Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 99.9%

Time: 11.8s

Alternatives: 17

Speedup: 0.2×

• 52.4% 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: 84.8% 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

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

FPCore

(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

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

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

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 83.6%

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

   1. sub-neg 83.6%

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

   2. +-commutative 83.6%

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

   3. associate--l+ 83.6%

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

   4. associate-/l* 86.9%

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

   5. distribute-neg-frac 86.9%

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

   6. associate-/r/ 86.9%

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

   7. fma-def 86.9%

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

   8. sub-neg 86.9%

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

3. Simplified 80.3%

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

   1. fma-udef 80.3%

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

   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 99.9%

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

   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. Final simplification 99.9%

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

Alternative 2: 99.5% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[N[(N[(1.0 - v), $MachinePrecision] / N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 83.6%

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

   1. sub-neg 83.6%

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

   2. associate-/l* 86.9%

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

   3. cancel-sign-sub-inv 86.9%

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

   4. metadata-eval 86.9%

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

   5. *-commutative 86.9%

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

   6. *-commutative 86.9%

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

   7. metadata-eval 86.9%

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

3. Simplified 86.9%

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

   1. add-cube-cbrt 86.8%

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

   2. pow3 86.8%

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

   3. associate-*r* 80.1%

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

   4. unswap-sqr 99.6%

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

   5. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{+307}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - {\left(r \cdot w\right)}^{2} \cdot 0.375\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 1e+307)
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))
     (+ -4.5 (- (+ 3.0 t_0) (* (pow (* r w) 2.0) 0.375))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e+307) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (pow((r * w), 2.0) * 0.375));
	}
	return tmp;
}

Fortran

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 ((w * w) <= 1d+307) then
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v))))
    else
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (((r * w) ** 2.0d0) * 0.375d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e+307) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (Math.pow((r * w), 2.0) * 0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 1e+307:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))))
	else:
		tmp = -4.5 + ((3.0 + t_0) - (math.pow((r * w), 2.0) * 0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 1e+307)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64((Float64(r * w) ^ 2.0) * 0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 1e+307)
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	else
		tmp = -4.5 + ((3.0 + t_0) - (((r * w) ^ 2.0) * 0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 1e+307], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 9.99999999999999986e306

   1. Initial program 93.6%

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

      1. associate--l- 93.6%

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

      2. +-commutative 93.6%

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

      3. associate--l+ 93.6%

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

      4. +-commutative 93.6%

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

      5. associate--r+ 93.6%

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

      6. metadata-eval 93.6%

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

      7. associate-*l/ 98.1%

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

      8. *-commutative 98.1%

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

      9. *-commutative 98.1%

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

      10. *-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in r around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      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%

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

   if 9.99999999999999986e306 < (*.f64 w w)

   1. Initial program 54.8%

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

      1. sub-neg 54.8%

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

      2. associate-/l* 54.8%

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

      3. cancel-sign-sub-inv 54.8%

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

      4. metadata-eval 54.8%

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

      5. *-commutative 54.8%

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

      6. *-commutative 54.8%

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

      7. metadata-eval 54.8%

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

   3. Simplified 54.8%

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

   4. Taylor expanded in v around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. *-commutative 54.8%

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

      3. unpow2 54.8%

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

      4. unpow2 54.8%

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

      5. swap-sqr 98.8%

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

      6. unpow2 98.8%

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

      7. *-commutative 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+307}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - {\left(r \cdot w\right)}^{2} \cdot 0.375\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+289}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left({\left(r \cdot w\right)}^{2} \cdot -0.25 - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 4e+289)
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))
     (+ t_0 (- (* (pow (* r w) 2.0) -0.25) 1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 4e+289) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = t_0 + ((pow((r * w), 2.0) * -0.25) - 1.5);
	}
	return tmp;
}

Fortran

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 ((w * w) <= 4d+289) then
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v))))
    else
        tmp = t_0 + ((((r * w) ** 2.0d0) * (-0.25d0)) - 1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 4e+289) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = t_0 + ((Math.pow((r * w), 2.0) * -0.25) - 1.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 4e+289:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))))
	else:
		tmp = t_0 + ((math.pow((r * w), 2.0) * -0.25) - 1.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 4e+289)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	else
		tmp = Float64(t_0 + Float64(Float64((Float64(r * w) ^ 2.0) * -0.25) - 1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 4e+289)
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	else
		tmp = t_0 + ((((r * w) ^ 2.0) * -0.25) - 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 4e+289], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * -0.25), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 4.0000000000000002e289

   1. Initial program 94.0%

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

      1. associate--l- 94.0%

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

      2. +-commutative 94.0%

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

      3. associate--l+ 94.0%

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

      4. +-commutative 94.0%

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

      5. associate--r+ 94.0%

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

      6. metadata-eval 94.0%

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

      7. associate-*l/ 98.1%

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

      8. *-commutative 98.1%

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

      9. *-commutative 98.1%

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

      10. *-commutative 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in r around 0 98.1%

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

      1. *-commutative 98.1%

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

      2. unpow2 98.1%

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

      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%

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

   if 4.0000000000000002e289 < (*.f64 w w)

   1. Initial program 55.4%

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

      1. sub-neg 55.4%

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

      2. +-commutative 55.4%

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

      3. associate--l+ 55.4%

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

      4. associate-/l* 56.8%

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

      5. distribute-neg-frac 56.8%

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

      6. associate-/r/ 56.8%

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

      7. fma-def 56.8%

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

      8. sub-neg 56.8%

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

   3. Simplified 56.8%

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

   4. Taylor expanded in v around inf 56.8%

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

      1. associate--l+ 56.8%

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

      2. associate-*r/ 56.8%

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

      3. metadata-eval 56.8%

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

      4. unpow2 56.8%

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

      5. *-commutative 56.8%

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

      6. unpow2 56.8%

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

      7. unpow2 56.8%

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

   6. Simplified 56.8%

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

      1. unswap-sqr 98.8%

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

      2. unpow2 98.8%

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

      3. *-commutative 98.8%

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

   8. Applied egg-rr 98.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+289}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left({\left(r \cdot w\right)}^{2} \cdot -0.25 - 1.5\right)\\ \end{array} \]

Alternative 5: 72.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{-140}:\\ \;\;\;\;2 \cdot {r}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7e-140)
   (* 2.0 (pow r -2.0))
   (+
    (/ 2.0 (* r r))
    (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7e-140) {
		tmp = 2.0 * pow(r, -2.0);
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	}
	return tmp;
}

Fortran

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 <= 7d-140) then
        tmp = 2.0d0 * (r ** (-2.0d0))
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((r * (w * (r * w))) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7e-140) {
		tmp = 2.0 * Math.pow(r, -2.0);
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7e-140:
		tmp = 2.0 * math.pow(r, -2.0)
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7e-140)
		tmp = Float64(2.0 * (r ^ -2.0));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7e-140)
		tmp = 2.0 * (r ^ -2.0);
	else
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7e-140], N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 6.9999999999999996e-140

