Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.9%

Time: 11.4s

Alternatives: 16

Speedup: 1.4×

• 53.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(2, {r}^{-2}, -1.5\right) + \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (if (<= r 2e+140)
   (+
    (fma 2.0 (pow r -2.0) -1.5)
    (* (/ (fma v 0.25 -0.375) (- 1.0 v)) (* w (* w (* r r)))))
   (+
    (/ 2.0 (* r r))
    (- -1.5 (* r (* (* w (* r w)) (/ (+ 0.375 (* v -0.25)) (- 1.0 v))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e+140) {
		tmp = fma(2.0, pow(r, -2.0), -1.5) + ((fma(v, 0.25, -0.375) / (1.0 - v)) * (w * (w * (r * r))));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - (r * ((w * (r * w)) * ((0.375 + (v * -0.25)) / (1.0 - v)))));
	}
	return tmp;
}

Julia

r = abs(r)
function code(v, w, r)
	tmp = 0.0
	if (r <= 2e+140)
		tmp = Float64(fma(2.0, (r ^ -2.0), -1.5) + Float64(Float64(fma(v, 0.25, -0.375) / Float64(1.0 - v)) * Float64(w * Float64(w * Float64(r * r)))));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(r * Float64(Float64(w * Float64(r * w)) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v))))));
	end
	return tmp
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := If[LessEqual[r, 2e+140], N[(N[(2.0 * N[Power[r, -2.0], $MachinePrecision] + -1.5), $MachinePrecision] + N[(N[(N[(v * 0.25 + -0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(r * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.00000000000000012e140

   1. Initial program 88.5%

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

      1. sub-neg 88.5%

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

      2. +-commutative 88.5%

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

      3. associate--l+ 88.5%

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

      4. associate-/l* 90.0%

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

      5. distribute-neg-frac 90.0%

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

      6. associate-/r/ 90.0%

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

      7. fma-def 90.0%

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

      8. sub-neg 90.0%

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

   3. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
   4. Step-by-step derivation

      1. fma-udef 84.7%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

      2. unswap-sqr 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      3. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      4. div-inv 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

      5. fma-def 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

      6. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

      7. pow-flip 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]
   6. Step-by-step derivation

      1. unpow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right) \]

      2. unswap-sqr 84.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right) \]

      3. associate-*r* 92.4%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right)} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right) \]

   7. Applied egg-rr 92.4%

      \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(\left(r \cdot r\right) \cdot w\right) \cdot w\right)} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right) \]

   if 2.00000000000000012e140 < r

   1. Initial program 91.1%

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

      1. associate--l- 91.1%

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

      2. +-commutative 91.1%

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

      3. associate--l+ 91.1%

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

      4. +-commutative 91.1%

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

      5. associate--r+ 91.1%

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

      6. metadata-eval 91.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 91.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 91.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 91.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 91.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 91.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]

   4. Taylor expanded in r around 0 91.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left({w}^{2} \cdot r\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   5. Step-by-step derivation

      1. unpow2 91.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot r\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. *-commutative 91.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. associate-*r* 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. *-commutative 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   6. Simplified 99.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   7. Step-by-step derivation

      1. pow1 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\left(r \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)}^{1}}\right) \]

      2. associate-*l* 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - {\color{blue}{\left(r \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)}}^{1}\right) \]

      3. associate-*l* 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - {\left(r \cdot \left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)}^{1}\right) \]

   8. Applied egg-rr 99.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)}^{1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(2, {r}^{-2}, -1.5\right) + \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right) \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (+
  (* (/ (fma v 0.25 -0.375) (- 1.0 v)) (pow (* r w) 2.0))
  (fma 2.0 (pow r -2.0) -1.5)))

C

r = abs(r);
double code(double v, double w, double r) {
	return ((fma(v, 0.25, -0.375) / (1.0 - v)) * pow((r * w), 2.0)) + fma(2.0, pow(r, -2.0), -1.5);
}

Julia

r = abs(r)
function code(v, w, r)
	return Float64(Float64(Float64(fma(v, 0.25, -0.375) / Float64(1.0 - v)) * (Float64(r * w) ^ 2.0)) + fma(2.0, (r ^ -2.0), -1.5))
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := N[(N[(N[(N[(v * 0.25 + -0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[r, -2.0], $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.9%

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

   1. sub-neg 88.9%

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

   2. +-commutative 88.9%

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

   3. associate--l+ 88.9%

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

   4. associate-/l* 90.2%

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

   5. distribute-neg-frac 90.2%

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

   6. associate-/r/ 90.2%

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

   7. fma-def 90.2%

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

   8. sub-neg 90.2%

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

3. Simplified 83.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
4. Step-by-step derivation

   1. fma-udef 83.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

   2. unswap-sqr 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

   3. pow2 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

   4. div-inv 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

   5. fma-def 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

   6. pow2 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

   7. pow-flip 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

   8. metadata-eval 99.8%

      \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]

6. Final simplification 99.8%

   \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right) \]

Alternative 3: 99.6% accurate, 0.1× speedup

Math

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

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (-
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (pow
    (cbrt (/ (* 0.125 (+ 3.0 (* v -2.0))) (/ (- 1.0 v) (pow (* r w) 2.0))))
    3.0))
  4.5))

C

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

Java

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

Julia

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

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Power[N[Power[N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 88.9%

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

   1. add-cube-cbrt 88.8%

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

   2. pow3 88.8%

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

   3. associate-/l* 90.0%

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

   4. cancel-sign-sub-inv 90.0%

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

   5. metadata-eval 90.0%

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

   6. associate-*l* 82.9%

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

   7. *-commutative 82.9%

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

   8. unswap-sqr 99.6%

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

   9. pow2 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2 \cdot 10^{-108}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2e-108)
     (+ t_0 (- -1.5 (* 0.25 (pow (* r w) 2.0))))
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* w (* r w)))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2e-108) {
		tmp = t_0 + (-1.5 - (0.25 * pow((r * w), 2.0)));
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2d-108) then
        tmp = t_0 + ((-1.5d0) - (0.25d0 * ((r * w) ** 2.0d0)))
    else
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (w * (r * w)))))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2e-108) {
		tmp = t_0 + (-1.5 - (0.25 * Math.pow((r * w), 2.0)));
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2e-108:
		tmp = t_0 + (-1.5 - (0.25 * math.pow((r * w), 2.0)))
	else:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2e-108)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * (Float64(r * w) ^ 2.0))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(r * w))))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2e-108)
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) ^ 2.0)));
	else
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2e-108], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.00000000000000008e-108

