. associate-*l* 96.5%

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

      6. associate-*l* 99.9%

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

      7. clear-num 99.9%

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

      8. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5200000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(\left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right) \cdot w\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(w \cdot \left(r \cdot \frac{w}{\frac{v + -1}{r}}\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{v + -1}{r\_m}\\ \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;3 + \left(\left(v \cdot -0.25\right) \cdot \frac{r\_m \cdot w}{\frac{t\_0}{w}} - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{w}{t\_0}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ (+ v -1.0) r_m)))
   (if (or (<= v -23.5) (not (<= v 7.8e-10)))
     (+ 3.0 (- (* (* v -0.25) (/ (* r_m w) (/ t_0 w))) 4.5))
     (- (+ 3.0 (* (* 0.375 (* r_m w)) (/ w t_0))) 4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = (v + -1.0) / r_m;
	double tmp;
	if ((v <= -23.5) || !(v <= 7.8e-10)) {
		tmp = 3.0 + (((v * -0.25) * ((r_m * w) / (t_0 / w))) - 4.5);
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / t_0))) - 4.5;
	}
	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) :: tmp
    t_0 = (v + (-1.0d0)) / r_m
    if ((v <= (-23.5d0)) .or. (.not. (v <= 7.8d-10))) then
        tmp = 3.0d0 + (((v * (-0.25d0)) * ((r_m * w) / (t_0 / w))) - 4.5d0)
    else
        tmp = (3.0d0 + ((0.375d0 * (r_m * w)) * (w / t_0))) - 4.5d0
    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 = (v + -1.0) / r_m;
	double tmp;
	if ((v <= -23.5) || !(v <= 7.8e-10)) {
		tmp = 3.0 + (((v * -0.25) * ((r_m * w) / (t_0 / w))) - 4.5);
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / t_0))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = (v + -1.0) / r_m
	tmp = 0
	if (v <= -23.5) or not (v <= 7.8e-10):
		tmp = 3.0 + (((v * -0.25) * ((r_m * w) / (t_0 / w))) - 4.5)
	else:
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / t_0))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(Float64(v + -1.0) / r_m)
	tmp = 0.0
	if ((v <= -23.5) || !(v <= 7.8e-10))
		tmp = Float64(3.0 + Float64(Float64(Float64(v * -0.25) * Float64(Float64(r_m * w) / Float64(t_0 / w))) - 4.5));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 * Float64(r_m * w)) * Float64(w / t_0))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = (v + -1.0) / r_m;
	tmp = 0.0;
	if ((v <= -23.5) || ~((v <= 7.8e-10)))
		tmp = 3.0 + (((v * -0.25) * ((r_m * w) / (t_0 / w))) - 4.5);
	else
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / t_0))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(N[(v + -1.0), $MachinePrecision] / r$95$m), $MachinePrecision]}, If[Or[LessEqual[v, -23.5], N[Not[LessEqual[v, 7.8e-10]], $MachinePrecision]], N[(3.0 + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(N[(r$95$m * w), $MachinePrecision] / N[(t$95$0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(w / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -23.5 or 7.7999999999999999e-10 < v

   1. Initial program 79.2%

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

      1. associate--l- 79.2%

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

      2. associate-*l* 75.0%

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

      3. sqr-neg 75.0%

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

      4. associate-*l* 79.2%

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

      5. associate-/l* 85.9%

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

      6. fma-define 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in r around inf 53.0%

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

      1. div-inv 85.9%

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

      2. *-commutative 85.9%

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

      3. associate-*r* 85.2%

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

      4. *-commutative 85.2%

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

      5. associate-*l* 92.0%

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

      6. add-sqr-sqrt 46.8%

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

      7. associate-*r* 46.9%

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

      8. add-sqr-sqrt 22.2%

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

      9. sqrt-prod 34.4%

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

      10. sqrt-prod 34.4%

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

      11. *-commutative 34.4%

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

      12. sqrt-prod 71.0%

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

      13. *-commutative 71.0%

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

      14. div-inv 70.9%

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

      15. associate-*l* 71.0%

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

   7. Applied egg-rr 57.3%

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

   8. Taylor expanded in v around inf 56.3%

      \[\leadsto 3 - \left(\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
   9. Step-by-step derivation

      1. *-commutative 56.3%

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

   10. Simplified 56.3%

       \[\leadsto 3 - \left(\color{blue}{\left(v \cdot -0.25\right)} \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
   11. Step-by-step derivation

       1. associate-*l* 56.3%

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

       2. clear-num 56.4%

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

       3. div-inv 56.4%

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

       4. associate-*r* 56.3%

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

       5. *-commutative 56.3%

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

       6. clear-num 56.3%

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

       7. un-div-inv 56.4%

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

   12. Applied egg-rr 56.4%

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

   if -23.5 < v < 7.7999999999999999e-10

   1. Initial program 85.4%

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

      1. associate-/l* 85.4%

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

      2. cancel-sign-sub-inv 85.4%

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

      3. metadata-eval 85.4%

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

      4. +-commutative 85.4%

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

      5. *-commutative 85.4%

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

      6. fma-undefine 85.4%

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

      7. *-commutative 85.4%

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

      8. *-commutative 85.4%

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

      9. associate-/l* 85.4%

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

      10. *-commutative 85.4%

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

      11. associate-*r/ 85.4%

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

      12. associate-*r* 85.5%

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

      13. associate-*l* 97.5%

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

      14. associate-*r* 99.9%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 99.6%