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. associate-/l* 81.6%

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

      3. cancel-sign-sub-inv 81.6%

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

      4. metadata-eval 81.6%

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

      5. *-commutative 81.6%

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

      6. *-commutative 81.6%

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

      7. metadata-eval 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in v around 0 72.4%

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

      1. *-commutative 72.4%

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

      2. *-commutative 72.4%

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

      3. unpow2 72.4%

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

      4. unpow2 72.4%

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

      5. swap-sqr 93.9%

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

      6. unpow2 93.9%

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

      7. *-commutative 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in r around 0 56.5%

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

      1. unpow2 56.5%

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

   9. Simplified 56.5%

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

       1. div-inv 56.5%

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

       2. pow2 56.5%

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

       3. pow-flip 56.6%

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

       4. metadata-eval 56.6%

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

   11. Applied egg-rr 56.6%

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

   if 6.9999999999999996e-140 < r

   1. Initial program 91.5%

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

      1. associate--l- 91.5%

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

      2. +-commutative 91.5%

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

      3. associate--l+ 91.5%

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

      4. +-commutative 91.5%

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

      5. associate--r+ 91.5%

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

      6. metadata-eval 91.5%

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

      7. associate-*l/ 96.6%

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

      8. *-commutative 96.6%

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

      9. *-commutative 96.6%

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

      10. *-commutative 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in r around 0 96.6%

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

      1. *-commutative 96.6%

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

      2. unpow2 96.6%

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

      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%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{-140}:\\ \;\;\;\;2 \cdot {r}^{-2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \end{array} \]

Alternative 6: 70.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.2 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.2e-134)
     t_0
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* r (* w w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.2e-134) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	}
	return tmp;
}

Fortran

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-134) 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

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.2e-134) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.2e-134:
		tmp = t_0
	else:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.2e-134)
		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

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.2e-134)
		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

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.2e-134], 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 < 2.2e-134

   1. Initial program 78.8%

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

      1. sub-neg 78.8%

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

      2. associate-/l* 81.1%

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

      3. cancel-sign-sub-inv 81.1%

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

      4. metadata-eval 81.1%

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

      5. *-commutative 81.1%

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

      6. *-commutative 81.1%

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

      7. metadata-eval 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in v around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. *-commutative 71.9%

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

      3. unpow2 71.9%

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

      4. unpow2 71.9%

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

      5. swap-sqr 93.9%

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

      6. unpow2 93.9%

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

      7. *-commutative 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in r around 0 56.8%

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

      1. unpow2 56.8%

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

   9. Simplified 56.8%

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

   if 2.2e-134 < r

   1. Initial program 92.5%

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

      1. associate--l- 92.5%

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

      2. +-commutative 92.5%

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

      3. associate--l+ 92.5%

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

      4. +-commutative 92.5%

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

      5. associate--r+ 92.5%

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

      6. metadata-eval 92.5%

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

      7. associate-*l/ 97.7%

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

      8. *-commutative 97.7%

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

      9. *-commutative 97.7%

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

      10. *-commutative 97.7%

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

   3. Simplified 97.7%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 71.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 8 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 8e-149)
     t_0
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 8e-149) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	}
	return tmp;
}

Fortran

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 <= 8d-149) then
        tmp = t_0
    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

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 8e-149) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 8e-149:
		tmp = t_0
	else:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 8e-149)
		tmp = t_0;
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 8e-149)
		tmp = t_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

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 8e-149], t$95$0, N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 7.99999999999999983e-149

   1. Initial program 79.2%

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

      1. sub-neg 79.2%

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

      2. associate-/l* 81.5%

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

      3. cancel-sign-sub-inv 81.5%

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

      4. metadata-eval 81.5%

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

      5. *-commutative 81.5%

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

      6. *-commutative 81.5%

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

      7. metadata-eval 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in v around 0 72.2%

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

      1. *-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

      4. unpow2 72.2%

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

      5. swap-sqr 93.8%

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

      6. unpow2 93.8%

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

      7. *-commutative 93.8%

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

   6. Simplified 93.8%

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

   7. Taylor expanded in r around 0 56.2%

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

      1. unpow2 56.2%

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

   9. Simplified 56.2%

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

   if 7.99999999999999983e-149 < r

   1. Initial program 91.5%

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

      1. associate--l- 91.5%

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

      2. +-commutative 91.5%

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

      3. associate--l+ 91.5%

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

      4. +-commutative 91.5%

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

      5. associate--r+ 91.5%

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

      6. metadata-eval 91.5%

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

      7. associate-*l/ 96.7%

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

      8. *-commutative 96.7%

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

      9. *-commutative 96.7%

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

      10. *-commutative 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in r around 0 96.7%

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

      1. *-commutative 96.7%

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

      2. unpow2 96.7%

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

      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%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \end{array} \]

Alternative 8: 66.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.35 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 2.5 \cdot 10^{+144}:\\ \;\;\;\;t_0 + \left(-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1.35e-132)
     t_0
     (if (<= r 2.5e+144)
       (+ t_0 (- (* -0.375 (* (* r r) (* w w))) 1.5))
       (* -0.25 (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.35e-132) {
		tmp = t_0;
	} else if (r <= 2.5e+144) {
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5);
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Fortran

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 <= 1.35d-132) then
        tmp = t_0
    else if (r <= 2.5d+144) 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

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.35e-132) {
		tmp = t_0;
	} else if (r <= 2.5e+144) {
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5);
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1.35e-132:
		tmp = t_0
	elif r <= 2.5e+144:
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5)
	else:
		tmp = -0.25 * ((r * w) * (r * w))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1.35e-132)
		tmp = t_0;
	elseif (r <= 2.5e+144)
		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

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 1.35e-132)
		tmp = t_0;
	elseif (r <= 2.5e+144)
		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

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.35e-132], t$95$0, If[LessEqual[r, 2.5e+144], 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 < 1.34999999999999995e-132

   1. Initial program 78.8%

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

      1. sub-neg 78.8%

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

      2. associate-/l* 81.1%

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

      3. cancel-sign-sub-inv 81.1%

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

      4. metadata-eval 81.1%

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

      5. *-commutative 81.1%

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

      6. *-commutative 81.1%

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

      7. metadata-eval 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in v around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. *-commutative 71.9%

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

      3. unpow2 71.9%

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

      4. unpow2 71.9%

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

      5. swap-sqr 93.9%

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

      6. unpow2 93.9%

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

      7. *-commutative 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in r around 0 56.8%