   1. Initial program 86.4%

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

      1. associate--l- 86.4%

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

      2. +-commutative 86.4%

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

      3. associate--l+ 86.4%

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

      4. +-commutative 86.4%

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

      5. associate--r+ 86.4%

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

      6. metadata-eval 86.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 86.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]

   4. Taylor expanded in v around inf 76.0%

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

      1. *-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. *-commutative 76.0%

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

      4. unpow2 76.0%

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

      5. swap-sqr 92.6%

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

      6. unpow2 92.6%

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

      7. *-commutative 92.6%

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

   6. Simplified 92.6%

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

   if 2.00000000000000008e-108 < r

   1. Initial program 93.1%

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

      1. associate--l- 93.1%

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

      2. +-commutative 93.1%

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

      3. associate--l+ 93.1%

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

      4. +-commutative 93.1%

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

      5. associate--r+ 93.1%

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

      6. metadata-eval 93.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 96.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]

   4. Taylor expanded in r around 0 96.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left({w}^{2} \cdot r\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   5. Step-by-step derivation

      1. unpow2 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot r\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. *-commutative 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. associate-*r* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   6. Simplified 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.4 \cdot 10^{-108}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.4e-108)
     (+ t_0 (- -1.5 (* 0.25 (pow (* r w) 2.0))))
     (+
      t_0
      (- -1.5 (* r (* (* w (* r w)) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.4e-108) {
		tmp = t_0 + (-1.5 - (0.25 * pow((r * w), 2.0)));
	} else {
		tmp = t_0 + (-1.5 - (r * ((w * (r * w)) * ((0.375 + (v * -0.25)) / (1.0 - v)))));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2.4d-108) then
        tmp = t_0 + ((-1.5d0) - (0.25d0 * ((r * w) ** 2.0d0)))
    else
        tmp = t_0 + ((-1.5d0) - (r * ((w * (r * w)) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)))))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.4e-108) {
		tmp = t_0 + (-1.5 - (0.25 * Math.pow((r * w), 2.0)));
	} else {
		tmp = t_0 + (-1.5 - (r * ((w * (r * w)) * ((0.375 + (v * -0.25)) / (1.0 - v)))));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.4e-108:
		tmp = t_0 + (-1.5 - (0.25 * math.pow((r * w), 2.0)))
	else:
		tmp = t_0 + (-1.5 - (r * ((w * (r * w)) * ((0.375 + (v * -0.25)) / (1.0 - v)))))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.4e-108)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.25 * (Float64(r * w) ^ 2.0))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r * Float64(Float64(w * Float64(r * w)) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v))))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.4e-108)
		tmp = t_0 + (-1.5 - (0.25 * ((r * w) ^ 2.0)));
	else
		tmp = t_0 + (-1.5 - (r * ((w * (r * w)) * ((0.375 + (v * -0.25)) / (1.0 - v)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.4e-108], N[(t$95$0 + N[(-1.5 - N[(0.25 * N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(r * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.40000000000000017e-108

   1. Initial program 86.4%

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

      1. associate--l- 86.4%

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

      2. +-commutative 86.4%

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

      3. associate--l+ 86.4%

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

      4. +-commutative 86.4%

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

      5. associate--r+ 86.4%

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

      6. metadata-eval 86.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 86.6%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 86.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]

   4. Taylor expanded in v around inf 76.0%

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

      1. *-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. *-commutative 76.0%

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

      4. unpow2 76.0%

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

      5. swap-sqr 92.6%

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

      6. unpow2 92.6%

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

      7. *-commutative 92.6%

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

   6. Simplified 92.6%

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

   if 2.40000000000000017e-108 < r

   1. Initial program 93.1%

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

      1. associate--l- 93.1%

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

      2. +-commutative 93.1%

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

      3. associate--l+ 93.1%

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

      4. +-commutative 93.1%

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

      5. associate--r+ 93.1%

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

      6. metadata-eval 93.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 96.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 96.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]

   4. Taylor expanded in r around 0 96.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left({w}^{2} \cdot r\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   5. Step-by-step derivation

      1. unpow2 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot r\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. *-commutative 96.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. associate-*r* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   6. Simplified 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   7. Step-by-step derivation

      1. pow1 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\left(r \cdot \left(\left(w \cdot r\right) \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)}^{1}}\right) \]

      2. associate-*l* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - {\color{blue}{\left(r \cdot \left(\left(\left(w \cdot r\right) \cdot w\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)}}^{1}\right) \]

      3. associate-*l* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - {\left(r \cdot \left(\color{blue}{\left(w \cdot \left(r \cdot w\right)\right)} \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)}^{1}\right) \]

   8. Applied egg-rr 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)}^{1}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.25 \cdot {\left(r \cdot w\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\right)\\ \end{array} \]

Alternative 6: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} \mathbf{if}\;r \leq 4.3 \cdot 10^{-117}:\\ \;\;\;\;2 \cdot {r}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (if (<= r 4.3e-117)
   (* 2.0 (pow r -2.0))
   (+
    (/ 2.0 (* r r))
    (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* w (* r w))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.3e-117) {
		tmp = 2.0 * pow(r, -2.0);
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 4.3d-117) then
        tmp = 2.0d0 * (r ** (-2.0d0))
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (w * (r * w)))))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.3e-117) {
		tmp = 2.0 * Math.pow(r, -2.0);
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	tmp = 0
	if r <= 4.3e-117:
		tmp = 2.0 * math.pow(r, -2.0)
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	tmp = 0.0
	if (r <= 4.3e-117)
		tmp = Float64(2.0 * (r ^ -2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(r * w))))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 4.3e-117)
		tmp = 2.0 * (r ^ -2.0);
	else
		tmp = (2.0 / (r * r)) + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := If[LessEqual[r, 4.3e-117], N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.3e-117

   1. Initial program 86.2%

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

      1. sub-neg 86.2%

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

      2. +-commutative 86.2%

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

      3. associate--l+ 86.2%

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

      4. associate-/l* 86.4%

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

      5. distribute-neg-frac 86.4%

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

      6. associate-/r/ 86.5%

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

      7. fma-def 86.5%

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

      8. sub-neg 86.5%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
   4. Step-by-step derivation

      1. fma-udef 79.3%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

      2. unswap-sqr 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      3. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      4. div-inv 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

      5. fma-def 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

      6. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

      7. pow-flip 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]

   6. Taylor expanded in r around 0 55.0%

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

      1. unpow2 55.0%

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

   8. Simplified 55.0%

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

      1. expm1-log1p-u 85.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-udef 85.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. div-inv 85.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. pow2 85.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      5. pow-flip 85.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      6. metadata-eval 85.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   10. Applied egg-rr 53.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 85.2%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

