Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.8%

Time: 11.3s

Alternatives: 11

Speedup: 1.4×

• 53.3% 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: 85.0% 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: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375 + -0.25 \cdot v}{\frac{1 - v}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}}\right) + -4.5 \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (+ 0.375 (* -0.25 v)) (/ (- 1.0 v) (* (* r w) (* r w)))))
  -4.5))

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.375 + (-0.25 * v)) / ((1.0 - v) / ((r * w) * (r * w))))) + -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))) - ((0.375d0 + ((-0.25d0) * v)) / ((1.0d0 - v) / ((r * w) * (r * w))))) + (-4.5d0)
end function

Java

public static double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.375 + (-0.25 * v)) / ((1.0 - v) / ((r * w) * (r * w))))) + -4.5;
}

Python

def code(v, w, r):
	return ((3.0 + (2.0 / (r * r))) - ((0.375 + (-0.25 * v)) / ((1.0 - v) / ((r * w) * (r * w))))) + -4.5

Julia

function code(v, w, r)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(0.375 + Float64(-0.25 * v)) / Float64(Float64(1.0 - v) / Float64(Float64(r * w) * Float64(r * w))))) + -4.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = ((3.0 + (2.0 / (r * r))) - ((0.375 + (-0.25 * v)) / ((1.0 - v) / ((r * w) * (r * w))))) + -4.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

3. Simplified 79.5%

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

   1. *-un-lft-identity 79.5%

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

   2. add-sqr-sqrt 79.5%

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

   3. times-frac 79.5%

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

   4. unswap-sqr 79.5%

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

   5. sqrt-prod 46.6%

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

   6. add-sqr-sqrt 64.6%

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

   7. unswap-sqr 80.3%

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

   8. sqrt-prod 56.5%

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

   9. add-sqr-sqrt 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in v around 0 92.0%

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

   1. +-commutative 92.0%

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

   2. *-commutative 92.0%

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

   3. mul-1-neg 92.0%

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

   4. sub-neg 92.0%

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

   5. *-commutative 92.0%

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

   6. div-sub 99.8%

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

   7. associate-/r* 99.0%

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

8. Simplified 99.0%

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

   1. div-inv 99.0%

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

   2. distribute-lft-in 99.0%

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

   3. metadata-eval 99.0%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(\color{blue}{0.375} + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \frac{1}{\frac{1}{w \cdot r} \cdot \frac{\frac{1 - v}{r}}{w}}\right) + -4.5 \]

   4. frac-times 95.2%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.375 + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \frac{1}{\color{blue}{\frac{1 \cdot \frac{1 - v}{r}}{\left(w \cdot r\right) \cdot w}}}\right) + -4.5 \]

   5. *-un-lft-identity 95.2%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.375 + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \frac{1}{\frac{\color{blue}{\frac{1 - v}{r}}}{\left(w \cdot r\right) \cdot w}}\right) + -4.5 \]

   6. *-commutative 95.2%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(0.375 + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \frac{1}{\frac{\frac{1 - v}{r}}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) + -4.5 \]

10. Applied egg-rr 95.2%

    \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.375 + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot \frac{1}{\frac{\frac{1 - v}{r}}{w \cdot \left(w \cdot r\right)}}}\right) + -4.5 \]
11. Step-by-step derivation

    1. associate-*r/ 95.2%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(0.375 + 0.125 \cdot \left(-2 \cdot v\right)\right) \cdot 1}{\frac{\frac{1 - v}{r}}{w \cdot \left(w \cdot r\right)}}}\right) + -4.5 \]

    2. associate-*r* 95.2%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.375 + \color{blue}{\left(0.125 \cdot -2\right) \cdot v}\right) \cdot 1}{\frac{\frac{1 - v}{r}}{w \cdot \left(w \cdot r\right)}}\right) + -4.5 \]

    3. metadata-eval 95.2%

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

    4. associate-/r* 96.0%

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

    5. associate-*r* 99.8%

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

    6. *-commutative 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{+240}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + -0.25 \cdot v}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 1e+240)
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* -0.25 v)) (- 1.0 v)))))
     (+ t_0 (- -1.5 (/ (* (* r w) (* r w)) 2.6666666666666665))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e+240) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (-0.25 * v)) / (1.0 - v))));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	}
	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+240) then
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * ((0.375d0 + ((-0.25d0) * v)) / (1.0d0 - v))))
    else
        tmp = t_0 + ((-1.5d0) - (((r * w) * (r * w)) / 2.6666666666666665d0))
    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+240) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (-0.25 * v)) / (1.0 - v))));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 1e+240:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (-0.25 * v)) / (1.0 - v))))
	else:
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 1e+240)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(-0.25 * v)) / Float64(1.0 - v)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) / 2.6666666666666665)));
	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+240)
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (-0.25 * v)) / (1.0 - v))));
	else
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	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+240], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(-0.25 * v), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.00000000000000001e240