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

   6. Taylor expanded in r around inf 57.0%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -23.5 \lor \neg \left(v \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;3 + \left(\left(v \cdot -0.25\right) \cdot \frac{r \cdot w}{\frac{\frac{v + -1}{r}}{w}} - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;v \leq -32 \lor \neg \left(v \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;3 + \left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(\left(r\_m \cdot w\right) \cdot \frac{r\_m}{v + -1}\right)\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r\_m}}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (or (<= v -32.0) (not (<= v 7.8e-10)))
   (+ 3.0 (- (* (* v -0.25) (* w (* (* r_m w) (/ r_m (+ v -1.0))))) 4.5))
   (- (+ 3.0 (* (* 0.375 (* r_m w)) (/ w (/ (+ v -1.0) r_m)))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if ((v <= -32.0) || !(v <= 7.8e-10)) {
		tmp = 3.0 + (((v * -0.25) * (w * ((r_m * w) * (r_m / (v + -1.0))))) - 4.5);
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	}
	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 ((v <= (-32.0d0)) .or. (.not. (v <= 7.8d-10))) then
        tmp = 3.0d0 + (((v * (-0.25d0)) * (w * ((r_m * w) * (r_m / (v + (-1.0d0)))))) - 4.5d0)
    else
        tmp = (3.0d0 + ((0.375d0 * (r_m * w)) * (w / ((v + (-1.0d0)) / r_m)))) - 4.5d0
    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 ((v <= -32.0) || !(v <= 7.8e-10)) {
		tmp = 3.0 + (((v * -0.25) * (w * ((r_m * w) * (r_m / (v + -1.0))))) - 4.5);
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if (v <= -32.0) or not (v <= 7.8e-10):
		tmp = 3.0 + (((v * -0.25) * (w * ((r_m * w) * (r_m / (v + -1.0))))) - 4.5)
	else:
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if ((v <= -32.0) || !(v <= 7.8e-10))
		tmp = Float64(3.0 + Float64(Float64(Float64(v * -0.25) * Float64(w * Float64(Float64(r_m * w) * Float64(r_m / Float64(v + -1.0))))) - 4.5));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 * Float64(r_m * w)) * Float64(w / Float64(Float64(v + -1.0) / r_m)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if ((v <= -32.0) || ~((v <= 7.8e-10)))
		tmp = 3.0 + (((v * -0.25) * (w * ((r_m * w) * (r_m / (v + -1.0))))) - 4.5);
	else
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[Or[LessEqual[v, -32.0], N[Not[LessEqual[v, 7.8e-10]], $MachinePrecision]], N[(3.0 + N[(N[(N[(v * -0.25), $MachinePrecision] * N[(w * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(w / N[(N[(v + -1.0), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -32 or 7.7999999999999999e-10 < v

   1. Initial program 79.2%

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

      1. associate--l- 79.2%

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

      2. associate-*l* 75.0%

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

      3. sqr-neg 75.0%

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

      4. associate-*l* 79.2%

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

      5. associate-/l* 85.9%

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

      6. fma-define 85.9%

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

   3. Simplified 85.9%

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

   5. Taylor expanded in r around inf 53.0%

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

      1. div-inv 85.9%

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

      2. *-commutative 85.9%

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

      3. associate-*r* 85.2%

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

      4. *-commutative 85.2%

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

      5. associate-*l* 92.0%

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

      6. add-sqr-sqrt 46.8%

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

      7. associate-*r* 46.9%

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

      8. add-sqr-sqrt 22.2%

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

      9. sqrt-prod 34.4%

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

      10. sqrt-prod 34.4%

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

      11. *-commutative 34.4%

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

      12. sqrt-prod 71.0%

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

      13. *-commutative 71.0%

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

      14. div-inv 70.9%

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

      15. associate-*l* 71.0%

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

   7. Applied egg-rr 57.3%

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

   8. Taylor expanded in v around inf 56.3%

      \[\leadsto 3 - \left(\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
   9. Step-by-step derivation

      1. *-commutative 56.3%

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

   10. Simplified 56.3%

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

   if -32 < v < 7.7999999999999999e-10

   1. Initial program 85.4%

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

      1. associate-/l* 85.4%

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

      2. cancel-sign-sub-inv 85.4%

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

      3. metadata-eval 85.4%

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

      4. +-commutative 85.4%

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

      5. *-commutative 85.4%

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

      6. fma-undefine 85.4%

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

      7. *-commutative 85.4%

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

      8. *-commutative 85.4%

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

      9. associate-/l* 85.4%

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

      10. *-commutative 85.4%

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

      11. associate-*r/ 85.4%

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

      12. associate-*r* 85.5%

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

      13. associate-*l* 97.5%

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

      14. associate-*r* 99.9%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 99.6%

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

   6. Taylor expanded in r around inf 57.0%

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

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -32 \lor \neg \left(v \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;3 + \left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{v + -1}\right)\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;v \leq -0.0135 \lor \neg \left(v \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;3 + \left(\left(\left(v \cdot -0.25\right) \cdot w\right) \cdot \left(r\_m \cdot \left(w \cdot \frac{r\_m}{v + -1}\right)\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r\_m}}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (or (<= v -0.0135) (not (<= v 7.8e-10)))
   (+ 3.0 (- (* (* (* v -0.25) w) (* r_m (* w (/ r_m (+ v -1.0))))) 4.5))
   (- (+ 3.0 (* (* 0.375 (* r_m w)) (/ w (/ (+ v -1.0) r_m)))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if ((v <= -0.0135) || !(v <= 7.8e-10)) {
		tmp = 3.0 + ((((v * -0.25) * w) * (r_m * (w * (r_m / (v + -1.0))))) - 4.5);
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	}
	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 ((v <= (-0.0135d0)) .or. (.not. (v <= 7.8d-10))) then
        tmp = 3.0d0 + ((((v * (-0.25d0)) * w) * (r_m * (w * (r_m / (v + (-1.0d0)))))) - 4.5d0)
    else
        tmp = (3.0d0 + ((0.375d0 * (r_m * w)) * (w / ((v + (-1.0d0)) / r_m)))) - 4.5d0
    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 ((v <= -0.0135) || !(v <= 7.8e-10)) {
		tmp = 3.0 + ((((v * -0.25) * w) * (r_m * (w * (r_m / (v + -1.0))))) - 4.5);
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if (v <= -0.0135) or not (v <= 7.8e-10):
		tmp = 3.0 + ((((v * -0.25) * w) * (r_m * (w * (r_m / (v + -1.0))))) - 4.5)
	else:
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if ((v <= -0.0135) || !(v <= 7.8e-10))
		tmp = Float64(3.0 + Float64(Float64(Float64(Float64(v * -0.25) * w) * Float64(r_m * Float64(w * Float64(r_m / Float64(v + -1.0))))) - 4.5));
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 * Float64(r_m * w)) * Float64(w / Float64(Float64(v + -1.0) / r_m)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if ((v <= -0.0135) || ~((v <= 7.8e-10)))
		tmp = 3.0 + ((((v * -0.25) * w) * (r_m * (w * (r_m / (v + -1.0))))) - 4.5);
	else
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[Or[LessEqual[v, -0.0135], N[Not[LessEqual[v, 7.8e-10]], $MachinePrecision]], N[(3.0 + N[(N[(N[(N[(v * -0.25), $MachinePrecision] * w), $MachinePrecision] * N[(r$95$m * N[(w * N[(r$95$m / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(w / N[(N[(v + -1.0), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -0.0134999999999999998 or 7.7999999999999999e-10 < v