       2. expm1-log1p 86.5%

          \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   12. Simplified 55.0%

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

   if 4.3e-117 < r

   1. Initial program 93.3%

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

      1. associate--l- 93.3%

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

      2. +-commutative 93.3%

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

      3. associate--l+ 93.3%

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

      4. +-commutative 93.3%

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

      5. associate--r+ 93.3%

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

      6. metadata-eval 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 96.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]

   4. Taylor expanded in r around 0 96.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left({w}^{2} \cdot r\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   5. Step-by-step derivation

      1. unpow2 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot r\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. *-commutative 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. associate-*r* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   6. Simplified 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.3 \cdot 10^{-117}:\\ \;\;\;\;2 \cdot {r}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 94.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2e-115)
     t_0
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* r (* w w)))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2e-115) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2d-115) then
        tmp = t_0
    else
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (r * (w * w)))))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2e-115) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2e-115:
		tmp = t_0
	else:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2e-115)
		tmp = t_0;
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2e-115)
		tmp = t_0;
	else
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2e-115], t$95$0, N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.0000000000000001e-115

   1. Initial program 86.2%

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

      1. sub-neg 86.2%

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

      2. +-commutative 86.2%

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

      3. associate--l+ 86.2%

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

      4. associate-/l* 86.4%

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

      5. distribute-neg-frac 86.4%

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

      6. associate-/r/ 86.5%

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

      7. fma-def 86.5%

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

      8. sub-neg 86.5%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
   4. Step-by-step derivation

      1. fma-udef 79.3%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

      2. unswap-sqr 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      3. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      4. div-inv 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

      5. fma-def 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

      6. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

      7. pow-flip 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]

   6. Taylor expanded in r around 0 55.0%

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

      1. unpow2 55.0%

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

   8. Simplified 55.0%

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

   if 2.0000000000000001e-115 < r

   1. Initial program 93.3%

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

      1. associate--l- 93.3%

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

      2. +-commutative 93.3%

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

      3. associate--l+ 93.3%

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

      4. +-commutative 93.3%

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

      5. associate--r+ 93.3%

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

      6. metadata-eval 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 96.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.2 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.2e-116)
     t_0
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* w (* r w)))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.2e-116) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2.2d-116) then
        tmp = t_0
    else
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (w * (r * w)))))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.2e-116) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.2e-116:
		tmp = t_0
	else:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.2e-116)
		tmp = t_0;
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(r * w))))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.2e-116)
		tmp = t_0;
	else
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.2e-116], t$95$0, N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 2.2000000000000001e-116

   1. Initial program 86.2%

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

      1. sub-neg 86.2%

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

      2. +-commutative 86.2%

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

      3. associate--l+ 86.2%

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

      4. associate-/l* 86.4%

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

      5. distribute-neg-frac 86.4%

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

      6. associate-/r/ 86.5%

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

      7. fma-def 86.5%

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

      8. sub-neg 86.5%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
   4. Step-by-step derivation

      1. fma-udef 79.3%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

      2. unswap-sqr 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      3. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      4. div-inv 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

      5. fma-def 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

      6. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

      7. pow-flip 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]

   6. Taylor expanded in r around 0 55.0%

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

      1. unpow2 55.0%

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

   8. Simplified 55.0%

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

   if 2.2000000000000001e-116 < r

   1. Initial program 93.3%

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

      1. associate--l- 93.3%

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

      2. +-commutative 93.3%

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

      3. associate--l+ 93.3%

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

      4. +-commutative 93.3%

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

      5. associate--r+ 93.3%

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

      6. metadata-eval 93.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 96.1%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 96.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]

   4. Taylor expanded in r around 0 96.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left({w}^{2} \cdot r\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   5. Step-by-step derivation

      1. unpow2 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot r\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. *-commutative 96.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. associate-*r* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. *-commutative 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot r\right)} \cdot w\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   6. Simplified 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot w\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 88.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 7.2 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 3 \cdot 10^{+152}:\\ \;\;\;\;t_0 + \left(\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 7.2e-116)
     t_0
     (if (<= r 3e+152)
       (+ t_0 (- (* (* r r) (* -0.25 (* w w))) 1.5))
       (* -0.25 (* (* r w) (* r w)))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 7.2e-116) {
		tmp = t_0;
	} else if (r <= 3e+152) {
		tmp = t_0 + (((r * r) * (-0.25 * (w * w))) - 1.5);
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 7.2d-116) then
        tmp = t_0
    else if (r <= 3d+152) then
        tmp = t_0 + (((r * r) * ((-0.25d0) * (w * w))) - 1.5d0)
    else
        tmp = (-0.25d0) * ((r * w) * (r * w))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 7.2e-116) {
		tmp = t_0;
	} else if (r <= 3e+152) {
		tmp = t_0 + (((r * r) * (-0.25 * (w * w))) - 1.5);
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 7.2e-116:
		tmp = t_0
	elif r <= 3e+152:
		tmp = t_0 + (((r * r) * (-0.25 * (w * w))) - 1.5)
	else:
		tmp = -0.25 * ((r * w) * (r * w))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 7.2e-116)
		tmp = t_0;
	elseif (r <= 3e+152)
		tmp = Float64(t_0 + Float64(Float64(Float64(r * r) * Float64(-0.25 * Float64(w * w))) - 1.5));
	else
		tmp = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 7.2e-116)
		tmp = t_0;
	elseif (r <= 3e+152)
		tmp = t_0 + (((r * r) * (-0.25 * (w * w))) - 1.5);
	else
		tmp = -0.25 * ((r * w) * (r * w));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 7.2e-116], t$95$0, If[LessEqual[r, 3e+152], N[(t$95$0 + N[(N[(N[(r * r), $MachinePrecision] * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.19999999999999951e-116

   1. Initial program 86.2%

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

      1. sub-neg 86.2%

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

      2. +-commutative 86.2%

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

      3. associate--l+ 86.2%

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

      4. associate-/l* 86.4%

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

      5. distribute-neg-frac 86.4%

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

      6. associate-/r/ 86.5%

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

      7. fma-def 86.5%

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

      8. sub-neg 86.5%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
   4. Step-by-step derivation

      1. fma-udef 79.3%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

      2. unswap-sqr 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      3. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      4. div-inv 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

      5. fma-def 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

      6. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

      7. pow-flip 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]

   6. Taylor expanded in r around 0 55.0%

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

      1. unpow2 55.0%

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

   8. Simplified 55.0%

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

   if 7.19999999999999951e-116 < r < 2.99999999999999991e152

   1. Initial program 94.9%

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

      1. sub-neg 94.9%

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

      2. +-commutative 94.9%

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

      3. associate--l+ 94.9%

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

      4. associate-/l* 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. associate-/r/ 99.8%