   1. Initial program 92.2%

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

      1. associate--l- 92.2%

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

      2. +-commutative 92.2%

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

      3. associate--l+ 92.3%

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

      4. +-commutative 92.3%

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

      5. associate--r+ 92.3%

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

      6. metadata-eval 92.3%

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

      7. associate-*l/ 95.3%

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

      8. *-commutative 95.3%

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

      9. *-commutative 95.3%

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

      10. *-commutative 95.3%

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

   3. Simplified 95.3%

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

   4. Taylor expanded in r around 0 95.3%

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

      1. unpow2 95.3%

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

      2. *-commutative 95.3%

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

      3. associate-*l* 99.8%

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

      4. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if 1.00000000000000001e240 < (*.f64 w w)

   1. Initial program 66.5%

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

      1. associate--l- 66.5%

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

      2. +-commutative 66.5%

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

      3. associate--l+ 66.5%

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

      4. +-commutative 66.5%

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

      5. associate--r+ 66.5%

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

      6. metadata-eval 66.5%

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

      7. associate-*r* 63.9%

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

      8. *-commutative 63.9%

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

      9. associate-/l* 63.9%

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

      10. *-commutative 63.9%

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

   3. Simplified 63.9%

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

   4. Taylor expanded in v around 0 66.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 66.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   6. Simplified 66.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot \left(w \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 66.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right) \]

      2. *-commutative 66.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot r\right)}\right) \]

      3. associate-*l* 87.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   8. Applied egg-rr 87.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*l/ 87.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{2.6666666666666665}}\right) \]

   10. Applied egg-rr 87.3%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{2.6666666666666665}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 99.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(w \cdot r\right)}}{2.6666666666666665}\right) \]

       2. *-commutative 99.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)}{2.6666666666666665}\right) \]

       3. *-commutative 99.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)}{2.6666666666666665}\right) \]

       4. *-commutative 99.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}}{2.6666666666666665}\right) \]

   12. Simplified 99.0%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{+240}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + -0.25 \cdot v}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}\right)\\ \end{array} \]

Alternative 3: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.2 \cdot 10^{+26} \lor \neg \left(v \leq 2 \cdot 10^{-30}\right):\\ \;\;\;\;t_1 + \left(-1.5 - \frac{r}{4} \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (or (<= v -3.2e+26) (not (<= v 2e-30)))
     (+ t_1 (- -1.5 (* (/ r 4.0) t_0)))
     (+ t_1 (- -1.5 (* t_0 (/ r 2.6666666666666665)))))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -3.2e+26) || !(v <= 2e-30)) {
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	} else {
		tmp = t_1 + (-1.5 - (t_0 * (r / 2.6666666666666665)));
	}
	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 = w * (r * w)
    t_1 = 2.0d0 / (r * r)
    if ((v <= (-3.2d+26)) .or. (.not. (v <= 2d-30))) then
        tmp = t_1 + ((-1.5d0) - ((r / 4.0d0) * t_0))
    else
        tmp = t_1 + ((-1.5d0) - (t_0 * (r / 2.6666666666666665d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if ((v <= -3.2e+26) || !(v <= 2e-30)) {
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	} else {
		tmp = t_1 + (-1.5 - (t_0 * (r / 2.6666666666666665)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if (v <= -3.2e+26) or not (v <= 2e-30):
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0))
	else:
		tmp = t_1 + (-1.5 - (t_0 * (r / 2.6666666666666665)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -3.2e+26) || !(v <= 2e-30))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(r / 4.0) * t_0)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * Float64(r / 2.6666666666666665))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -3.2e+26) || ~((v <= 2e-30)))
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	else
		tmp = t_1 + (-1.5 - (t_0 * (r / 2.6666666666666665)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -3.2e+26], N[Not[LessEqual[v, 2e-30]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 - N[(N[(r / 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * N[(r / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.20000000000000029e26 or 2e-30 < v