   1. Initial program 78.8%

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

      1. associate--l- 78.8%

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

      2. associate-*l* 74.6%

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

      3. sqr-neg 74.6%

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

      4. associate-*l* 78.8%

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

      5. associate-/l* 85.4%

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

      6. fma-define 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in r around inf 52.2%

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

      1. div-inv 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-*r* 84.6%

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

      4. *-commutative 84.6%

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

      5. associate-*l* 92.1%

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

      6. add-sqr-sqrt 46.9%

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

      7. associate-*r* 46.9%

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

      8. add-sqr-sqrt 22.6%

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

      9. sqrt-prod 34.7%

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

      10. sqrt-prod 34.7%

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

      11. *-commutative 34.7%

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

      12. sqrt-prod 70.6%

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

      13. *-commutative 70.6%

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

      14. div-inv 70.6%

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

      15. associate-*l* 70.6%

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

   7. Applied egg-rr 56.4%

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

   8. Taylor expanded in v around inf 55.4%

      \[\leadsto 3 - \left(\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
   9. Step-by-step derivation

      1. *-commutative 55.4%

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

   10. Simplified 55.4%

       \[\leadsto 3 - \left(\color{blue}{\left(v \cdot -0.25\right)} \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
   11. Step-by-step derivation

       1. pow1 55.4%

          \[\leadsto 3 - \left(\color{blue}{{\left(\left(v \cdot -0.25\right) \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)}^{1}} + 4.5\right) \]

       2. associate-*r* 53.9%

          \[\leadsto 3 - \left({\color{blue}{\left(\left(\left(v \cdot -0.25\right) \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}}^{1} + 4.5\right) \]

   12. Applied egg-rr 53.9%

       \[\leadsto 3 - \left(\color{blue}{{\left(\left(\left(v \cdot -0.25\right) \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)}^{1}} + 4.5\right) \]
   13. Step-by-step derivation

       1. unpow1 53.9%

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

       2. *-commutative 53.9%

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

       3. associate-*l* 54.0%

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

       4. *-commutative 54.0%

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

       5. *-commutative 54.0%

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

   14. Simplified 54.0%

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

   if -0.0134999999999999998 < v < 7.7999999999999999e-10

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

      2. cancel-sign-sub-inv 86.0%

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

      3. metadata-eval 86.0%

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

      4. +-commutative 86.0%

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

      5. *-commutative 86.0%

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

      6. fma-undefine 86.0%

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

      7. *-commutative 86.0%

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

      8. *-commutative 86.0%

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

      9. associate-/l* 86.0%

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

      10. *-commutative 86.0%

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

      11. associate-*r/ 86.0%

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

      12. associate-*r* 86.0%

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

      13. associate-*l* 97.5%

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

      14. associate-*r* 99.9%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 99.6%

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

   6. Taylor expanded in r around inf 57.8%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -0.0135 \lor \neg \left(v \leq 7.8 \cdot 10^{-10}\right):\\ \;\;\;\;3 + \left(\left(\left(v \cdot -0.25\right) \cdot w\right) \cdot \left(r \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = (3.0d0 + (2.0d0 / (r_m * r_m))) - (4.5d0 - ((w * (r_m * (w / ((v + (-1.0d0)) / r_m)))) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))))
end function

Java

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

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (3.0 + (2.0 / (r_m * r_m))) - (4.5 - ((w * (r_m * (w / ((v + -1.0) / r_m)))) * (0.125 * (3.0 + (-2.0 * v)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate--l- 82.4%

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

   2. associate-*l* 77.6%

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

   3. sqr-neg 77.6%

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

   4. associate-*l* 82.4%

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

   5. associate-/l* 85.7%

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

   6. fma-define 85.7%

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

3. Simplified 85.7%

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

   1. associate-/l* 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-*r/ 85.3%

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

   4. *-commutative 85.3%

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

   5. associate-*l* 95.9%

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

   6. associate-*l* 97.9%

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

   7. clear-num 97.9%

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

   8. un-div-inv 97.9%

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 8: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

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 = (3.0d0 + (2.0d0 / (r_m * r_m))) + (((w * ((r_m * w) * (r_m / (v + (-1.0d0))))) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))) - 4.5d0)
end function

Java

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

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (3.0 + (2.0 / (r_m * r_m))) + (((w * ((r_m * w) * (r_m / (v + -1.0)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate--l- 82.4%