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

      7. fma-def 99.8%

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

      8. sub-neg 99.8%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   4. Taylor expanded in v around inf 91.6%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} + -0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)\right) - 1.5} \]
   5. Step-by-step derivation

      1. associate--l+ 91.6%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(-0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right) - 1.5\right)} \]

      2. associate-*r/ 91.6%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(-0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right) - 1.5\right) \]

      3. metadata-eval 91.6%

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

      4. unpow2 91.6%

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

      5. unpow2 91.6%

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

      6. associate-*r* 91.6%

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

      7. unpow2 91.6%

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

   6. Simplified 91.6%

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

   if 2.99999999999999991e152 < r

   1. Initial program 90.9%

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

      1. associate--l- 90.9%

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

      2. +-commutative 90.9%

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

      3. associate--l+ 90.9%

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

      4. +-commutative 90.9%

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

      5. associate--r+ 90.9%

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

      6. metadata-eval 90.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 90.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 90.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 90.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 90.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-udef 90.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. div-inv 90.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. pow2 90.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      5. pow-flip 90.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      6. metadata-eval 90.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   5. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   6. Step-by-step derivation

      1. expm1-def 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-log1p 90.9%

         \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   7. Simplified 90.9%

      \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   8. Taylor expanded in v around inf 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. *-commutative 73.6%

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

      4. unpow2 73.6%

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

   10. Simplified 73.6%

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

   11. Taylor expanded in r around inf 73.6%

       \[\leadsto \color{blue}{-0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 73.6%

          \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right) \cdot -0.25} \]

       2. *-commutative 73.6%

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \cdot -0.25 \]

       3. unpow2 73.6%

          \[\leadsto \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25 \]

       4. unpow2 73.6%

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

       5. swap-sqr 84.8%

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

       6. unpow2 84.8%

          \[\leadsto \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25 \]

       7. *-commutative 84.8%

          \[\leadsto {\color{blue}{\left(w \cdot r\right)}}^{2} \cdot -0.25 \]

   13. Simplified 84.8%

       \[\leadsto \color{blue}{{\left(w \cdot r\right)}^{2} \cdot -0.25} \]
   14. Step-by-step derivation

       1. *-commutative 84.8%

          \[\leadsto {\color{blue}{\left(r \cdot w\right)}}^{2} \cdot -0.25 \]

       2. unpow2 84.8%

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

   15. Applied egg-rr 84.8%

       \[\leadsto \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25 \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 3 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(\left(r \cdot r\right) \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 10: 88.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.3 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 3 \cdot 10^{+152}:\\ \;\;\;\;t_0 + \left(-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.3e-114)
     t_0
     (if (<= r 3e+152)
       (+ t_0 (- (* -0.375 (* (* r r) (* w w))) 1.5))
       (* -0.25 (* (* r w) (* r w)))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.3e-114) {
		tmp = t_0;
	} else if (r <= 3e+152) {
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5);
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2.3d-114) then
        tmp = t_0
    else if (r <= 3d+152) then
        tmp = t_0 + (((-0.375d0) * ((r * r) * (w * w))) - 1.5d0)
    else
        tmp = (-0.25d0) * ((r * w) * (r * w))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.3e-114) {
		tmp = t_0;
	} else if (r <= 3e+152) {
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5);
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.3e-114:
		tmp = t_0
	elif r <= 3e+152:
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5)
	else:
		tmp = -0.25 * ((r * w) * (r * w))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.3e-114)
		tmp = t_0;
	elseif (r <= 3e+152)
		tmp = Float64(t_0 + Float64(Float64(-0.375 * Float64(Float64(r * r) * Float64(w * w))) - 1.5));
	else
		tmp = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.3e-114)
		tmp = t_0;
	elseif (r <= 3e+152)
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5);
	else
		tmp = -0.25 * ((r * w) * (r * w));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.3e-114], t$95$0, If[LessEqual[r, 3e+152], N[(t$95$0 + N[(N[(-0.375 * N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.2999999999999999e-114

   1. Initial program 86.2%

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

      1. sub-neg 86.2%

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

      2. +-commutative 86.2%

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

      3. associate--l+ 86.2%

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

      4. associate-/l* 86.4%

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

      5. distribute-neg-frac 86.4%

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

      6. associate-/r/ 86.5%

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

      7. fma-def 86.5%

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

      8. sub-neg 86.5%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
   4. Step-by-step derivation

      1. fma-udef 79.3%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

      2. unswap-sqr 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      3. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      4. div-inv 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

      5. fma-def 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

      6. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

      7. pow-flip 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

      8. metadata-eval 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]

   6. Taylor expanded in r around 0 55.0%

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

      1. unpow2 55.0%

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

   8. Simplified 55.0%

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

   if 2.2999999999999999e-114 < r < 2.99999999999999991e152

   1. Initial program 94.9%

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

      1. sub-neg 94.9%

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

      2. +-commutative 94.9%

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

      3. associate--l+ 94.9%

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

      4. associate-/l* 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. associate-/r/ 99.8%

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

      7. fma-def 99.8%

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

      8. sub-neg 99.8%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   4. Taylor expanded in v around 0 90.5%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}} + -0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)\right) - 1.5} \]
   5. Step-by-step derivation

      1. associate--l+ 90.5%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(-0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right) - 1.5\right)} \]

      2. associate-*r/ 90.5%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(-0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right) - 1.5\right) \]

      3. metadata-eval 90.5%

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

      4. unpow2 90.5%

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

      5. *-commutative 90.5%

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

      6. unpow2 90.5%

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

      7. *-commutative 90.5%

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

      8. unpow2 90.5%

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

   6. Simplified 90.5%

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

   if 2.99999999999999991e152 < r

   1. Initial program 90.9%

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

      1. associate--l- 90.9%

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

      2. +-commutative 90.9%

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

      3. associate--l+ 90.9%

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

      4. +-commutative 90.9%

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

      5. associate--r+ 90.9%

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

      6. metadata-eval 90.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 90.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 90.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 90.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 90.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 90.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-udef 90.9%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. div-inv 90.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. pow2 90.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      5. pow-flip 90.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      6. metadata-eval 90.9%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   5. Applied egg-rr 90.9%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   6. Step-by-step derivation

      1. expm1-def 90.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-log1p 90.9%

         \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   7. Simplified 90.9%

      \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   8. Taylor expanded in v around inf 73.6%