   1. Initial program 82.2%

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

      1. associate--l- 82.2%

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

      2. +-commutative 82.2%

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

      3. associate--l+ 82.2%

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

      4. +-commutative 82.2%

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

      5. associate--r+ 82.2%

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

      6. metadata-eval 82.2%

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

      7. associate-*r* 80.5%

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

      8. *-commutative 80.5%

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

      9. associate-/l* 82.8%

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

      10. *-commutative 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in v around inf 86.7%

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

      1. unpow2 86.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 86.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 95.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 95.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 95.7%

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

      1. associate-/r/ 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. *-commutative 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \]

   8. Applied egg-rr 95.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   if -3.20000000000000029e26 < v < 2e-30

   1. Initial program 86.6%

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

      1. associate--l- 86.6%

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

      2. +-commutative 86.6%

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

      3. associate--l+ 86.6%

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

      4. +-commutative 86.6%

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

      5. associate--r+ 86.6%

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

      6. metadata-eval 86.6%

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

      7. associate-*r* 86.6%

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

      8. *-commutative 86.6%

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

      9. associate-/l* 86.6%

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

      10. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in v around 0 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   6. Simplified 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot \left(w \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right) \]

      2. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot r\right)}\right) \]

      3. associate-*l* 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   8. Applied egg-rr 95.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.2 \cdot 10^{+26} \lor \neg \left(v \leq 2 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \]

Alternative 4: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -5 \cdot 10^{+26} \lor \neg \left(v \leq 3.6 \cdot 10^{-26}\right):\\ \;\;\;\;\left(t_0 + -1.5\right) - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= v -5e+26) (not (<= v 3.6e-26)))
     (- (+ t_0 -1.5) (* (* r w) (/ r (/ 4.0 w))))
     (+ t_0 (- -1.5 (/ (* (* r w) (* r w)) 2.6666666666666665))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((v <= -5e+26) || !(v <= 3.6e-26)) {
		tmp = (t_0 + -1.5) - ((r * w) * (r / (4.0 / w)));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	}
	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 ((v <= (-5d+26)) .or. (.not. (v <= 3.6d-26))) then
        tmp = (t_0 + (-1.5d0)) - ((r * w) * (r / (4.0d0 / w)))
    else
        tmp = t_0 + ((-1.5d0) - (((r * w) * (r * w)) / 2.6666666666666665d0))
    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 ((v <= -5e+26) || !(v <= 3.6e-26)) {
		tmp = (t_0 + -1.5) - ((r * w) * (r / (4.0 / w)));
	} else {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (v <= -5e+26) or not (v <= 3.6e-26):
		tmp = (t_0 + -1.5) - ((r * w) * (r / (4.0 / w)))
	else:
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((v <= -5e+26) || !(v <= 3.6e-26))
		tmp = Float64(Float64(t_0 + -1.5) - Float64(Float64(r * w) * Float64(r / Float64(4.0 / w))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) / 2.6666666666666665)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((v <= -5e+26) || ~((v <= 3.6e-26)))
		tmp = (t_0 + -1.5) - ((r * w) * (r / (4.0 / w)));
	else
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[v, -5e+26], N[Not[LessEqual[v, 3.6e-26]], $MachinePrecision]], N[(N[(t$95$0 + -1.5), $MachinePrecision] - N[(N[(r * w), $MachinePrecision] * N[(r / N[(4.0 / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -5.0000000000000001e26 or 3.6000000000000001e-26 < v

   1. Initial program 82.2%

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

      1. associate--l- 82.2%

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

      2. +-commutative 82.2%

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

      3. associate--l+ 82.2%

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

      4. +-commutative 82.2%

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

      5. associate--r+ 82.2%

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

      6. metadata-eval 82.2%

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

      7. associate-*r* 80.5%

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

      8. *-commutative 80.5%

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

      9. associate-/l* 82.8%

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

      10. *-commutative 82.8%

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

   3. Simplified 82.8%

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

   4. Taylor expanded in v around inf 86.7%

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

      1. unpow2 86.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 86.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 95.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 95.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 95.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

   7. Taylor expanded in w around 0 86.7%

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

      1. *-commutative 86.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

      2. unpow2 86.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      3. associate-*r* 95.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. associate-/r* 97.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      5. *-commutative 97.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

   9. Simplified 97.4%

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

       1. associate-+r- 97.4%

          \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r}{\frac{\frac{4}{w}}{r \cdot w}}} \]

       2. associate-/r/ 99.6%

          \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\frac{r}{\frac{4}{w}} \cdot \left(r \cdot w\right)} \]

       3. *-commutative 99.6%

          \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r}{\frac{4}{w}} \cdot \color{blue}{\left(w \cdot r\right)} \]