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

      1. unpow2 56.8%

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

   9. Simplified 56.8%

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

   if 1.34999999999999995e-132 < r < 2.5e144

   1. Initial program 95.2%

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

      1. sub-neg 95.2%

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

      2. +-commutative 95.2%

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

      3. associate--l+ 95.2%

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

      4. associate-/l* 98.3%

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

      5. distribute-neg-frac 98.3%

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

      6. associate-/r/ 98.3%

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

      7. fma-def 98.3%

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

      8. sub-neg 98.3%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in v around 0 93.2%

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

      1. associate--l+ 93.2%

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

      2. associate-*r/ 93.2%

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

      3. metadata-eval 93.2%

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

      4. unpow2 93.2%

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

      5. *-commutative 93.2%

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

      6. unpow2 93.2%

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

      7. unpow2 93.2%

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

   6. Simplified 93.2%

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

   if 2.5e144 < r

   1. Initial program 86.1%

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

      1. sub-neg 86.1%

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

      2. +-commutative 86.1%

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

      3. associate--l+ 86.1%

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

      4. associate-/l* 96.4%

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

      5. distribute-neg-frac 96.4%

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

      6. associate-/r/ 96.4%

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

      7. fma-def 96.4%

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

      8. sub-neg 96.4%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in r around inf 67.8%

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

   5. Taylor expanded in v around inf 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. *-commutative 67.8%

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

      4. unpow2 67.8%

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

      5. swap-sqr 86.4%

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

      6. unpow2 86.4%

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

      7. *-commutative 86.4%

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

   7. Simplified 86.4%

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

      1. unpow2 86.4%

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

      2. *-commutative 86.4%

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

      3. *-commutative 86.4%

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

   9. Applied egg-rr 86.4%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.35 \cdot 10^{-132}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 2.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 9: 66.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 3.6 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 2.5 \cdot 10^{+144}:\\ \;\;\;\;t_0 + \left(-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 3.6e-136)
     t_0
     (if (<= r 2.5e+144)
       (+ t_0 (- (* -0.25 (* (* r r) (* w w))) 1.5))
       (* -0.25 (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.6e-136) {
		tmp = t_0;
	} else if (r <= 2.5e+144) {
		tmp = t_0 + ((-0.25 * ((r * r) * (w * w))) - 1.5);
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Fortran

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 <= 3.6d-136) then
        tmp = t_0
    else if (r <= 2.5d+144) then
        tmp = t_0 + (((-0.25d0) * ((r * r) * (w * w))) - 1.5d0)
    else
        tmp = (-0.25d0) * ((r * w) * (r * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 3.6e-136) {
		tmp = t_0;
	} else if (r <= 2.5e+144) {
		tmp = t_0 + ((-0.25 * ((r * r) * (w * w))) - 1.5);
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 3.6e-136:
		tmp = t_0
	elif r <= 2.5e+144:
		tmp = t_0 + ((-0.25 * ((r * r) * (w * w))) - 1.5)
	else:
		tmp = -0.25 * ((r * w) * (r * w))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 3.6e-136)
		tmp = t_0;
	elseif (r <= 2.5e+144)
		tmp = Float64(t_0 + Float64(Float64(-0.25 * 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

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 3.6e-136)
		tmp = t_0;
	elseif (r <= 2.5e+144)
		tmp = t_0 + ((-0.25 * ((r * r) * (w * w))) - 1.5);
	else
		tmp = -0.25 * ((r * w) * (r * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 3.6e-136], t$95$0, If[LessEqual[r, 2.5e+144], N[(t$95$0 + N[(N[(-0.25 * 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 < 3.5999999999999998e-136

   1. Initial program 78.8%

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

      1. sub-neg 78.8%

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

      2. associate-/l* 81.1%

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

      3. cancel-sign-sub-inv 81.1%

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

      4. metadata-eval 81.1%

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

      5. *-commutative 81.1%

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

      6. *-commutative 81.1%

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

      7. metadata-eval 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in v around 0 71.9%

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

      1. *-commutative 71.9%

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

      2. *-commutative 71.9%

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

      3. unpow2 71.9%

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

      4. unpow2 71.9%

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

      5. swap-sqr 93.9%

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

      6. unpow2 93.9%

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

      7. *-commutative 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in r around 0 56.8%

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

      1. unpow2 56.8%

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

   9. Simplified 56.8%

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

   if 3.5999999999999998e-136 < r < 2.5e144

   1. Initial program 95.2%

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

      1. sub-neg 95.2%

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

      2. +-commutative 95.2%

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

      3. associate--l+ 95.2%

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

      4. associate-/l* 98.3%

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

      5. distribute-neg-frac 98.3%

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

      6. associate-/r/ 98.3%

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

      7. fma-def 98.3%

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

      8. sub-neg 98.3%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in v around inf 95.7%

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

      1. associate--l+ 95.7%

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

      2. associate-*r/ 95.7%

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

      3. metadata-eval 95.7%

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

      4. unpow2 95.7%

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

      5. *-commutative 95.7%

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

      6. unpow2 95.7%

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

      7. unpow2 95.7%

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

   6. Simplified 95.7%

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

   if 2.5e144 < r

   1. Initial program 86.1%

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

      1. sub-neg 86.1%

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

      2. +-commutative 86.1%

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

      3. associate--l+ 86.1%

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

      4. associate-/l* 96.4%

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

      5. distribute-neg-frac 96.4%

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

      6. associate-/r/ 96.4%

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

      7. fma-def 96.4%

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

      8. sub-neg 96.4%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in r around inf 67.8%

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

   5. Taylor expanded in v around inf 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. *-commutative 67.8%

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

      4. unpow2 67.8%

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

      5. swap-sqr 86.4%

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

      6. unpow2 86.4%

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

      7. *-commutative 86.4%

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

   7. Simplified 86.4%

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

      1. unpow2 86.4%

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

      2. *-commutative 86.4%

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

      3. *-commutative 86.4%

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

   9. Applied egg-rr 86.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 2.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 10: 65.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{+23} \lor \neg \left(r \leq 4.2 \cdot 10^{+63}\right) \land \left(r \leq 2 \cdot 10^{+95} \lor \neg \left(r \leq 4.6 \cdot 10^{+123}\right) \land r \leq 5.2 \cdot 10^{+142}\right):\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= r 5.5e+23)
         (and (not (<= r 4.2e+63))
              (or (<= r 2e+95) (and (not (<= r 4.6e+123)) (<= r 5.2e+142)))))
   (+ -1.5 (/ 2.0 (* r r)))
   (* -0.25 (* r (* r (* w w))))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((r <= 5.5e+23) || (!(r <= 4.2e+63) && ((r <= 2e+95) || (!(r <= 4.6e+123) && (r <= 5.2e+142))))) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -0.25 * (r * (r * (w * w)));
	}
	return tmp;
}