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

      1. *-commutative 73.6%

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

      2. unpow2 73.6%

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

      3. *-commutative 73.6%

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

      4. unpow2 73.6%

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

   10. Simplified 73.6%

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

   11. Taylor expanded in r around inf 73.6%

       \[\leadsto \color{blue}{-0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 73.6%

          \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right) \cdot -0.25} \]

       2. *-commutative 73.6%

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \cdot -0.25 \]

       3. unpow2 73.6%

          \[\leadsto \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25 \]

       4. unpow2 73.6%

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

       5. swap-sqr 84.8%

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

       6. unpow2 84.8%

          \[\leadsto \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25 \]

       7. *-commutative 84.8%

          \[\leadsto {\color{blue}{\left(w \cdot r\right)}}^{2} \cdot -0.25 \]

   13. Simplified 84.8%

       \[\leadsto \color{blue}{{\left(w \cdot r\right)}^{2} \cdot -0.25} \]
   14. Step-by-step derivation

       1. *-commutative 84.8%

          \[\leadsto {\color{blue}{\left(r \cdot w\right)}}^{2} \cdot -0.25 \]

       2. unpow2 84.8%

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

   15. Applied egg-rr 84.8%

       \[\leadsto \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25 \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.3 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 3 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 11: 76.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := -1.5 + \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.2:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 2.45 \cdot 10^{+35}:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{elif}\;r \leq 5.6 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ -1.5 (/ 2.0 (* r r)))))
   (if (<= r 1.2)
     t_0
     (if (<= r 2.45e+35)
       (* -0.25 (* w (* w (* r r))))
       (if (<= r 5.6e+59) t_0 (* -0.25 (* (* r w) (* r w))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double tmp;
	if (r <= 1.2) {
		tmp = t_0;
	} else if (r <= 2.45e+35) {
		tmp = -0.25 * (w * (w * (r * r)));
	} else if (r <= 5.6e+59) {
		tmp = t_0;
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.5d0) + (2.0d0 / (r * r))
    if (r <= 1.2d0) then
        tmp = t_0
    else if (r <= 2.45d+35) then
        tmp = (-0.25d0) * (w * (w * (r * r)))
    else if (r <= 5.6d+59) then
        tmp = t_0
    else
        tmp = (-0.25d0) * ((r * w) * (r * w))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double tmp;
	if (r <= 1.2) {
		tmp = t_0;
	} else if (r <= 2.45e+35) {
		tmp = -0.25 * (w * (w * (r * r)));
	} else if (r <= 5.6e+59) {
		tmp = t_0;
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = -1.5 + (2.0 / (r * r))
	tmp = 0
	if r <= 1.2:
		tmp = t_0
	elif r <= 2.45e+35:
		tmp = -0.25 * (w * (w * (r * r)))
	elif r <= 5.6e+59:
		tmp = t_0
	else:
		tmp = -0.25 * ((r * w) * (r * w))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(-1.5 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (r <= 1.2)
		tmp = t_0;
	elseif (r <= 2.45e+35)
		tmp = Float64(-0.25 * Float64(w * Float64(w * Float64(r * r))));
	elseif (r <= 5.6e+59)
		tmp = t_0;
	else
		tmp = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = -1.5 + (2.0 / (r * r));
	tmp = 0.0;
	if (r <= 1.2)
		tmp = t_0;
	elseif (r <= 2.45e+35)
		tmp = -0.25 * (w * (w * (r * r)));
	elseif (r <= 5.6e+59)
		tmp = t_0;
	else
		tmp = -0.25 * ((r * w) * (r * w));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.2], t$95$0, If[LessEqual[r, 2.45e+35], N[(-0.25 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 5.6e+59], t$95$0, N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.19999999999999996 or 2.45000000000000013e35 < r < 5.5999999999999996e59

   1. Initial program 88.6%

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

      1. sub-neg 88.6%

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

      2. +-commutative 88.6%

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

      3. associate--l+ 88.6%

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

      4. associate-/l* 88.8%

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

      5. distribute-neg-frac 88.8%

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

      6. associate-/r/ 88.8%

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

      7. fma-def 88.8%

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

      8. sub-neg 88.8%

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

   3. Simplified 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   4. Taylor expanded in r around 0 65.9%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   5. Step-by-step derivation

      1. sub-neg 65.9%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(-1.5\right)} \]

      2. associate-*r/ 65.9%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(-1.5\right) \]

      3. metadata-eval 65.9%

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(-1.5\right) \]

      4. unpow2 65.9%

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(-1.5\right) \]

      5. metadata-eval 65.9%

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

   6. Simplified 65.9%

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

   if 1.19999999999999996 < r < 2.45000000000000013e35

   1. Initial program 81.3%

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

      1. associate--l- 81.3%

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

      2. +-commutative 81.3%

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

      3. associate--l+ 81.3%

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

      4. +-commutative 81.3%

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

      5. associate--r+ 81.3%

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

      6. metadata-eval 81.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 100.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 100.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 100.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 100.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-udef 100.0%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. div-inv 100.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. pow2 100.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      5. pow-flip 100.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      6. metadata-eval 100.0%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   6. Step-by-step derivation

      1. expm1-def 100.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-log1p 100.0%

         \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   8. Taylor expanded in v around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. unpow2 99.7%

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

      3. *-commutative 99.7%

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

      4. unpow2 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in r around inf 99.7%

       \[\leadsto \color{blue}{-0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 99.7%

          \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right) \cdot -0.25} \]

       2. *-commutative 99.7%

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \cdot -0.25 \]

       3. unpow2 99.7%

          \[\leadsto \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25 \]

       4. unpow2 99.7%

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

       5. swap-sqr 100.0%

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

       6. unpow2 100.0%

          \[\leadsto \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25 \]

       7. *-commutative 100.0%

          \[\leadsto {\color{blue}{\left(w \cdot r\right)}}^{2} \cdot -0.25 \]

   13. Simplified 100.0%

       \[\leadsto \color{blue}{{\left(w \cdot r\right)}^{2} \cdot -0.25} \]
   14. Step-by-step derivation

       1. unpow2 100.0%

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

       2. unswap-sqr 99.7%

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

       3. associate-*l* 100.0%

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

   15. Applied egg-rr 100.0%

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

   if 5.5999999999999996e59 < r

   1. Initial program 90.6%

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

      1. associate--l- 90.6%

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

      2. +-commutative 90.6%

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

      3. associate--l+ 90.6%

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

      4. +-commutative 90.6%

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

      5. associate--r+ 90.6%

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

      6. metadata-eval 90.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 93.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 93.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 93.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 93.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 93.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-udef 93.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. div-inv 93.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. pow2 93.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      5. pow-flip 93.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      6. metadata-eval 93.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   5. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   6. Step-by-step derivation