   11. Applied egg-rr 99.6%

       \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + -1.5\right) - \frac{r}{\frac{4}{w}} \cdot \left(w \cdot r\right)} \]

   if -5.0000000000000001e26 < v < 3.6000000000000001e-26

   1. Initial program 86.6%

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

      1. associate--l- 86.6%

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

      2. +-commutative 86.6%

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

      3. associate--l+ 86.6%

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

      4. +-commutative 86.6%

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

      5. associate--r+ 86.6%

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

      6. metadata-eval 86.6%

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

      7. associate-*r* 86.6%

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

      8. *-commutative 86.6%

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

      9. associate-/l* 86.6%

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

      10. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in v around 0 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   6. Simplified 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot \left(w \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right) \]

      2. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot r\right)}\right) \]

      3. associate-*l* 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   8. Applied egg-rr 95.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*l/ 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{2.6666666666666665}}\right) \]

   10. Applied egg-rr 95.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{2.6666666666666665}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 99.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(w \cdot r\right)}}{2.6666666666666665}\right) \]

       2. *-commutative 99.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)}{2.6666666666666665}\right) \]

       3. *-commutative 99.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)}{2.6666666666666665}\right) \]

       4. *-commutative 99.3%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}}{2.6666666666666665}\right) \]

   12. Simplified 99.3%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -5 \cdot 10^{+26} \lor \neg \left(v \leq 3.6 \cdot 10^{-26}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(r \cdot w\right) \cdot \frac{r}{\frac{4}{w}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}\right)\\ \end{array} \]

Alternative 5: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2 \cdot 10^{+27}:\\ \;\;\;\;t_1 + \left(-1.5 - \frac{r}{4} \cdot t_0\right)\\ \mathbf{elif}\;v \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot \frac{r}{2.6666666666666665}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (<= v -2e+27)
     (+ t_1 (- -1.5 (* (/ r 4.0) t_0)))
     (if (<= v 3.6e-26)
       (+ t_1 (- -1.5 (* t_0 (/ r 2.6666666666666665))))
       (+ t_1 (- -1.5 (/ r (/ (/ 4.0 w) (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -2e+27) {
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	} else if (v <= 3.6e-26) {
		tmp = t_1 + (-1.5 - (t_0 * (r / 2.6666666666666665)));
	} else {
		tmp = t_1 + (-1.5 - (r / ((4.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = w * (r * w)
    t_1 = 2.0d0 / (r * r)
    if (v <= (-2d+27)) then
        tmp = t_1 + ((-1.5d0) - ((r / 4.0d0) * t_0))
    else if (v <= 3.6d-26) then
        tmp = t_1 + ((-1.5d0) - (t_0 * (r / 2.6666666666666665d0)))
    else
        tmp = t_1 + ((-1.5d0) - (r / ((4.0d0 / w) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -2e+27) {
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	} else if (v <= 3.6e-26) {
		tmp = t_1 + (-1.5 - (t_0 * (r / 2.6666666666666665)));
	} else {
		tmp = t_1 + (-1.5 - (r / ((4.0 / w) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= -2e+27:
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0))
	elif v <= 3.6e-26:
		tmp = t_1 + (-1.5 - (t_0 * (r / 2.6666666666666665)))
	else:
		tmp = t_1 + (-1.5 - (r / ((4.0 / w) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -2e+27)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(r / 4.0) * t_0)));
	elseif (v <= 3.6e-26)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * Float64(r / 2.6666666666666665))));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(r / Float64(Float64(4.0 / w) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -2e+27)
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	elseif (v <= 3.6e-26)
		tmp = t_1 + (-1.5 - (t_0 * (r / 2.6666666666666665)));
	else
		tmp = t_1 + (-1.5 - (r / ((4.0 / w) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2e+27], N[(t$95$1 + N[(-1.5 - N[(N[(r / 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 3.6e-26], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * N[(r / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(r / N[(N[(4.0 / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2e27

   1. Initial program 82.0%

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

      1. associate--l- 82.0%

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

      2. +-commutative 82.0%

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

      3. associate--l+ 82.0%

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

      4. +-commutative 82.0%

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

      5. associate--r+ 82.0%

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

      6. metadata-eval 82.0%

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

      7. associate-*r* 80.2%

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

      8. *-commutative 80.2%

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

      9. associate-/l* 83.5%

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

      10. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in v around inf 87.1%

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

      1. unpow2 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 98.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 98.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 98.0%

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

      1. associate-/r/ 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. *-commutative 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \]