Fortran

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 <= 5.5d+23) .or. (.not. (r <= 4.2d+63)) .and. (r <= 2d+95) .or. (.not. (r <= 4.6d+123)) .and. (r <= 5.2d+142)) then
        tmp = (-1.5d0) + (2.0d0 / (r * r))
    else
        tmp = (-0.25d0) * (r * (r * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((r <= 5.5e+23) || (!(r <= 4.2e+63) && ((r <= 2e+95) || (!(r <= 4.6e+123) && (r <= 5.2e+142))))) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -0.25 * (r * (r * (w * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (r <= 5.5e+23) or (not (r <= 4.2e+63) and ((r <= 2e+95) or (not (r <= 4.6e+123) and (r <= 5.2e+142)))):
		tmp = -1.5 + (2.0 / (r * r))
	else:
		tmp = -0.25 * (r * (r * (w * w)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((r <= 5.5e+23) || (!(r <= 4.2e+63) && ((r <= 2e+95) || (!(r <= 4.6e+123) && (r <= 5.2e+142)))))
		tmp = Float64(-1.5 + Float64(2.0 / Float64(r * r)));
	else
		tmp = Float64(-0.25 * Float64(r * Float64(r * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((r <= 5.5e+23) || (~((r <= 4.2e+63)) && ((r <= 2e+95) || (~((r <= 4.6e+123)) && (r <= 5.2e+142)))))
		tmp = -1.5 + (2.0 / (r * r));
	else
		tmp = -0.25 * (r * (r * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[r, 5.5e+23], And[N[Not[LessEqual[r, 4.2e+63]], $MachinePrecision], Or[LessEqual[r, 2e+95], And[N[Not[LessEqual[r, 4.6e+123]], $MachinePrecision], LessEqual[r, 5.2e+142]]]]], N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.50000000000000004e23 or 4.2000000000000004e63 < r < 2.00000000000000004e95 or 4.59999999999999981e123 < r < 5.20000000000000043e142

   1. Initial program 82.9%

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

      1. sub-neg 82.9%

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

      2. +-commutative 82.9%

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

      3. associate--l+ 82.9%

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

      4. associate-/l* 85.1%

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

      5. distribute-neg-frac 85.1%

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

      6. associate-/r/ 85.1%

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

      7. fma-def 85.1%

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

      8. sub-neg 85.1%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in r around 0 71.4%

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

      1. sub-neg 71.4%

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

      2. associate-*r/ 71.4%

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

      3. metadata-eval 71.4%

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

      4. unpow2 71.4%

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

      5. metadata-eval 71.4%

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

   6. Simplified 71.4%

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

   if 5.50000000000000004e23 < r < 4.2000000000000004e63 or 2.00000000000000004e95 < r < 4.59999999999999981e123 or 5.20000000000000043e142 < r

   1. Initial program 87.6%

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

      1. sub-neg 87.6%

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

      2. +-commutative 87.6%

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

      3. associate--l+ 87.6%

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

      4. associate-/l* 97.4%

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

      5. distribute-neg-frac 97.4%

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

      6. associate-/r/ 97.4%

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

      7. fma-def 97.4%

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

      8. sub-neg 97.4%

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

   3. Simplified 77.1%

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

   4. Taylor expanded in r around inf 64.4%

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

   5. Taylor expanded in v around inf 72.1%

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

      1. *-commutative 72.1%

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

      2. unpow2 72.1%

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

      3. *-commutative 72.1%

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

      4. unpow2 72.1%

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

      5. swap-sqr 85.3%

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

      6. unpow2 85.3%

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

      7. *-commutative 85.3%

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

   7. Simplified 85.3%

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

      1. unpow2 85.3%

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

      2. *-commutative 85.3%

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

      3. *-commutative 85.3%

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

      4. unswap-sqr 72.1%

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

      5. associate-*l* 82.5%

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

   9. Applied egg-rr 82.5%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.5 \cdot 10^{+23} \lor \neg \left(r \leq 4.2 \cdot 10^{+63}\right) \land \left(r \leq 2 \cdot 10^{+95} \lor \neg \left(r \leq 4.6 \cdot 10^{+123}\right) \land r \leq 5.2 \cdot 10^{+142}\right):\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 11: 66.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 6.5 \cdot 10^{+22} \lor \neg \left(r \leq 1.15 \cdot 10^{+71}\right) \land \left(r \leq 5.8 \cdot 10^{+98} \lor \neg \left(r \leq 4.8 \cdot 10^{+123}\right) \land r \leq 3.1 \cdot 10^{+142}\right):\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= r 6.5e+22)
         (and (not (<= r 1.15e+71))
              (or (<= r 5.8e+98) (and (not (<= r 4.8e+123)) (<= r 3.1e+142)))))
   (+ -1.5 (/ 2.0 (* r r)))
   (* -0.25 (* (* r w) (* r w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((r <= 6.5e+22) || (!(r <= 1.15e+71) && ((r <= 5.8e+98) || (!(r <= 4.8e+123) && (r <= 3.1e+142))))) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Fortran

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 <= 6.5d+22) .or. (.not. (r <= 1.15d+71)) .and. (r <= 5.8d+98) .or. (.not. (r <= 4.8d+123)) .and. (r <= 3.1d+142)) then
        tmp = (-1.5d0) + (2.0d0 / (r * r))
    else
        tmp = (-0.25d0) * ((r * w) * (r * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((r <= 6.5e+22) || (!(r <= 1.15e+71) && ((r <= 5.8e+98) || (!(r <= 4.8e+123) && (r <= 3.1e+142))))) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (r <= 6.5e+22) or (not (r <= 1.15e+71) and ((r <= 5.8e+98) or (not (r <= 4.8e+123) and (r <= 3.1e+142)))):
		tmp = -1.5 + (2.0 / (r * r))
	else:
		tmp = -0.25 * ((r * w) * (r * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((r <= 6.5e+22) || (!(r <= 1.15e+71) && ((r <= 5.8e+98) || (!(r <= 4.8e+123) && (r <= 3.1e+142)))))
		tmp = Float64(-1.5 + Float64(2.0 / Float64(r * r)));
	else
		tmp = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((r <= 6.5e+22) || (~((r <= 1.15e+71)) && ((r <= 5.8e+98) || (~((r <= 4.8e+123)) && (r <= 3.1e+142)))))
		tmp = -1.5 + (2.0 / (r * r));
	else
		tmp = -0.25 * ((r * w) * (r * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[r, 6.5e+22], And[N[Not[LessEqual[r, 1.15e+71]], $MachinePrecision], Or[LessEqual[r, 5.8e+98], And[N[Not[LessEqual[r, 4.8e+123]], $MachinePrecision], LessEqual[r, 3.1e+142]]]]], N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 6.49999999999999979e22 or 1.1500000000000001e71 < r < 5.8000000000000002e98 or 4.79999999999999978e123 < r < 3.0999999999999999e142