      1. expm1-def 93.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-log1p 93.7%

         \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   7. Simplified 93.7%

      \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   8. Taylor expanded in v around inf 76.6%

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

      1. *-commutative 76.6%

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

      2. unpow2 76.6%

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

      3. *-commutative 76.6%

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

      4. unpow2 76.6%

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

   10. Simplified 76.6%

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

   11. Taylor expanded in r around inf 68.5%

       \[\leadsto \color{blue}{-0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 68.5%

          \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right) \cdot -0.25} \]

       2. *-commutative 68.5%

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \cdot -0.25 \]

       3. unpow2 68.5%

          \[\leadsto \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25 \]

       4. unpow2 68.5%

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

       5. swap-sqr 76.2%

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

       6. unpow2 76.2%

          \[\leadsto \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25 \]

       7. *-commutative 76.2%

          \[\leadsto {\color{blue}{\left(w \cdot r\right)}}^{2} \cdot -0.25 \]

   13. Simplified 76.2%

       \[\leadsto \color{blue}{{\left(w \cdot r\right)}^{2} \cdot -0.25} \]
   14. Step-by-step derivation

       1. *-commutative 76.2%

          \[\leadsto {\color{blue}{\left(r \cdot w\right)}}^{2} \cdot -0.25 \]

       2. unpow2 76.2%

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

   15. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25 \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.2:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 2.45 \cdot 10^{+35}:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{elif}\;r \leq 5.6 \cdot 10^{+59}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 12: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} t_0 := -1.5 + \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.25:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 3.65 \cdot 10^{+34}:\\ \;\;\;\;\frac{w \cdot w}{\frac{-2.6666666666666665}{r \cdot r}}\\ \mathbf{elif}\;r \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ -1.5 (/ 2.0 (* r r)))))
   (if (<= r 1.25)
     t_0
     (if (<= r 3.65e+34)
       (/ (* w w) (/ -2.6666666666666665 (* r r)))
       (if (<= r 7.5e+63) t_0 (* -0.25 (* (* r w) (* r w))))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double tmp;
	if (r <= 1.25) {
		tmp = t_0;
	} else if (r <= 3.65e+34) {
		tmp = (w * w) / (-2.6666666666666665 / (r * r));
	} else if (r <= 7.5e+63) {
		tmp = t_0;
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.5d0) + (2.0d0 / (r * r))
    if (r <= 1.25d0) then
        tmp = t_0
    else if (r <= 3.65d+34) then
        tmp = (w * w) / ((-2.6666666666666665d0) / (r * r))
    else if (r <= 7.5d+63) then
        tmp = t_0
    else
        tmp = (-0.25d0) * ((r * w) * (r * w))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double tmp;
	if (r <= 1.25) {
		tmp = t_0;
	} else if (r <= 3.65e+34) {
		tmp = (w * w) / (-2.6666666666666665 / (r * r));
	} else if (r <= 7.5e+63) {
		tmp = t_0;
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	t_0 = -1.5 + (2.0 / (r * r))
	tmp = 0
	if r <= 1.25:
		tmp = t_0
	elif r <= 3.65e+34:
		tmp = (w * w) / (-2.6666666666666665 / (r * r))
	elif r <= 7.5e+63:
		tmp = t_0
	else:
		tmp = -0.25 * ((r * w) * (r * w))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	t_0 = Float64(-1.5 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (r <= 1.25)
		tmp = t_0;
	elseif (r <= 3.65e+34)
		tmp = Float64(Float64(w * w) / Float64(-2.6666666666666665 / Float64(r * r)));
	elseif (r <= 7.5e+63)
		tmp = t_0;
	else
		tmp = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	t_0 = -1.5 + (2.0 / (r * r));
	tmp = 0.0;
	if (r <= 1.25)
		tmp = t_0;
	elseif (r <= 3.65e+34)
		tmp = (w * w) / (-2.6666666666666665 / (r * r));
	elseif (r <= 7.5e+63)
		tmp = t_0;
	else
		tmp = -0.25 * ((r * w) * (r * w));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := Block[{t$95$0 = N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.25], t$95$0, If[LessEqual[r, 3.65e+34], N[(N[(w * w), $MachinePrecision] / N[(-2.6666666666666665 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 7.5e+63], t$95$0, N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.25 or 3.6499999999999998e34 < r < 7.5000000000000005e63

   1. Initial program 88.6%

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

      1. sub-neg 88.6%

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

      2. +-commutative 88.6%

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

      3. associate--l+ 88.6%

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

      4. associate-/l* 88.8%

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

      5. distribute-neg-frac 88.8%

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

      6. associate-/r/ 88.8%

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

      7. fma-def 88.8%

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

      8. sub-neg 88.8%

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

   3. Simplified 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   4. Taylor expanded in r around 0 65.9%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   5. Step-by-step derivation

      1. sub-neg 65.9%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(-1.5\right)} \]

      2. associate-*r/ 65.9%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(-1.5\right) \]

      3. metadata-eval 65.9%

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(-1.5\right) \]

      4. unpow2 65.9%

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(-1.5\right) \]

      5. metadata-eval 65.9%

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

   6. Simplified 65.9%

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

   if 1.25 < r < 3.6499999999999998e34

   1. Initial program 81.3%

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

      1. sub-neg 81.3%

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

      2. +-commutative 81.3%

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

      3. associate--l+ 81.3%

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

      4. associate-/l* 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. associate-/r/ 100.0%

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

      7. fma-def 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.7%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

      2. unswap-sqr 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      3. pow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      4. div-inv 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

      5. fma-def 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

      6. pow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

      7. pow-flip 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

      8. metadata-eval 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]

   6. Taylor expanded in w around inf 81.3%

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

   7. Taylor expanded in r around 0 81.3%

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

      1. associate-/l* 81.3%

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

      2. unpow2 81.3%

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

      3. fma-neg 81.3%

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

      4. metadata-eval 81.3%

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

      5. *-commutative 81.3%

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

      6. unpow2 81.3%

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

      7. associate-*l* 81.3%

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

   9. Simplified 81.3%

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

   10. Taylor expanded in v around 0 84.0%

       \[\leadsto \frac{w \cdot w}{\color{blue}{\frac{-2.6666666666666665}{{r}^{2}}}} \]
   11. Step-by-step derivation

       1. unpow2 84.0%

          \[\leadsto \frac{w \cdot w}{\frac{-2.6666666666666665}{\color{blue}{r \cdot r}}} \]