   8. Applied egg-rr 98.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   if -2e27 < v < 3.6000000000000001e-26

   1. Initial program 86.6%

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

      1. associate--l- 86.6%

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

      2. +-commutative 86.6%

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

      3. associate--l+ 86.6%

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

      4. +-commutative 86.6%

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

      5. associate--r+ 86.6%

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

      6. metadata-eval 86.6%

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

      7. associate-*r* 86.6%

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

      8. *-commutative 86.6%

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

      9. associate-/l* 86.6%

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

      10. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in v around 0 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   6. Simplified 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot \left(w \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right) \]

      2. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot r\right)}\right) \]

      3. associate-*l* 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   8. Applied egg-rr 95.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   if 3.6000000000000001e-26 < v

   1. Initial program 82.4%

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

      1. associate--l- 82.4%

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

      2. +-commutative 82.4%

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

      3. associate--l+ 82.4%

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

      4. +-commutative 82.4%

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

      5. associate--r+ 82.4%

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

      6. metadata-eval 82.4%

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

      7. associate-*r* 80.8%

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

      8. *-commutative 80.8%

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

      9. associate-/l* 82.2%

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

      10. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in v around inf 86.5%

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

      1. unpow2 86.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 86.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 93.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 93.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 93.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

   7. Taylor expanded in w around 0 86.5%

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

      1. *-commutative 86.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

      2. unpow2 86.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      3. associate-*r* 93.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. associate-/r* 96.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      5. *-commutative 96.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

   9. Simplified 96.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\\ \end{array} \]

Alternative 6: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.2 \cdot 10^{+26}:\\ \;\;\;\;t_1 + \left(-1.5 - \frac{r}{4} \cdot t_0\right)\\ \mathbf{elif}\;v \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;t_1 + \left(-1.5 - \frac{r \cdot t_0}{2.6666666666666665}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (<= v -3.2e+26)
     (+ t_1 (- -1.5 (* (/ r 4.0) t_0)))
     (if (<= v 3.6e-26)
       (+ t_1 (- -1.5 (/ (* r t_0) 2.6666666666666665)))
       (+ t_1 (- -1.5 (/ r (/ (/ 4.0 w) (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -3.2e+26) {
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	} else if (v <= 3.6e-26) {
		tmp = t_1 + (-1.5 - ((r * t_0) / 2.6666666666666665));
	} else {
		tmp = t_1 + (-1.5 - (r / ((4.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = w * (r * w)
    t_1 = 2.0d0 / (r * r)
    if (v <= (-3.2d+26)) then
        tmp = t_1 + ((-1.5d0) - ((r / 4.0d0) * t_0))
    else if (v <= 3.6d-26) then
        tmp = t_1 + ((-1.5d0) - ((r * t_0) / 2.6666666666666665d0))
    else
        tmp = t_1 + ((-1.5d0) - (r / ((4.0d0 / w) / (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -3.2e+26) {
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	} else if (v <= 3.6e-26) {
		tmp = t_1 + (-1.5 - ((r * t_0) / 2.6666666666666665));
	} else {
		tmp = t_1 + (-1.5 - (r / ((4.0 / w) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= -3.2e+26:
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0))
	elif v <= 3.6e-26:
		tmp = t_1 + (-1.5 - ((r * t_0) / 2.6666666666666665))
	else:
		tmp = t_1 + (-1.5 - (r / ((4.0 / w) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -3.2e+26)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(r / 4.0) * t_0)));
	elseif (v <= 3.6e-26)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(r * t_0) / 2.6666666666666665)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(r / Float64(Float64(4.0 / w) / Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -3.2e+26)
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	elseif (v <= 3.6e-26)
		tmp = t_1 + (-1.5 - ((r * t_0) / 2.6666666666666665));
	else
		tmp = t_1 + (-1.5 - (r / ((4.0 / w) / (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.2e+26], N[(t$95$1 + N[(-1.5 - N[(N[(r / 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 3.6e-26], N[(t$95$1 + N[(-1.5 - N[(N[(r * t$95$0), $MachinePrecision] / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(r / N[(N[(4.0 / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.20000000000000029e26

   1. Initial program 82.0%

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

      1. associate--l- 82.0%

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

      2. +-commutative 82.0%

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

      3. associate--l+ 82.0%

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

      4. +-commutative 82.0%

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

      5. associate--r+ 82.0%

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

      6. metadata-eval 82.0%

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

      7. associate-*r* 80.2%

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

      8. *-commutative 80.2%

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

      9. associate-/l* 83.5%

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

      10. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in v around inf 87.1%

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

      1. unpow2 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 98.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 98.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 98.0%