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

   2. associate-*l* 77.6%

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

   3. sqr-neg 77.6%

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

   4. associate-*l* 82.4%

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

   5. associate-/l* 85.7%

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

   6. fma-define 85.7%

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

3. Simplified 85.7%

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

   1. div-inv 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-*r* 85.3%

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

   4. *-commutative 85.3%

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

   5. associate-*l* 94.8%

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

   6. add-sqr-sqrt 46.1%

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

   7. associate-*r* 46.1%

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

   8. add-sqr-sqrt 25.0%

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

   9. sqrt-prod 33.7%

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

   10. sqrt-prod 33.7%

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

   11. *-commutative 33.7%

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

   12. sqrt-prod 71.8%

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

   13. *-commutative 71.8%

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

   14. div-inv 71.8%

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

   15. associate-*l* 71.4%

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

6. Applied egg-rr 97.9%

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

7. Final simplification 97.9%

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

Alternative 9: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 4600000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(r\_m \cdot w\right) \cdot \left(0.375 \cdot \left(r\_m \cdot w\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(w \cdot \left(r\_m \cdot \frac{w}{\frac{v + -1}{r\_m}}\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 4600000.0)
   (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* r_m w) (* 0.375 (* r_m w)))) 4.5)
   (+
    3.0
    (-
     (* (* w (* r_m (/ w (/ (+ v -1.0) r_m)))) (* 0.125 (+ 3.0 (* -2.0 v))))
     4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 4600000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	} else {
		tmp = 3.0 + (((w * (r_m * (w / ((v + -1.0) / r_m)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	}
	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 <= 4600000.0d0) then
        tmp = ((3.0d0 + (2.0d0 / (r_m * r_m))) - ((r_m * w) * (0.375d0 * (r_m * w)))) - 4.5d0
    else
        tmp = 3.0d0 + (((w * (r_m * (w / ((v + (-1.0d0)) / r_m)))) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))) - 4.5d0)
    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 <= 4600000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	} else {
		tmp = 3.0 + (((w * (r_m * (w / ((v + -1.0) / r_m)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 4600000.0:
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5
	else:
		tmp = 3.0 + (((w * (r_m * (w / ((v + -1.0) / r_m)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5)
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 4600000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(r_m * w) * Float64(0.375 * Float64(r_m * w)))) - 4.5);
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(w * Float64(r_m * Float64(w / Float64(Float64(v + -1.0) / r_m)))) * Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))) - 4.5));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 4600000.0)
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	else
		tmp = 3.0 + (((w * (r_m * (w / ((v + -1.0) / r_m)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 4600000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r$95$m * w), $MachinePrecision] * N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 + N[(N[(N[(w * N[(r$95$m * N[(w / N[(N[(v + -1.0), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.6e6

   1. Initial program 80.8%

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

      1. associate-/l* 83.3%

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

      2. cancel-sign-sub-inv 83.3%

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

      3. metadata-eval 83.3%

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

      4. +-commutative 83.3%

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

      5. *-commutative 83.3%

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

      6. fma-undefine 83.3%

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

      7. *-commutative 83.3%

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

      8. *-commutative 83.3%

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

      9. associate-/l* 83.3%

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

      10. *-commutative 83.3%

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

      11. associate-*r/ 82.8%

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

      12. associate-*r* 80.1%

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

      13. associate-*l* 93.1%

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

      14. associate-*r* 94.7%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in v around 0 87.4%

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

   6. Taylor expanded in v around 0 95.1%

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

   if 4.6e6 < r

   1. Initial program 87.0%

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

      1. associate--l- 87.0%

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

      2. associate-*l* 77.3%

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

      3. sqr-neg 77.3%

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

      4. associate-*l* 87.0%

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

      5. associate-/l* 92.7%

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

      6. fma-define 92.7%

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

   3. Simplified 92.7%

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

      1. associate-/l* 92.7%

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

      2. *-commutative 92.7%

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

      3. associate-*r/ 92.7%

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

      4. *-commutative 92.7%

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

      5. associate-*l* 96.5%

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

      6. associate-*l* 99.9%

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

      7. clear-num 99.9%

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

      8. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4600000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(w \cdot \left(r \cdot \frac{w}{\frac{v + -1}{r}}\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 9000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(r\_m \cdot w\right) \cdot \left(0.375 \cdot \left(r\_m \cdot w\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(w \cdot \left(\left(r\_m \cdot w\right) \cdot \frac{r\_m}{v + -1}\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 9000000.0)
   (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* r_m w) (* 0.375 (* r_m w)))) 4.5)
   (+
    3.0
    (-
     (* (* w (* (* r_m w) (/ r_m (+ v -1.0)))) (* 0.125 (+ 3.0 (* -2.0 v))))
     4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 9000000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	} else {
		tmp = 3.0 + (((w * ((r_m * w) * (r_m / (v + -1.0)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	}
	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 <= 9000000.0d0) then
        tmp = ((3.0d0 + (2.0d0 / (r_m * r_m))) - ((r_m * w) * (0.375d0 * (r_m * w)))) - 4.5d0
    else
        tmp = 3.0d0 + (((w * ((r_m * w) * (r_m / (v + (-1.0d0))))) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))) - 4.5d0)
    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 <= 9000000.0) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	} else {
		tmp = 3.0 + (((w * ((r_m * w) * (r_m / (v + -1.0)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 9000000.0:
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5
	else:
		tmp = 3.0 + (((w * ((r_m * w) * (r_m / (v + -1.0)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5)
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 9000000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(r_m * w) * Float64(0.375 * Float64(r_m * w)))) - 4.5);
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(w * Float64(Float64(r_m * w) * Float64(r_m / Float64(v + -1.0)))) * Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))) - 4.5));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 9000000.0)
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	else
		tmp = 3.0 + (((w * ((r_m * w) * (r_m / (v + -1.0)))) * (0.125 * (3.0 + (-2.0 * v)))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 9000000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r$95$m * w), $MachinePrecision] * N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 + N[(N[(N[(w * N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9e6

   1. Initial program 80.8%

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

      1. associate-/l* 83.3%

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

      2. cancel-sign-sub-inv 83.3%

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

      3. metadata-eval 83.3%

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

      4. +-commutative 83.3%

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

      5. *-commutative 83.3%

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

      6. fma-undefine 83.3%

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

      7. *-commutative 83.3%

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

      8. *-commutative 83.3%

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

      9. associate-/l* 83.3%

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

      10. *-commutative 83.3%

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

      11. associate-*r/ 82.8%

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

      12. associate-*r* 80.1%

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

      13. associate-*l* 93.1%

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

      14. associate-*r* 94.7%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in v around 0 87.4%

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

   6. Taylor expanded in v around 0 95.1%

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

   if 9e6 < r

   1. Initial program 87.0%

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

      1. associate--l- 87.0%

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

      2. associate-*l* 77.3%

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

      3. sqr-neg 77.3%

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

      4. associate-*l* 87.0%

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

      5. associate-/l* 92.7%

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

      6. fma-define 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in r around inf 92.7%