   1. Initial program 82.8%

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

      1. sub-neg 82.8%

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

      2. +-commutative 82.8%

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

      3. associate--l+ 82.8%

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

      4. associate-/l* 85.0%

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

      5. distribute-neg-frac 85.0%

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

      6. associate-/r/ 85.0%

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

      7. fma-def 85.0%

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

      8. sub-neg 85.0%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in r around 0 71.5%

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

      1. sub-neg 71.5%

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

      2. associate-*r/ 71.5%

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

      3. metadata-eval 71.5%

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

      4. unpow2 71.5%

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

      5. metadata-eval 71.5%

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

   6. Simplified 71.5%

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

   if 6.49999999999999979e22 < r < 1.1500000000000001e71 or 5.8000000000000002e98 < r < 4.79999999999999978e123 or 3.0999999999999999e142 < r

   1. Initial program 88.2%

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

      1. sub-neg 88.2%

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

      2. +-commutative 88.2%

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

      3. associate--l+ 88.2%

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

      4. associate-/l* 97.5%

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

      5. distribute-neg-frac 97.5%

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

      6. associate-/r/ 97.5%

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

      7. fma-def 97.5%

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

      8. sub-neg 97.5%

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

   3. Simplified 78.2%

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

   4. Taylor expanded in r around inf 63.8%

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

   5. Taylor expanded in v around inf 71.0%

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

      1. *-commutative 71.0%

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

      2. unpow2 71.0%

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

      3. *-commutative 71.0%

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

      4. unpow2 71.0%

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

      5. swap-sqr 83.6%

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

      6. unpow2 83.6%

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

      7. *-commutative 83.6%

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

   7. Simplified 83.6%

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

      1. unpow2 83.6%

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

      2. *-commutative 83.6%

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

      3. *-commutative 83.6%

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

   9. Applied egg-rr 83.6%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.5 \cdot 10^{+22} \lor \neg \left(r \leq 1.15 \cdot 10^{+71}\right) \land \left(r \leq 5.8 \cdot 10^{+98} \lor \neg \left(r \leq 4.8 \cdot 10^{+123}\right) \land r \leq 3.1 \cdot 10^{+142}\right):\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 12: 65.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1.5 + \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.18 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 3.6 \cdot 10^{+62}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;r \leq 1.9 \cdot 10^{+95} \lor \neg \left(r \leq 1.4 \cdot 10^{+123}\right) \land r \leq 3.1 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ -1.5 (/ 2.0 (* r r)))))
   (if (<= r 1.18e+24)
     t_0
     (if (<= r 3.6e+62)
       (* -0.25 (* (* r r) (* w w)))
       (if (or (<= r 1.9e+95) (and (not (<= r 1.4e+123)) (<= r 3.1e+142)))
         t_0
         (* -0.25 (* r (* r (* w w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double tmp;
	if (r <= 1.18e+24) {
		tmp = t_0;
	} else if (r <= 3.6e+62) {
		tmp = -0.25 * ((r * r) * (w * w));
	} else if ((r <= 1.9e+95) || (!(r <= 1.4e+123) && (r <= 3.1e+142))) {
		tmp = t_0;
	} else {
		tmp = -0.25 * (r * (r * (w * w)));
	}
	return tmp;
}

Fortran

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 <= 1.18d+24) then
        tmp = t_0
    else if (r <= 3.6d+62) then
        tmp = (-0.25d0) * ((r * r) * (w * w))
    else if ((r <= 1.9d+95) .or. (.not. (r <= 1.4d+123)) .and. (r <= 3.1d+142)) then
        tmp = t_0
    else
        tmp = (-0.25d0) * (r * (r * (w * w)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double tmp;
	if (r <= 1.18e+24) {
		tmp = t_0;
	} else if (r <= 3.6e+62) {
		tmp = -0.25 * ((r * r) * (w * w));
	} else if ((r <= 1.9e+95) || (!(r <= 1.4e+123) && (r <= 3.1e+142))) {
		tmp = t_0;
	} else {
		tmp = -0.25 * (r * (r * (w * w)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = -1.5 + (2.0 / (r * r))
	tmp = 0
	if r <= 1.18e+24:
		tmp = t_0
	elif r <= 3.6e+62:
		tmp = -0.25 * ((r * r) * (w * w))
	elif (r <= 1.9e+95) or (not (r <= 1.4e+123) and (r <= 3.1e+142)):
		tmp = t_0
	else:
		tmp = -0.25 * (r * (r * (w * w)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(-1.5 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (r <= 1.18e+24)
		tmp = t_0;
	elseif (r <= 3.6e+62)
		tmp = Float64(-0.25 * Float64(Float64(r * r) * Float64(w * w)));
	elseif ((r <= 1.9e+95) || (!(r <= 1.4e+123) && (r <= 3.1e+142)))
		tmp = t_0;
	else
		tmp = Float64(-0.25 * Float64(r * Float64(r * Float64(w * w))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = -1.5 + (2.0 / (r * r));
	tmp = 0.0;
	if (r <= 1.18e+24)
		tmp = t_0;
	elseif (r <= 3.6e+62)
		tmp = -0.25 * ((r * r) * (w * w));
	elseif ((r <= 1.9e+95) || (~((r <= 1.4e+123)) && (r <= 3.1e+142)))
		tmp = t_0;
	else
		tmp = -0.25 * (r * (r * (w * w)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.18e+24], t$95$0, If[LessEqual[r, 3.6e+62], N[(-0.25 * N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[r, 1.9e+95], And[N[Not[LessEqual[r, 1.4e+123]], $MachinePrecision], LessEqual[r, 3.1e+142]]], t$95$0, N[(-0.25 * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.17999999999999997e24 or 3.6e62 < r < 1.9e95 or 1.40000000000000006e123 < r < 3.0999999999999999e142

   1. Initial program 82.9%

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

      1. sub-neg 82.9%

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

      2. +-commutative 82.9%

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

      3. associate--l+ 82.9%

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

      4. associate-/l* 85.1%

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

      5. distribute-neg-frac 85.1%

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

      6. associate-/r/ 85.1%

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

      7. fma-def 85.1%

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

      8. sub-neg 85.1%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in r around 0 71.4%

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

      1. sub-neg 71.4%

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

      2. associate-*r/ 71.4%

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

      3. metadata-eval 71.4%

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

      4. unpow2 71.4%

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

      5. metadata-eval 71.4%

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

   6. Simplified 71.4%

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

   if 1.17999999999999997e24 < r < 3.6e62

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.2%

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

      4. associate-/l* 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. associate-/r/ 99.2%