   12. Simplified 84.0%

       \[\leadsto \frac{w \cdot w}{\color{blue}{\frac{-2.6666666666666665}{r \cdot r}}} \]

   if 7.5000000000000005e63 < r

   1. Initial program 90.6%

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

      1. associate--l- 90.6%

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

      2. +-commutative 90.6%

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

      3. associate--l+ 90.6%

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

      4. +-commutative 90.6%

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

      5. associate--r+ 90.6%

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

      6. metadata-eval 90.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 93.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 93.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 93.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 93.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 93.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-udef 93.7%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. div-inv 93.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. pow2 93.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      5. pow-flip 93.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      6. metadata-eval 93.7%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   5. Applied egg-rr 93.7%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   6. Step-by-step derivation

      1. expm1-def 93.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-log1p 93.7%

         \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   7. Simplified 93.7%

      \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   8. Taylor expanded in v around inf 76.6%

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

      1. *-commutative 76.6%

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

      2. unpow2 76.6%

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

      3. *-commutative 76.6%

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

      4. unpow2 76.6%

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

   10. Simplified 76.6%

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

   11. Taylor expanded in r around inf 68.5%

       \[\leadsto \color{blue}{-0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 68.5%

          \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right) \cdot -0.25} \]

       2. *-commutative 68.5%

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \cdot -0.25 \]

       3. unpow2 68.5%

          \[\leadsto \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25 \]

       4. unpow2 68.5%

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

       5. swap-sqr 76.2%

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

       6. unpow2 76.2%

          \[\leadsto \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25 \]

       7. *-commutative 76.2%

          \[\leadsto {\color{blue}{\left(w \cdot r\right)}}^{2} \cdot -0.25 \]

   13. Simplified 76.2%

       \[\leadsto \color{blue}{{\left(w \cdot r\right)}^{2} \cdot -0.25} \]
   14. Step-by-step derivation

       1. *-commutative 76.2%

          \[\leadsto {\color{blue}{\left(r \cdot w\right)}}^{2} \cdot -0.25 \]

       2. unpow2 76.2%

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

   15. Applied egg-rr 76.2%

       \[\leadsto \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25 \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.25:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 3.65 \cdot 10^{+34}:\\ \;\;\;\;\frac{w \cdot w}{\frac{-2.6666666666666665}{r \cdot r}}\\ \mathbf{elif}\;r \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 13: 62.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} r = |r|\\ \\ \begin{array}{l} \mathbf{if}\;w \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r)
 :precision binary64
 (if (<= w 6.6e+44) (+ -1.5 (/ 2.0 (* r r))) (* -0.25 (* w (* w (* r r))))))

C

r = abs(r);
double code(double v, double w, double r) {
	double tmp;
	if (w <= 6.6e+44) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -0.25 * (w * (w * (r * r)));
	}
	return tmp;
}

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (w <= 6.6d+44) then
        tmp = (-1.5d0) + (2.0d0 / (r * r))
    else
        tmp = (-0.25d0) * (w * (w * (r * r)))
    end if
    code = tmp
end function

Java

r = Math.abs(r);
public static double code(double v, double w, double r) {
	double tmp;
	if (w <= 6.6e+44) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -0.25 * (w * (w * (r * r)));
	}
	return tmp;
}

Python

r = abs(r)
def code(v, w, r):
	tmp = 0
	if w <= 6.6e+44:
		tmp = -1.5 + (2.0 / (r * r))
	else:
		tmp = -0.25 * (w * (w * (r * r)))
	return tmp

Julia

r = abs(r)
function code(v, w, r)
	tmp = 0.0
	if (w <= 6.6e+44)
		tmp = Float64(-1.5 + Float64(2.0 / Float64(r * r)));
	else
		tmp = Float64(-0.25 * Float64(w * Float64(w * Float64(r * r))));
	end
	return tmp
end

MATLAB

r = abs(r)
function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (w <= 6.6e+44)
		tmp = -1.5 + (2.0 / (r * r));
	else
		tmp = -0.25 * (w * (w * (r * r)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := If[LessEqual[w, 6.6e+44], N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 6.60000000000000027e44

   1. Initial program 90.2%

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

      1. sub-neg 90.2%

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

      2. +-commutative 90.2%

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

      3. associate--l+ 90.2%

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

      4. associate-/l* 91.7%

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

      5. distribute-neg-frac 91.7%

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

      6. associate-/r/ 91.8%

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

      7. fma-def 91.7%

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

      8. sub-neg 91.7%

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

   3. Simplified 82.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   4. Taylor expanded in r around 0 60.7%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   5. Step-by-step derivation

      1. sub-neg 60.7%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(-1.5\right)} \]

      2. associate-*r/ 60.7%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(-1.5\right) \]

      3. metadata-eval 60.7%

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(-1.5\right) \]

      4. unpow2 60.7%

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(-1.5\right) \]

      5. metadata-eval 60.7%

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

   6. Simplified 60.7%

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

   if 6.60000000000000027e44 < w

   1. Initial program 83.2%

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

      1. associate--l- 83.2%

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

      2. +-commutative 83.2%

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

      3. associate--l+ 83.2%

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

      4. +-commutative 83.2%

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

      5. associate--r+ 83.2%

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

      6. metadata-eval 83.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]

      7. associate-*l/ 83.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]

      8. *-commutative 83.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]

      9. *-commutative 83.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

      10. *-commutative 83.2%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Simplified 83.2%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 83.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-udef 83.2%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{r \cdot r}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      3. div-inv 83.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \frac{1}{r \cdot r}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      4. pow2 83.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      5. pow-flip 83.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      6. metadata-eval 83.2%

         \[\leadsto \left(e^{\mathsf{log1p}\left(2 \cdot {r}^{\color{blue}{-2}}\right)} - 1\right) + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   5. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)} - 1\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
   6. Step-by-step derivation

      1. expm1-def 83.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot {r}^{-2}\right)\right)} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

      2. expm1-log1p 83.2%

         \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   7. Simplified 83.2%

      \[\leadsto \color{blue}{2 \cdot {r}^{-2}} + \left(-1.5 - \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   8. Taylor expanded in v around inf 78.2%

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

      1. *-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. *-commutative 78.2%

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

      4. unpow2 78.2%

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

   10. Simplified 78.2%

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

   11. Taylor expanded in r around inf 71.9%

       \[\leadsto \color{blue}{-0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   12. Step-by-step derivation

       1. *-commutative 71.9%

          \[\leadsto \color{blue}{\left({w}^{2} \cdot {r}^{2}\right) \cdot -0.25} \]