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

      1. div-inv 92.7%

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

      2. *-commutative 92.7%

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

      3. associate-*r* 92.8%

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

      4. *-commutative 92.8%

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

      5. associate-*l* 99.9%

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

      6. add-sqr-sqrt 99.8%

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

      7. associate-*r* 99.9%

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

      8. add-sqr-sqrt 56.9%

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

      9. sqrt-prod 68.2%

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

      10. sqrt-prod 68.2%

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

      11. *-commutative 68.2%

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

      12. sqrt-prod 68.2%

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

      13. *-commutative 68.2%

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

      14. div-inv 68.2%

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

      15. associate-*l* 68.2%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9000000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{v + -1}\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;v \leq -24 \lor \neg \left(v \leq 5.2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(3 + \left(\left(v \cdot w\right) \cdot \left(r\_m \cdot -0.25\right)\right) \cdot \frac{w}{\frac{v}{r\_m}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r\_m}}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (or (<= v -24.0) (not (<= v 5.2e-11)))
   (- (+ 3.0 (* (* (* v w) (* r_m -0.25)) (/ w (/ v r_m)))) 4.5)
   (- (+ 3.0 (* (* 0.375 (* r_m w)) (/ w (/ (+ v -1.0) r_m)))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if ((v <= -24.0) || !(v <= 5.2e-11)) {
		tmp = (3.0 + (((v * w) * (r_m * -0.25)) * (w / (v / r_m)))) - 4.5;
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	}
	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 ((v <= (-24.0d0)) .or. (.not. (v <= 5.2d-11))) then
        tmp = (3.0d0 + (((v * w) * (r_m * (-0.25d0))) * (w / (v / r_m)))) - 4.5d0
    else
        tmp = (3.0d0 + ((0.375d0 * (r_m * w)) * (w / ((v + (-1.0d0)) / r_m)))) - 4.5d0
    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 ((v <= -24.0) || !(v <= 5.2e-11)) {
		tmp = (3.0 + (((v * w) * (r_m * -0.25)) * (w / (v / r_m)))) - 4.5;
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if (v <= -24.0) or not (v <= 5.2e-11):
		tmp = (3.0 + (((v * w) * (r_m * -0.25)) * (w / (v / r_m)))) - 4.5
	else:
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if ((v <= -24.0) || !(v <= 5.2e-11))
		tmp = Float64(Float64(3.0 + Float64(Float64(Float64(v * w) * Float64(r_m * -0.25)) * Float64(w / Float64(v / r_m)))) - 4.5);
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 * Float64(r_m * w)) * Float64(w / Float64(Float64(v + -1.0) / r_m)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if ((v <= -24.0) || ~((v <= 5.2e-11)))
		tmp = (3.0 + (((v * w) * (r_m * -0.25)) * (w / (v / r_m)))) - 4.5;
	else
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[Or[LessEqual[v, -24.0], N[Not[LessEqual[v, 5.2e-11]], $MachinePrecision]], N[(N[(3.0 + N[(N[(N[(v * w), $MachinePrecision] * N[(r$95$m * -0.25), $MachinePrecision]), $MachinePrecision] * N[(w / N[(v / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(w / N[(N[(v + -1.0), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if v < -24 or 5.2000000000000001e-11 < v

   1. Initial program 79.2%

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

      1. associate-/l* 85.9%

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

      2. cancel-sign-sub-inv 85.9%

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

      3. metadata-eval 85.9%

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

      4. +-commutative 85.9%

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

      5. *-commutative 85.9%

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

      6. fma-undefine 85.9%

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

      7. *-commutative 85.9%

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

      8. *-commutative 85.9%

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

      9. associate-/l* 85.9%

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

      10. *-commutative 85.9%

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

      11. associate-*r/ 85.1%

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

      12. associate-*r* 77.2%

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

      13. associate-*l* 85.1%

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

      14. associate-*r* 84.5%

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in r around inf 43.6%

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

   6. Taylor expanded in v around inf 42.6%

      \[\leadsto \left(3 - \left(\left(\left(v \cdot -0.25 + 0.375\right) \cdot r\right) \cdot w\right) \cdot \frac{w}{\frac{\color{blue}{-1 \cdot v}}{r}}\right) - 4.5 \]
   7. Step-by-step derivation

      1. neg-mul-1 42.6%

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

   8. Simplified 42.6%

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

   9. Taylor expanded in v around inf 50.3%

      \[\leadsto \left(3 - \color{blue}{\left(-0.25 \cdot \left(r \cdot \left(v \cdot w\right)\right)\right)} \cdot \frac{w}{\frac{-v}{r}}\right) - 4.5 \]
   10. Step-by-step derivation

       1. *-commutative 50.3%

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

       2. *-commutative 50.3%

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

       3. associate-*l* 50.3%

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

       4. *-commutative 50.3%

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

   11. Simplified 50.3%

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

   if -24 < v < 5.2000000000000001e-11

   1. Initial program 85.4%

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

      1. associate-/l* 85.4%

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

      2. cancel-sign-sub-inv 85.4%

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

      3. metadata-eval 85.4%

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

      4. +-commutative 85.4%

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

      5. *-commutative 85.4%

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

      6. fma-undefine 85.4%

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

      7. *-commutative 85.4%

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

      8. *-commutative 85.4%

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

      9. associate-/l* 85.4%

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

      10. *-commutative 85.4%

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

      11. associate-*r/ 85.4%

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

      12. associate-*r* 85.5%

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

      13. associate-*l* 97.5%

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

      14. associate-*r* 99.9%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in v around 0 99.6%