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

      7. fma-def 99.2%

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

      8. sub-neg 99.2%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in r around inf 51.7%

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

   5. Taylor expanded in v around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. *-commutative 100.0%

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

      4. unpow2 100.0%

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

   7. Simplified 100.0%

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

   if 1.9e95 < r < 1.40000000000000006e123 or 3.0999999999999999e142 < r

   1. Initial program 87.0%

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

      1. sub-neg 87.0%

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

      2. +-commutative 87.0%

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

      3. associate--l+ 87.0%

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

      4. associate-/l* 97.3%

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

      5. distribute-neg-frac 97.3%

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

      6. associate-/r/ 97.3%

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

      7. fma-def 97.3%

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

      8. sub-neg 97.3%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in r around inf 65.1%

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

   5. Taylor expanded in v around inf 70.5%

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

      1. *-commutative 70.5%

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

      2. unpow2 70.5%

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

      3. *-commutative 70.5%

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

      4. unpow2 70.5%

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

      5. swap-sqr 84.4%

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

      6. unpow2 84.4%

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

      7. *-commutative 84.4%

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

   7. Simplified 84.4%

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

      1. unpow2 84.4%

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

      2. *-commutative 84.4%

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

      3. *-commutative 84.4%

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

      4. unswap-sqr 70.5%

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

      5. associate-*l* 81.6%

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

   9. Applied egg-rr 81.6%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.18 \cdot 10^{+24}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 3.6 \cdot 10^{+62}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;r \leq 1.9 \cdot 10^{+95} \lor \neg \left(r \leq 1.4 \cdot 10^{+123}\right) \land r \leq 3.1 \cdot 10^{+142}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 13: 66.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1.5 + \frac{2}{r \cdot r}\\ t_1 := -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{if}\;r \leq 1.9 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 1.25 \cdot 10^{+71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 6 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 4.8 \cdot 10^{+123}:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{elif}\;r \leq 3.1 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ -1.5 (/ 2.0 (* r r)))) (t_1 (* -0.25 (* (* r w) (* r w)))))
   (if (<= r 1.9e+23)
     t_0
     (if (<= r 1.25e+71)
       t_1
       (if (<= r 6e+95)
         t_0
         (if (<= r 4.8e+123)
           (* -0.25 (* w (* w (* r r))))
           (if (<= r 3.1e+142) t_0 t_1)))))))

C

double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double t_1 = -0.25 * ((r * w) * (r * w));
	double tmp;
	if (r <= 1.9e+23) {
		tmp = t_0;
	} else if (r <= 1.25e+71) {
		tmp = t_1;
	} else if (r <= 6e+95) {
		tmp = t_0;
	} else if (r <= 4.8e+123) {
		tmp = -0.25 * (w * (w * (r * r)));
	} else if (r <= 3.1e+142) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

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) :: t_1
    real(8) :: tmp
    t_0 = (-1.5d0) + (2.0d0 / (r * r))
    t_1 = (-0.25d0) * ((r * w) * (r * w))
    if (r <= 1.9d+23) then
        tmp = t_0
    else if (r <= 1.25d+71) then
        tmp = t_1
    else if (r <= 6d+95) then
        tmp = t_0
    else if (r <= 4.8d+123) then
        tmp = (-0.25d0) * (w * (w * (r * r)))
    else if (r <= 3.1d+142) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double t_1 = -0.25 * ((r * w) * (r * w));
	double tmp;
	if (r <= 1.9e+23) {
		tmp = t_0;
	} else if (r <= 1.25e+71) {
		tmp = t_1;
	} else if (r <= 6e+95) {
		tmp = t_0;
	} else if (r <= 4.8e+123) {
		tmp = -0.25 * (w * (w * (r * r)));
	} else if (r <= 3.1e+142) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = -1.5 + (2.0 / (r * r))
	t_1 = -0.25 * ((r * w) * (r * w))
	tmp = 0
	if r <= 1.9e+23:
		tmp = t_0
	elif r <= 1.25e+71:
		tmp = t_1
	elif r <= 6e+95:
		tmp = t_0
	elif r <= 4.8e+123:
		tmp = -0.25 * (w * (w * (r * r)))
	elif r <= 3.1e+142:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(-1.5 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)))
	tmp = 0.0
	if (r <= 1.9e+23)
		tmp = t_0;
	elseif (r <= 1.25e+71)
		tmp = t_1;
	elseif (r <= 6e+95)
		tmp = t_0;
	elseif (r <= 4.8e+123)
		tmp = Float64(-0.25 * Float64(w * Float64(w * Float64(r * r))));
	elseif (r <= 3.1e+142)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = -1.5 + (2.0 / (r * r));
	t_1 = -0.25 * ((r * w) * (r * w));
	tmp = 0.0;
	if (r <= 1.9e+23)
		tmp = t_0;
	elseif (r <= 1.25e+71)
		tmp = t_1;
	elseif (r <= 6e+95)
		tmp = t_0;
	elseif (r <= 4.8e+123)
		tmp = -0.25 * (w * (w * (r * r)));
	elseif (r <= 3.1e+142)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.9e+23], t$95$0, If[LessEqual[r, 1.25e+71], t$95$1, If[LessEqual[r, 6e+95], t$95$0, If[LessEqual[r, 4.8e+123], N[(-0.25 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 3.1e+142], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.89999999999999987e23 or 1.24999999999999993e71 < r < 5.99999999999999982e95 or 4.79999999999999978e123 < r < 3.0999999999999999e142

   1. Initial program 82.8%

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

      1. sub-neg 82.8%

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

      2. +-commutative 82.8%

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

      3. associate--l+ 82.8%

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

      4. associate-/l* 85.0%

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

      5. distribute-neg-frac 85.0%

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

      6. associate-/r/ 85.0%

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

      7. fma-def 85.0%

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

      8. sub-neg 85.0%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in r around 0 71.5%

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

      1. sub-neg 71.5%

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

      2. associate-*r/ 71.5%

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

      3. metadata-eval 71.5%

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

      4. unpow2 71.5%

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

      5. metadata-eval 71.5%

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

   6. Simplified 71.5%

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

   if 1.89999999999999987e23 < r < 1.24999999999999993e71 or 3.0999999999999999e142 < r

   1. Initial program 87.9%

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

      1. sub-neg 87.9%

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

      2. +-commutative 87.9%

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

      3. associate--l+ 87.9%

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

      4. associate-/l* 96.8%

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

      5. distribute-neg-frac 96.8%

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

      6. associate-/r/ 96.8%

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

      7. fma-def 96.8%

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

      8. sub-neg 96.8%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in r around inf 65.7%