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

      1. associate-/r/ 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. *-commutative 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \]

   8. Applied egg-rr 98.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   if -3.20000000000000029e26 < v < 3.6000000000000001e-26

   1. Initial program 86.6%

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

      1. associate--l- 86.6%

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

      2. +-commutative 86.6%

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

      3. associate--l+ 86.6%

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

      4. +-commutative 86.6%

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

      5. associate--r+ 86.6%

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

      6. metadata-eval 86.6%

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

      7. associate-*r* 86.6%

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

      8. *-commutative 86.6%

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

      9. associate-/l* 86.6%

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

      10. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in v around 0 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   6. Simplified 86.6%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot \left(w \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right) \]

      2. *-commutative 86.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot r\right)}\right) \]

      3. associate-*l* 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   8. Applied egg-rr 95.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*l/ 95.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{2.6666666666666665}}\right) \]

   10. Applied egg-rr 95.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{2.6666666666666665}}\right) \]

   if 3.6000000000000001e-26 < v

   1. Initial program 82.4%

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

      1. associate--l- 82.4%

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

      2. +-commutative 82.4%

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

      3. associate--l+ 82.4%

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

      4. +-commutative 82.4%

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

      5. associate--r+ 82.4%

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

      6. metadata-eval 82.4%

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

      7. associate-*r* 80.8%

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

      8. *-commutative 80.8%

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

      9. associate-/l* 82.2%

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

      10. *-commutative 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in v around inf 86.5%

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

      1. unpow2 86.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 86.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 93.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 93.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 93.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

   7. Taylor expanded in w around 0 86.5%

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

      1. *-commutative 86.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

      2. unpow2 86.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      3. associate-*r* 93.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. associate-/r* 96.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      5. *-commutative 96.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

   9. Simplified 96.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 3.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{2.6666666666666665}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\\ \end{array} \]

Alternative 7: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.12 \cdot 10^{+27}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 1950000000:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -1.12e+27)
     (+ t_0 (- -1.5 (* (/ r 4.0) (* w (* r w)))))
     (if (<= v 1950000000.0)
       (+ t_0 (- -1.5 (/ (* (* r w) (* r w)) 2.6666666666666665)))
       (+ t_0 (- -1.5 (/ r (/ (/ 4.0 w) (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -1.12e+27) {
		tmp = t_0 + (-1.5 - ((r / 4.0) * (w * (r * w))));
	} else if (v <= 1950000000.0) {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	} else {
		tmp = t_0 + (-1.5 - (r / ((4.0 / 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 (v <= (-1.12d+27)) then
        tmp = t_0 + ((-1.5d0) - ((r / 4.0d0) * (w * (r * w))))
    else if (v <= 1950000000.0d0) then
        tmp = t_0 + ((-1.5d0) - (((r * w) * (r * w)) / 2.6666666666666665d0))
    else
        tmp = t_0 + ((-1.5d0) - (r / ((4.0d0 / 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 (v <= -1.12e+27) {
		tmp = t_0 + (-1.5 - ((r / 4.0) * (w * (r * w))));
	} else if (v <= 1950000000.0) {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	} else {
		tmp = t_0 + (-1.5 - (r / ((4.0 / w) / (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -1.12e+27:
		tmp = t_0 + (-1.5 - ((r / 4.0) * (w * (r * w))))
	elif v <= 1950000000.0:
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665))
	else:
		tmp = t_0 + (-1.5 - (r / ((4.0 / w) / (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -1.12e+27)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r / 4.0) * Float64(w * Float64(r * w)))));
	elseif (v <= 1950000000.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) / 2.6666666666666665)));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r / Float64(Float64(4.0 / 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 (v <= -1.12e+27)
		tmp = t_0 + (-1.5 - ((r / 4.0) * (w * (r * w))));
	elseif (v <= 1950000000.0)
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) / 2.6666666666666665));
	else
		tmp = t_0 + (-1.5 - (r / ((4.0 / 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[v, -1.12e+27], N[(t$95$0 + N[(-1.5 - N[(N[(r / 4.0), $MachinePrecision] * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1950000000.0], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(r / N[(N[(4.0 / w), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.12e27

   1. Initial program 82.0%

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

      1. associate--l- 82.0%

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

      2. +-commutative 82.0%

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

      3. associate--l+ 82.0%

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

      4. +-commutative 82.0%

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

      5. associate--r+ 82.0%

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

      6. metadata-eval 82.0%

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

      7. associate-*r* 80.2%

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

      8. *-commutative 80.2%

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

      9. associate-/l* 83.5%

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

      10. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in v around inf 87.1%

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

      1. unpow2 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 87.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 98.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 98.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 98.0%