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

   6. Taylor expanded in r around inf 57.0%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -24 \lor \neg \left(v \leq 5.2 \cdot 10^{-11}\right):\\ \;\;\;\;\left(3 + \left(\left(v \cdot w\right) \cdot \left(r \cdot -0.25\right)\right) \cdot \frac{w}{\frac{v}{r}}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 96.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 5.5 \cdot 10^{+42}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(r\_m \cdot w\right) \cdot \left(0.375 \cdot \left(r\_m \cdot w\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{w}{\frac{1 - v}{r\_m}} \cdot \left(r\_m \cdot \left(\left(v \cdot -0.25 + 0.375\right) \cdot w\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 5.5e+42)
   (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* r_m w) (* 0.375 (* r_m w)))) 4.5)
   (-
    (- 3.0 (* (/ w (/ (- 1.0 v) r_m)) (* r_m (* (+ (* v -0.25) 0.375) w))))
    4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 5.5e+42) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	} else {
		tmp = (3.0 - ((w / ((1.0 - v) / r_m)) * (r_m * (((v * -0.25) + 0.375) * w)))) - 4.5;
	}
	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 <= 5.5d+42) then
        tmp = ((3.0d0 + (2.0d0 / (r_m * r_m))) - ((r_m * w) * (0.375d0 * (r_m * w)))) - 4.5d0
    else
        tmp = (3.0d0 - ((w / ((1.0d0 - v) / r_m)) * (r_m * (((v * (-0.25d0)) + 0.375d0) * w)))) - 4.5d0
    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 <= 5.5e+42) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	} else {
		tmp = (3.0 - ((w / ((1.0 - v) / r_m)) * (r_m * (((v * -0.25) + 0.375) * w)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 5.5e+42:
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5
	else:
		tmp = (3.0 - ((w / ((1.0 - v) / r_m)) * (r_m * (((v * -0.25) + 0.375) * w)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 5.5e+42)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(r_m * w) * Float64(0.375 * Float64(r_m * w)))) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(w / Float64(Float64(1.0 - v) / r_m)) * Float64(r_m * Float64(Float64(Float64(v * -0.25) + 0.375) * w)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 5.5e+42)
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
	else
		tmp = (3.0 - ((w / ((1.0 - v) / r_m)) * (r_m * (((v * -0.25) + 0.375) * w)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 5.5e+42], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r$95$m * w), $MachinePrecision] * N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(w / N[(N[(1.0 - v), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.50000000000000001e42

   1. Initial program 81.0%

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

      1. associate-/l* 83.9%

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

      2. cancel-sign-sub-inv 83.9%

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

      3. metadata-eval 83.9%

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

      4. +-commutative 83.9%

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

      5. *-commutative 83.9%

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

      6. fma-undefine 83.9%

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

      7. *-commutative 83.9%

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

      8. *-commutative 83.9%

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

      9. associate-/l* 83.9%

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

      10. *-commutative 83.9%

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

      11. associate-*r/ 83.4%

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

      12. associate-*r* 80.8%

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

      13. associate-*l* 93.3%

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

      14. associate-*r* 94.4%

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

   4. Applied egg-rr 94.9%

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

   5. Taylor expanded in v around 0 86.8%

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

   6. Taylor expanded in v around 0 94.9%

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

   if 5.50000000000000001e42 < r

   1. Initial program 87.0%

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

      1. associate-/l* 91.8%

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

      2. cancel-sign-sub-inv 91.8%

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

      3. metadata-eval 91.8%

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

      4. +-commutative 91.8%

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

      5. *-commutative 91.8%

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

      6. fma-undefine 91.8%

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

      7. *-commutative 91.8%

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

      8. *-commutative 91.8%

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

      9. associate-/l* 91.8%

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

      10. *-commutative 91.8%

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

      11. associate-*r/ 91.8%

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

      12. associate-*r* 83.2%

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

      13. associate-*l* 85.0%

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

      14. associate-*r* 85.1%

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

   4. Applied egg-rr 85.1%

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

   5. Taylor expanded in r around 0 95.1%

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

   6. Taylor expanded in r around inf 95.1%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{+42}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \frac{w}{\frac{1 - v}{r}} \cdot \left(r \cdot \left(\left(v \cdot -0.25 + 0.375\right) \cdot w\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.2:\\ \;\;\;\;\left(3 - \left(\left(r\_m \cdot \left(v \cdot -0.25 + 0.375\right)\right) \cdot w\right) \cdot \left(r\_m \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r\_m}}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.2)
   (- (- 3.0 (* (* (* r_m (+ (* v -0.25) 0.375)) w) (* r_m w))) 4.5)
   (- (+ 3.0 (* (* 0.375 (* r_m w)) (/ w (/ (+ v -1.0) r_m)))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.2) {
		tmp = (3.0 - (((r_m * ((v * -0.25) + 0.375)) * w) * (r_m * w))) - 4.5;
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	}
	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 <= 1.2d0) then
        tmp = (3.0d0 - (((r_m * ((v * (-0.25d0)) + 0.375d0)) * w) * (r_m * w))) - 4.5d0
    else
        tmp = (3.0d0 + ((0.375d0 * (r_m * w)) * (w / ((v + (-1.0d0)) / r_m)))) - 4.5d0
    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 <= 1.2) {
		tmp = (3.0 - (((r_m * ((v * -0.25) + 0.375)) * w) * (r_m * w))) - 4.5;
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.2:
		tmp = (3.0 - (((r_m * ((v * -0.25) + 0.375)) * w) * (r_m * w))) - 4.5
	else:
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.2)
		tmp = Float64(Float64(3.0 - Float64(Float64(Float64(r_m * Float64(Float64(v * -0.25) + 0.375)) * w) * Float64(r_m * w))) - 4.5);
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 * Float64(r_m * w)) * Float64(w / Float64(Float64(v + -1.0) / r_m)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.2)
		tmp = (3.0 - (((r_m * ((v * -0.25) + 0.375)) * w) * (r_m * w))) - 4.5;
	else
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / ((v + -1.0) / r_m)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.2], N[(N[(3.0 - N[(N[(N[(r$95$m * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision] * w), $MachinePrecision] * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 + N[(N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(w / N[(N[(v + -1.0), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.19999999999999996