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

   5. Taylor expanded in v around inf 68.8%

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

      1. *-commutative 68.8%

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

      2. unpow2 68.8%

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

      3. *-commutative 68.8%

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

      4. unpow2 68.8%

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

      5. swap-sqr 85.0%

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

      6. unpow2 85.0%

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

      7. *-commutative 85.0%

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

   7. Simplified 85.0%

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

      1. unpow2 85.0%

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

      2. *-commutative 85.0%

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

      3. *-commutative 85.0%

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

   9. Applied egg-rr 85.0%

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

   if 5.99999999999999982e95 < r < 4.79999999999999978e123

   1. Initial program 89.5%

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

      1. sub-neg 89.5%

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

      2. +-commutative 89.5%

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

      3. associate--l+ 89.5%

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

      4. associate-/l* 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 99.8%

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

   4. Taylor expanded in r around inf 57.3%

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

   5. Taylor expanded in v around inf 78.7%

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

      1. *-commutative 78.7%

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

      2. unpow2 78.7%

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

      3. *-commutative 78.7%

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

      4. unpow2 78.7%

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

      5. swap-sqr 78.6%

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

      6. unpow2 78.6%

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

      7. *-commutative 78.6%

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

   7. Simplified 78.6%

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

      1. unpow2 78.6%

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

      2. *-commutative 78.6%

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

      3. *-commutative 78.6%

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

      4. unswap-sqr 78.7%

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

      5. associate-*r* 79.0%

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

   9. Applied egg-rr 79.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.9 \cdot 10^{+23}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 1.25 \cdot 10^{+71}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{elif}\;r \leq 6 \cdot 10^{+95}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 4.8 \cdot 10^{+123}:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{elif}\;r \leq 3.1 \cdot 10^{+142}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 14: 65.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1.5 + \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 7.6 \cdot 10^{+23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 3.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)}{1 - v}\\ \mathbf{elif}\;r \leq 3 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 4.2 \cdot 10^{+123}:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{elif}\;r \leq 9.2 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (+ -1.5 (/ 2.0 (* r r)))))
   (if (<= r 7.6e+23)
     t_0
     (if (<= r 3.8e+58)
       (/ (* -0.375 (* (* r r) (* w w))) (- 1.0 v))
       (if (<= r 3e+95)
         t_0
         (if (<= r 4.2e+123)
           (* -0.25 (* w (* w (* r r))))
           (if (<= r 9.2e+142) t_0 (* -0.25 (* (* r w) (* r w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double tmp;
	if (r <= 7.6e+23) {
		tmp = t_0;
	} else if (r <= 3.8e+58) {
		tmp = (-0.375 * ((r * r) * (w * w))) / (1.0 - v);
	} else if (r <= 3e+95) {
		tmp = t_0;
	} else if (r <= 4.2e+123) {
		tmp = -0.25 * (w * (w * (r * r)));
	} else if (r <= 9.2e+142) {
		tmp = t_0;
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Fortran

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 <= 7.6d+23) then
        tmp = t_0
    else if (r <= 3.8d+58) then
        tmp = ((-0.375d0) * ((r * r) * (w * w))) / (1.0d0 - v)
    else if (r <= 3d+95) then
        tmp = t_0
    else if (r <= 4.2d+123) then
        tmp = (-0.25d0) * (w * (w * (r * r)))
    else if (r <= 9.2d+142) then
        tmp = t_0
    else
        tmp = (-0.25d0) * ((r * w) * (r * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = -1.5 + (2.0 / (r * r));
	double tmp;
	if (r <= 7.6e+23) {
		tmp = t_0;
	} else if (r <= 3.8e+58) {
		tmp = (-0.375 * ((r * r) * (w * w))) / (1.0 - v);
	} else if (r <= 3e+95) {
		tmp = t_0;
	} else if (r <= 4.2e+123) {
		tmp = -0.25 * (w * (w * (r * r)));
	} else if (r <= 9.2e+142) {
		tmp = t_0;
	} else {
		tmp = -0.25 * ((r * w) * (r * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = -1.5 + (2.0 / (r * r))
	tmp = 0
	if r <= 7.6e+23:
		tmp = t_0
	elif r <= 3.8e+58:
		tmp = (-0.375 * ((r * r) * (w * w))) / (1.0 - v)
	elif r <= 3e+95:
		tmp = t_0
	elif r <= 4.2e+123:
		tmp = -0.25 * (w * (w * (r * r)))
	elif r <= 9.2e+142:
		tmp = t_0
	else:
		tmp = -0.25 * ((r * w) * (r * w))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(-1.5 + Float64(2.0 / Float64(r * r)))
	tmp = 0.0
	if (r <= 7.6e+23)
		tmp = t_0;
	elseif (r <= 3.8e+58)
		tmp = Float64(Float64(-0.375 * Float64(Float64(r * r) * Float64(w * w))) / Float64(1.0 - v));
	elseif (r <= 3e+95)
		tmp = t_0;
	elseif (r <= 4.2e+123)
		tmp = Float64(-0.25 * Float64(w * Float64(w * Float64(r * r))));
	elseif (r <= 9.2e+142)
		tmp = t_0;
	else
		tmp = Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = -1.5 + (2.0 / (r * r));
	tmp = 0.0;
	if (r <= 7.6e+23)
		tmp = t_0;
	elseif (r <= 3.8e+58)
		tmp = (-0.375 * ((r * r) * (w * w))) / (1.0 - v);
	elseif (r <= 3e+95)
		tmp = t_0;
	elseif (r <= 4.2e+123)
		tmp = -0.25 * (w * (w * (r * r)));
	elseif (r <= 9.2e+142)
		tmp = t_0;
	else
		tmp = -0.25 * ((r * w) * (r * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 7.6e+23], t$95$0, If[LessEqual[r, 3.8e+58], N[(N[(-0.375 * N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 3e+95], t$95$0, If[LessEqual[r, 4.2e+123], N[(-0.25 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 9.2e+142], t$95$0, N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if r < 7.5999999999999995e23 or 3.7999999999999999e58 < r < 2.99999999999999991e95 or 4.19999999999999988e123 < r < 9.20000000000000009e142

   1. Initial program 82.9%

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

      1. sub-neg 82.9%

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

      2. +-commutative 82.9%

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

      3. associate--l+ 82.9%

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

      4. associate-/l* 85.1%

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

      5. distribute-neg-frac 85.1%

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

      6. associate-/r/ 85.1%

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

      7. fma-def 85.1%

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

      8. sub-neg 85.1%

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

   3. Simplified 80.8%

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

   4. Taylor expanded in r around 0 71.4%

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

      1. sub-neg 71.4%

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

      2. associate-*r/ 71.4%

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

      3. metadata-eval 71.4%

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

      4. unpow2 71.4%

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

      5. metadata-eval 71.4%

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

   6. Simplified 71.4%

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

   if 7.5999999999999995e23 < r < 3.7999999999999999e58

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.2%

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

      4. associate-/l* 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. associate-/r/ 99.2%