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

      1. associate-/r/ 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)}\right) \]

      2. *-commutative 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \]

   8. Applied egg-rr 98.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{4} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   if -1.12e27 < v < 1.95e9

   1. Initial program 85.2%

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

      1. associate--l- 85.2%

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

      2. +-commutative 85.2%

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

      3. associate--l+ 85.2%

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

      4. +-commutative 85.2%

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

      5. associate--r+ 85.2%

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

      6. metadata-eval 85.2%

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

      7. associate-*r* 85.2%

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

      8. *-commutative 85.2%

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

      9. associate-/l* 85.2%

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

      10. *-commutative 85.2%

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

   3. Simplified 85.2%

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

   4. Taylor expanded in v around 0 85.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   6. Simplified 85.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot \left(w \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right) \]

      2. *-commutative 85.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot r\right)}\right) \]

      3. associate-*l* 93.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   8. Applied egg-rr 93.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
   9. Step-by-step derivation

      1. associate-*l/ 93.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{2.6666666666666665}}\right) \]

   10. Applied egg-rr 93.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{2.6666666666666665}}\right) \]
   11. Step-by-step derivation

       1. associate-*r* 98.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(r \cdot w\right) \cdot \left(w \cdot r\right)}}{2.6666666666666665}\right) \]

       2. *-commutative 98.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(w \cdot r\right)} \cdot \left(w \cdot r\right)}{2.6666666666666665}\right) \]

       3. *-commutative 98.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{\left(r \cdot w\right)} \cdot \left(w \cdot r\right)}{2.6666666666666665}\right) \]

       4. *-commutative 98.8%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \color{blue}{\left(r \cdot w\right)}}{2.6666666666666665}\right) \]

   12. Simplified 98.8%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}}\right) \]

   if 1.95e9 < v

   1. Initial program 85.2%

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

      1. associate--l- 85.2%

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

      2. +-commutative 85.2%

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

      3. associate--l+ 85.2%

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

      4. +-commutative 85.2%

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

      5. associate--r+ 85.2%

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

      6. metadata-eval 85.2%

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

      7. associate-*r* 83.5%

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

      8. *-commutative 83.5%

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

      9. associate-/l* 85.1%

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

      10. *-commutative 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in v around inf 89.8%

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

      1. unpow2 89.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 89.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 98.1%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

   7. Taylor expanded in w around 0 89.8%

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

      1. *-commutative 89.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

      2. unpow2 89.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      3. associate-*r* 98.1%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. associate-/r* 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      5. *-commutative 99.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

   9. Simplified 99.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -1.12 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 1950000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{2.6666666666666665}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{r \cdot w}}\right)\\ \end{array} \]

Alternative 8: 67.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1e-143)
     t_0
     (+ t_0 (- -1.5 (* (* w (* r w)) (/ r 2.6666666666666665)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1e-143) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	}
	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 <= 1d-143) then
        tmp = t_0
    else
        tmp = t_0 + ((-1.5d0) - ((w * (r * w)) * (r / 2.6666666666666665d0)))
    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 <= 1e-143) {
		tmp = t_0;
	} else {
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1e-143:
		tmp = t_0
	else:
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1e-143)
		tmp = t_0;
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(w * Float64(r * w)) * Float64(r / 2.6666666666666665))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 1e-143)
		tmp = t_0;
	else
		tmp = t_0 + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	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, 1e-143], t$95$0, N[(t$95$0 + N[(-1.5 - N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 9.9999999999999995e-144

   1. Initial program 79.4%

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

      1. associate--l- 79.4%

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

      2. +-commutative 79.4%

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

      3. associate--l+ 79.5%

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

      4. +-commutative 79.5%

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

      5. associate--r+ 79.5%

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

      6. metadata-eval 79.5%

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

      7. associate-*r* 78.1%

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

      8. *-commutative 78.1%

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

      9. associate-/l* 78.7%

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

      10. *-commutative 78.7%

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

   3. Simplified 78.7%

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

   4. Taylor expanded in v around inf 77.0%

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

      1. unpow2 77.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 77.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 88.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 88.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 88.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

   7. Taylor expanded in w around 0 77.0%

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

      1. *-commutative 77.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

      2. unpow2 77.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      3. associate-*r* 88.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. associate-/r* 90.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      5. *-commutative 90.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