   1. Initial program 80.8%

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

      1. associate-/l* 83.3%

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

      2. cancel-sign-sub-inv 83.3%

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

      3. metadata-eval 83.3%

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

      4. +-commutative 83.3%

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

      5. *-commutative 83.3%

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

      6. fma-undefine 83.3%

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

      7. *-commutative 83.3%

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

      8. *-commutative 83.3%

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

      9. associate-/l* 83.3%

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

      10. *-commutative 83.3%

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

      11. associate-*r/ 82.8%

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

      12. associate-*r* 80.1%

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

      13. associate-*l* 93.1%

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

      14. associate-*r* 94.7%

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

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in r around inf 38.7%

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

   6. Taylor expanded in v around 0 29.2%

      \[\leadsto \left(3 - \left(\left(\left(v \cdot -0.25 + 0.375\right) \cdot r\right) \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}\right) - 4.5 \]

   if 1.19999999999999996 < r

   1. Initial program 87.0%

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

      1. associate-/l* 92.7%

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

      2. cancel-sign-sub-inv 92.7%

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

      3. metadata-eval 92.7%

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

      4. +-commutative 92.7%

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

      5. *-commutative 92.7%

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

      6. fma-undefine 92.7%

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

      7. *-commutative 92.7%

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

      8. *-commutative 92.7%

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

      9. associate-/l* 92.7%

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

      10. *-commutative 92.7%

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

      11. associate-*r/ 92.7%

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

      12. associate-*r* 85.0%

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

      13. associate-*l* 86.6%

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

      14. associate-*r* 85.2%

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

   4. Applied egg-rr 85.2%

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

   5. Taylor expanded in v around 0 77.6%

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

   6. Taylor expanded in r around inf 77.6%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.2:\\ \;\;\;\;\left(3 - \left(\left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right) \cdot w\right) \cdot \left(r \cdot w\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \frac{w}{\frac{v + -1}{r}}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 20.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;w \leq 9.5 \cdot 10^{-49}:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r\_m \cdot w\right)\right) \cdot \frac{w}{\frac{v}{r\_m}}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= w 9.5e-49)
   -1.5
   (- (+ 3.0 (* (* 0.375 (* r_m w)) (/ w (/ v r_m)))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (w <= 9.5e-49) {
		tmp = -1.5;
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / (v / r_m)))) - 4.5;
	}
	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 (w <= 9.5d-49) then
        tmp = -1.5d0
    else
        tmp = (3.0d0 + ((0.375d0 * (r_m * w)) * (w / (v / r_m)))) - 4.5d0
    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 (w <= 9.5e-49) {
		tmp = -1.5;
	} else {
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / (v / r_m)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if w <= 9.5e-49:
		tmp = -1.5
	else:
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / (v / r_m)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (w <= 9.5e-49)
		tmp = -1.5;
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(0.375 * Float64(r_m * w)) * Float64(w / Float64(v / r_m)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (w <= 9.5e-49)
		tmp = -1.5;
	else
		tmp = (3.0 + ((0.375 * (r_m * w)) * (w / (v / r_m)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[w, 9.5e-49], -1.5, N[(N[(3.0 + N[(N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision] * N[(w / N[(v / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 9.50000000000000006e-49

   1. Initial program 86.0%

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

      1. associate--l- 86.0%

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

      2. associate-*l* 79.6%

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

      3. sqr-neg 79.6%

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

      4. associate-*l* 86.0%

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

      5. associate-/l* 89.6%

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

      6. fma-define 89.6%

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

   3. Simplified 89.6%

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

   5. Taylor expanded in r around inf 51.0%

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

      1. div-inv 89.6%

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

      2. *-commutative 89.6%

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

      3. associate-*r* 89.6%

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

      4. *-commutative 89.6%

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

      5. associate-*l* 96.5%

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

      6. add-sqr-sqrt 45.2%

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

      7. associate-*r* 45.3%

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

      8. add-sqr-sqrt 15.1%

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

      9. sqrt-prod 31.0%

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

      10. sqrt-prod 31.0%

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

      11. *-commutative 31.0%

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

      12. sqrt-prod 77.9%

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

      13. *-commutative 77.9%

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

      14. div-inv 77.9%

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

      15. associate-*l* 77.4%

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

   7. Applied egg-rr 55.1%

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

   8. Taylor expanded in v around inf 45.1%

      \[\leadsto 3 - \left(\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
   9. Step-by-step derivation

      1. *-commutative 45.1%

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

   10. Simplified 45.1%

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

   11. Taylor expanded in v around 0 22.7%

       \[\leadsto \color{blue}{-1.5} \]

   if 9.50000000000000006e-49 < w

   1. Initial program 74.1%

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

      1. associate-/l* 76.6%

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

      2. cancel-sign-sub-inv 76.6%

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

      3. metadata-eval 76.6%

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

      4. +-commutative 76.6%

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

      5. *-commutative 76.6%

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

      6. fma-undefine 76.6%

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

      7. *-commutative 76.6%

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

      8. *-commutative 76.6%

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

      9. associate-/l* 76.5%

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

      10. *-commutative 76.5%

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

      11. associate-*r/ 75.3%

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

      12. associate-*r* 75.3%

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

      13. associate-*l* 94.7%

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

      14. associate-*r* 96.1%

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

   4. Applied egg-rr 96.1%

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

   5. Taylor expanded in r around inf 56.1%

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

   6. Taylor expanded in v around inf 38.0%

      \[\leadsto \left(3 - \left(\left(\left(v \cdot -0.25 + 0.375\right) \cdot r\right) \cdot w\right) \cdot \frac{w}{\frac{\color{blue}{-1 \cdot v}}{r}}\right) - 4.5 \]
   7. Step-by-step derivation

      1. neg-mul-1 38.0%

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

   8. Simplified 38.0%

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

   9. Taylor expanded in v around 0 23.4%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 9.5 \cdot 10^{-49}:\\ \;\;\;\;-1.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(0.375 \cdot \left(r \cdot w\right)\right) \cdot \frac{w}{\frac{v}{r}}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 93.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(r\_m \cdot w\right) \cdot \left(0.375 \cdot \left(r\_m \cdot w\right)\right)\right) - 4.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* r_m w) (* 0.375 (* r_m w)))) 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))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
}