       2. *-commutative 71.9%

          \[\leadsto \color{blue}{\left({r}^{2} \cdot {w}^{2}\right)} \cdot -0.25 \]

       3. unpow2 71.9%

          \[\leadsto \left(\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}\right) \cdot -0.25 \]

       4. unpow2 71.9%

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

       5. swap-sqr 75.7%

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

       6. unpow2 75.7%

          \[\leadsto \color{blue}{{\left(r \cdot w\right)}^{2}} \cdot -0.25 \]

       7. *-commutative 75.7%

          \[\leadsto {\color{blue}{\left(w \cdot r\right)}}^{2} \cdot -0.25 \]

   13. Simplified 75.7%

       \[\leadsto \color{blue}{{\left(w \cdot r\right)}^{2} \cdot -0.25} \]
   14. Step-by-step derivation

       1. unpow2 75.7%

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

       2. unswap-sqr 71.9%

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

       3. associate-*l* 75.8%

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

   15. Applied egg-rr 75.8%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 6.6 \cdot 10^{+44}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \end{array} \]

Alternative 14: 57.1% accurate, 4.1× speedup

Math

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

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r) :precision binary64 (if (<= r 1.15) (/ 2.0 (* r r)) -1.5))

C

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

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.15d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := If[LessEqual[r, 1.15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 1.1499999999999999

   1. Initial program 88.4%

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

      1. sub-neg 88.4%

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

      2. +-commutative 88.4%

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

      3. associate--l+ 88.4%

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

      4. associate-/l* 88.6%

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

      5. distribute-neg-frac 88.6%

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

      6. associate-/r/ 88.7%

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

      7. fma-def 88.6%

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

      8. sub-neg 88.6%

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

   3. Simplified 82.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]
   4. Step-by-step derivation

      1. fma-udef 82.7%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]

      2. unswap-sqr 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      3. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \color{blue}{{\left(r \cdot w\right)}^{2}} + \left(\frac{2}{r \cdot r} + -1.5\right) \]

      4. div-inv 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \left(\color{blue}{2 \cdot \frac{1}{r \cdot r}} + -1.5\right) \]

      5. fma-def 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \color{blue}{\mathsf{fma}\left(2, \frac{1}{r \cdot r}, -1.5\right)} \]

      6. pow2 99.8%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \frac{1}{\color{blue}{{r}^{2}}}, -1.5\right) \]

      7. pow-flip 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, \color{blue}{{r}^{\left(-2\right)}}, -1.5\right) \]

      8. metadata-eval 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{\color{blue}{-2}}, -1.5\right) \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot {\left(r \cdot w\right)}^{2} + \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)} \]

   6. Taylor expanded in r around 0 57.3%

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

      1. unpow2 57.3%

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

   8. Simplified 57.3%

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

   if 1.1499999999999999 < r

   1. Initial program 90.3%

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

      1. sub-neg 90.3%

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

      2. +-commutative 90.3%

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

      3. associate--l+ 90.3%

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

      4. associate-/l* 94.4%

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

      5. distribute-neg-frac 94.4%

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

      6. associate-/r/ 94.4%

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

      7. fma-def 94.4%

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

      8. sub-neg 94.4%

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

   3. Simplified 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

   4. Taylor expanded in r around 0 19.3%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
   5. Step-by-step derivation

      1. sub-neg 19.3%

         \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(-1.5\right)} \]

      2. associate-*r/ 19.3%

         \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(-1.5\right) \]

      3. metadata-eval 19.3%

         \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(-1.5\right) \]

      4. unpow2 19.3%

         \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(-1.5\right) \]

      5. metadata-eval 19.3%

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

   6. Simplified 19.3%

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

   7. Taylor expanded in r around inf 19.3%

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

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.15:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]

Alternative 15: 57.6% accurate, 4.1× speedup

Math

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

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r) :precision binary64 (+ -1.5 (/ 2.0 (* r r))))

C

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

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (-1.5d0) + (2.0d0 / (r * r))
end function

Java

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

Python

r = abs(r)
def code(v, w, r):
	return -1.5 + (2.0 / (r * r))

Julia

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

MATLAB

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

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.9%

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

   1. sub-neg 88.9%

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

   2. +-commutative 88.9%

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

   3. associate--l+ 88.9%

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

   4. associate-/l* 90.2%

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

   5. distribute-neg-frac 90.2%

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

   6. associate-/r/ 90.2%

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

   7. fma-def 90.2%

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

   8. sub-neg 90.2%

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

3. Simplified 83.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

4. Taylor expanded in r around 0 53.5%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
5. Step-by-step derivation

   1. sub-neg 53.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(-1.5\right)} \]

   2. associate-*r/ 53.5%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(-1.5\right) \]

   3. metadata-eval 53.5%

      \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(-1.5\right) \]

   4. unpow2 53.5%

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(-1.5\right) \]

   5. metadata-eval 53.5%

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

6. Simplified 53.5%

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

7. Final simplification 53.5%

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

Alternative 16: 14.0% accurate, 29.0× speedup

Math

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

FPCore

NOTE: r should be positive before calling this function
(FPCore (v w r) :precision binary64 -1.5)

C

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

Fortran

NOTE: r should be positive before calling this function
real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

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

Python

r = abs(r)
def code(v, w, r):
	return -1.5

Julia

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

MATLAB

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

Wolfram

NOTE: r should be positive before calling this function
code[v_, w_, r_] := -1.5

Derivation

1. Initial program 88.9%

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

   1. sub-neg 88.9%

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

   2. +-commutative 88.9%

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

   3. associate--l+ 88.9%

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

   4. associate-/l* 90.2%

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

   5. distribute-neg-frac 90.2%

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

   6. associate-/r/ 90.2%

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

   7. fma-def 90.2%

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

   8. sub-neg 90.2%

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

3. Simplified 83.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \]

4. Taylor expanded in r around 0 53.5%

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - 1.5} \]
5. Step-by-step derivation

   1. sub-neg 53.5%

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} + \left(-1.5\right)} \]

   2. associate-*r/ 53.5%

      \[\leadsto \color{blue}{\frac{2 \cdot 1}{{r}^{2}}} + \left(-1.5\right) \]

   3. metadata-eval 53.5%

      \[\leadsto \frac{\color{blue}{2}}{{r}^{2}} + \left(-1.5\right) \]

   4. unpow2 53.5%

      \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + \left(-1.5\right) \]

   5. metadata-eval 53.5%

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

6. Simplified 53.5%

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

7. Taylor expanded in r around inf 12.0%

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

8. Final simplification 12.0%

   \[\leadsto -1.5 \]

Reproduce

herbie shell --seed 2023213 
(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))