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

      7. fma-def 99.2%

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

      8. sub-neg 99.2%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in r around inf 51.7%

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

   5. Taylor expanded in v around 0 51.1%

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

      1. *-commutative 51.1%

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

      2. unpow2 51.1%

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

      3. *-commutative 51.1%

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

      4. unpow2 51.1%

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

   7. Simplified 51.1%

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

   if 2.99999999999999991e95 < r < 4.19999999999999988e123

   1. Initial program 89.5%

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

      1. sub-neg 89.5%

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

      2. +-commutative 89.5%

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

      3. associate--l+ 89.5%

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

      4. associate-/l* 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 99.8%

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

   4. Taylor expanded in r around inf 57.3%

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

   5. Taylor expanded in v around inf 78.7%

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

      1. *-commutative 78.7%

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

      2. unpow2 78.7%

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

      3. *-commutative 78.7%

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

      4. unpow2 78.7%

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

      5. swap-sqr 78.6%

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

      6. unpow2 78.6%

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

      7. *-commutative 78.6%

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

   7. Simplified 78.6%

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

      1. unpow2 78.6%

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

      2. *-commutative 78.6%

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

      3. *-commutative 78.6%

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

      4. unswap-sqr 78.7%

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

      5. associate-*r* 79.0%

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

   9. Applied egg-rr 79.0%

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

   if 9.20000000000000009e142 < r

   1. Initial program 86.1%

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

      1. sub-neg 86.1%

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

      2. +-commutative 86.1%

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

      3. associate--l+ 86.1%

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

      4. associate-/l* 96.4%

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

      5. distribute-neg-frac 96.4%

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

      6. associate-/r/ 96.4%

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

      7. fma-def 96.4%

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

      8. sub-neg 96.4%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in r around inf 67.8%

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

   5. Taylor expanded in v around inf 67.8%

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

      1. *-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. *-commutative 67.8%

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

      4. unpow2 67.8%

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

      5. swap-sqr 86.4%

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

      6. unpow2 86.4%

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

      7. *-commutative 86.4%

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

   7. Simplified 86.4%

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

      1. unpow2 86.4%

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

      2. *-commutative 86.4%

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

      3. *-commutative 86.4%

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

   9. Applied egg-rr 86.4%

      \[\leadsto \color{blue}{\left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)} \cdot -0.25 \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.6 \cdot 10^{+23}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 3.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)}{1 - v}\\ \mathbf{elif}\;r \leq 3 \cdot 10^{+95}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 4.2 \cdot 10^{+123}:\\ \;\;\;\;-0.25 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \mathbf{elif}\;r \leq 9.2 \cdot 10^{+142}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 15: 47.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -4.5 \end{array} \]

FPCore

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

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + -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 = (2.0d0 / (r * r)) + (-4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.6%

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

   1. sub-neg 83.6%

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

   2. associate-/l* 86.9%

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

   3. cancel-sign-sub-inv 86.9%

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

   4. metadata-eval 86.9%

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

   5. *-commutative 86.9%

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

   6. *-commutative 86.9%

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

   7. metadata-eval 86.9%

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

3. Simplified 86.9%

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

4. Taylor expanded in v around 0 69.6%

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

   1. unpow2 69.6%

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

   2. unpow2 69.6%

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

6. Simplified 69.6%

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

   1. add-sqr-sqrt 69.6%

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

   2. sqrt-div 69.6%

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

   3. metadata-eval 69.6%

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

   4. unswap-sqr 69.6%

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

   5. *-commutative 69.6%

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

   6. *-commutative 69.6%

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

   7. sqrt-prod 39.1%

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

   8. add-sqr-sqrt 59.9%

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

   9. sqrt-div 59.9%

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

   10. metadata-eval 59.9%

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

   11. unswap-sqr 70.6%

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

   12. *-commutative 70.6%

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

   13. *-commutative 70.6%

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

   14. sqrt-prod 44.4%

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

   15. add-sqr-sqrt 81.9%

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

8. Applied egg-rr 81.9%

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

   1. un-div-inv 81.9%

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

10. Applied egg-rr 81.9%

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

11. Taylor expanded in r around 0 50.9%

    \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + -4.5 \]
12. Step-by-step derivation

    1. unpow2 50.9%

       \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + -4.5 \]

13. Simplified 50.9%

    \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + -4.5 \]

14. Final simplification 50.9%

    \[\leadsto \frac{2}{r \cdot r} + -4.5 \]

Alternative 16: 57.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ -1.5 + \frac{2}{r \cdot r} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (+ -1.5 (/ 2.0 (* r r))))

C

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

Fortran

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

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

Python

def code(v, w, r):
	return -1.5 + (2.0 / (r * r))

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.6%

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

   1. sub-neg 83.6%

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

   2. +-commutative 83.6%

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

   3. associate--l+ 83.6%

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

   4. associate-/l* 86.9%

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

   5. distribute-neg-frac 86.9%

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

   6. associate-/r/ 86.9%

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

   7. fma-def 86.9%

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

   8. sub-neg 86.9%

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

3. Simplified 80.3%

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

4. Taylor expanded in r around 0 63.2%

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

   1. sub-neg 63.2%

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

   2. associate-*r/ 63.2%

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

   3. metadata-eval 63.2%

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

   4. unpow2 63.2%

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

   5. metadata-eval 63.2%

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

6. Simplified 63.2%

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

7. Final simplification 63.2%

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

Alternative 17: 44.6% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (/ 2.0 (* r r)))

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = 2.0d0 / (r * r)
end function

Java

public static double code(double v, double w, double r) {
	return 2.0 / (r * r);
}

Python

def code(v, w, r):
	return 2.0 / (r * r)

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.6%

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

   1. sub-neg 83.6%

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

   2. associate-/l* 86.9%

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

   3. cancel-sign-sub-inv 86.9%

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

   4. metadata-eval 86.9%

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

   5. *-commutative 86.9%

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

   6. *-commutative 86.9%

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

   7. metadata-eval 86.9%

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

3. Simplified 86.9%

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

4. Taylor expanded in v around 0 76.7%

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

   1. *-commutative 76.7%

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

   2. *-commutative 76.7%

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

   3. unpow2 76.7%

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

   4. unpow2 76.7%

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

   5. swap-sqr 93.2%

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

   6. unpow2 93.2%

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

   7. *-commutative 93.2%

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

6. Simplified 93.2%

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

7. Taylor expanded in r around 0 47.8%

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

   1. unpow2 47.8%

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

9. Simplified 47.8%

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

10. Final simplification 47.8%

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

Reproduce

herbie shell --seed 2023200 
(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))