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 = ((3.0d0 + (2.0d0 / (r_m * r_m))) - ((r_m * w) * (0.375d0 * (r_m * w)))) - 4.5d0
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 4.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 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(Float64(r_m * w) * Float64(0.375 * Float64(r_m * w)))) - 4.5)
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (0.375 * (r_m * w)))) - 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[(r$95$m * w), $MachinePrecision] * N[(0.375 * N[(r$95$m * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-/l* 85.7%

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

   2. cancel-sign-sub-inv 85.7%

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

   3. metadata-eval 85.7%

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

   4. +-commutative 85.7%

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

   5. *-commutative 85.7%

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

   6. fma-undefine 85.7%

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

   7. *-commutative 85.7%

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

   8. *-commutative 85.7%

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

   9. associate-/l* 85.7%

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

   10. *-commutative 85.7%

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

   11. associate-*r/ 85.3%

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

   12. associate-*r* 81.4%

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

   13. associate-*l* 91.4%

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

   14. associate-*r* 92.3%

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

4. Applied egg-rr 92.7%

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

5. Taylor expanded in v around 0 84.9%

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

6. Taylor expanded in v around 0 93.5%

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

7. Final simplification 93.5%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(0.375 \cdot \left(r \cdot w\right)\right)\right) - 4.5 \]
8. Add Preprocessing

Alternative 16: 39.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ 3 - \left(4.5 - 0.375 \cdot \left(r\_m \cdot \frac{w}{\frac{v + -1}{r\_m \cdot w}}\right)\right) \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (- 3.0 (- 4.5 (* 0.375 (* r_m (/ w (/ (+ v -1.0) (* r_m w))))))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return 3.0 - (4.5 - (0.375 * (r_m * (w / ((v + -1.0) / (r_m * w))))));
}

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 = 3.0d0 - (4.5d0 - (0.375d0 * (r_m * (w / ((v + (-1.0d0)) / (r_m * w))))))
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return 3.0 - (4.5 - (0.375 * (r_m * (w / ((v + -1.0) / (r_m * w))))));
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return 3.0 - (4.5 - (0.375 * (r_m * (w / ((v + -1.0) / (r_m * w))))))

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(3.0 - Float64(4.5 - Float64(0.375 * Float64(r_m * Float64(w / Float64(Float64(v + -1.0) / Float64(r_m * w)))))))
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = 3.0 - (4.5 - (0.375 * (r_m * (w / ((v + -1.0) / (r_m * w))))));
end

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate--l- 82.4%

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

   2. associate-*l* 77.6%

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

   3. sqr-neg 77.6%

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

   4. associate-*l* 82.4%

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

   5. associate-/l* 85.7%

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

   6. fma-define 85.7%

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

3. Simplified 85.7%

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

5. Taylor expanded in r around inf 53.6%

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

   1. div-inv 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-*r* 85.3%

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

   4. *-commutative 85.3%

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

   5. associate-*l* 94.8%

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

   6. add-sqr-sqrt 46.1%

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

   7. associate-*r* 46.1%

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

   8. add-sqr-sqrt 25.0%

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

   9. sqrt-prod 33.7%

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

   10. sqrt-prod 33.7%

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

   11. *-commutative 33.7%

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

   12. sqrt-prod 71.8%

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

   13. *-commutative 71.8%

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

   14. div-inv 71.8%

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

   15. associate-*l* 71.4%

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

7. Applied egg-rr 56.1%

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

8. Taylor expanded in v around 0 41.8%

   \[\leadsto 3 - \left(\color{blue}{0.375} \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
9. Step-by-step derivation

   1. associate-*l* 43.6%

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

   2. clear-num 43.6%

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

   3. div-inv 43.6%

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

   4. associate-*r* 43.6%

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

   5. *-commutative 43.6%

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

   6. clear-num 43.6%

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

   7. un-div-inv 43.6%

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

10. Applied egg-rr 42.8%

    \[\leadsto 3 - \left(0.375 \cdot \color{blue}{\frac{r \cdot w}{\frac{\frac{1 - v}{r}}{w}}} + 4.5\right) \]
11. Step-by-step derivation

    1. associate-/l* 42.8%

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

    2. associate-/l/ 42.9%

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

    3. *-commutative 42.9%

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

12. Simplified 42.9%

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

13. Final simplification 42.9%

    \[\leadsto 3 - \left(4.5 - 0.375 \cdot \left(r \cdot \frac{w}{\frac{v + -1}{r \cdot w}}\right)\right) \]
14. Add Preprocessing

Alternative 17: 14.0% accurate, 29.0× 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 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. Step-by-step derivation

   1. associate--l- 82.4%

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

   2. associate-*l* 77.6%

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

   3. sqr-neg 77.6%

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

   4. associate-*l* 82.4%

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

   5. associate-/l* 85.7%

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

   6. fma-define 85.7%

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

3. Simplified 85.7%

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

5. Taylor expanded in r around inf 53.6%

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

   1. div-inv 85.7%

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

   2. *-commutative 85.7%

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

   3. associate-*r* 85.3%

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

   4. *-commutative 85.3%

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

   5. associate-*l* 94.8%

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

   6. add-sqr-sqrt 46.1%

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

   7. associate-*r* 46.1%

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

   8. add-sqr-sqrt 25.0%

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

   9. sqrt-prod 33.7%

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

   10. sqrt-prod 33.7%

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

   11. *-commutative 33.7%

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

   12. sqrt-prod 71.8%

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

   13. *-commutative 71.8%

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

   14. div-inv 71.8%

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

   15. associate-*l* 71.4%

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

7. Applied egg-rr 56.1%

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

8. Taylor expanded in v around inf 43.6%

   \[\leadsto 3 - \left(\color{blue}{\left(-0.25 \cdot v\right)} \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{1 - v}\right)\right) + 4.5\right) \]
9. Step-by-step derivation

   1. *-commutative 43.6%

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

10. Simplified 43.6%

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

11. Taylor expanded in v around 0 17.3%

    \[\leadsto \color{blue}{-1.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(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))