   9. Simplified 90.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

   10. Taylor expanded in r around 0 56.0%

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

       1. unpow2 56.0%

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

   12. Simplified 56.0%

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

   if 9.9999999999999995e-144 < r

   1. Initial program 91.9%

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

      1. associate--l- 91.9%

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

      2. +-commutative 91.9%

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

      3. associate--l+ 91.9%

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

      4. +-commutative 91.9%

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

      5. associate--r+ 91.9%

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

      6. metadata-eval 91.9%

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

      7. associate-*r* 91.9%

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

      8. *-commutative 91.9%

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

      9. associate-/l* 93.7%

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

      10. *-commutative 93.7%

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

   3. Simplified 93.7%

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

   4. Taylor expanded in v around 0 85.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot {w}^{2}}}}\right) \]
   5. Step-by-step derivation

      1. unpow2 85.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{2.6666666666666665}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   6. Simplified 85.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot \left(w \cdot w\right)}}}\right) \]
   7. Step-by-step derivation

      1. associate-/r/ 85.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(r \cdot \left(w \cdot w\right)\right)}\right) \]

      2. *-commutative 85.5%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(\left(w \cdot w\right) \cdot r\right)}\right) \]

      3. associate-*l* 89.0%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{2.6666666666666665} \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \]

   8. Applied egg-rr 89.0%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{r}{2.6666666666666665} \cdot \left(w \cdot \left(w \cdot r\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10^{-143}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \]

Alternative 9: 57.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.45 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 4.5 \cdot 10^{-40} \lor \neg \left(r \leq 1.8 \cdot 10^{+103}\right):\\ \;\;\;\;\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.45e-57)
     t_0
     (if (or (<= r 4.5e-40) (not (<= r 1.8e+103)))
       (* (* w w) (* (* r r) -0.25))
       (+ t_0 -1.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.45e-57) {
		tmp = t_0;
	} else if ((r <= 4.5e-40) || !(r <= 1.8e+103)) {
		tmp = (w * w) * ((r * r) * -0.25);
	} else {
		tmp = t_0 + -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 (r <= 2.45d-57) then
        tmp = t_0
    else if ((r <= 4.5d-40) .or. (.not. (r <= 1.8d+103))) then
        tmp = (w * w) * ((r * r) * (-0.25d0))
    else
        tmp = t_0 + (-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 (r <= 2.45e-57) {
		tmp = t_0;
	} else if ((r <= 4.5e-40) || !(r <= 1.8e+103)) {
		tmp = (w * w) * ((r * r) * -0.25);
	} else {
		tmp = t_0 + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.45e-57:
		tmp = t_0
	elif (r <= 4.5e-40) or not (r <= 1.8e+103):
		tmp = (w * w) * ((r * r) * -0.25)
	else:
		tmp = t_0 + -1.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.45e-57)
		tmp = t_0;
	elseif ((r <= 4.5e-40) || !(r <= 1.8e+103))
		tmp = Float64(Float64(w * w) * Float64(Float64(r * r) * -0.25));
	else
		tmp = Float64(t_0 + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.45e-57)
		tmp = t_0;
	elseif ((r <= 4.5e-40) || ~((r <= 1.8e+103)))
		tmp = (w * w) * ((r * r) * -0.25);
	else
		tmp = t_0 + -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[r, 2.45e-57], t$95$0, If[Or[LessEqual[r, 4.5e-40], N[Not[LessEqual[r, 1.8e+103]], $MachinePrecision]], N[(N[(w * w), $MachinePrecision] * N[(N[(r * r), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -1.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.44999999999999994e-57

   1. Initial program 80.8%

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

      1. associate--l- 80.8%

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

      2. +-commutative 80.8%

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

      3. associate--l+ 80.8%

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

      4. +-commutative 80.8%

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

      5. associate--r+ 80.8%

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

      6. metadata-eval 80.8%

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

      7. associate-*r* 79.7%

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

      8. *-commutative 79.7%

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

      9. associate-/l* 80.2%

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

      10. *-commutative 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in v around inf 78.6%

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

      1. unpow2 78.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 78.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 89.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 89.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 89.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

   7. Taylor expanded in w around 0 78.6%

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

      1. *-commutative 78.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

      2. unpow2 78.6%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      3. associate-*r* 89.4%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. associate-/r* 90.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      5. *-commutative 90.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

   9. Simplified 90.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

   10. Taylor expanded in r around 0 58.4%

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

       1. unpow2 58.4%

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

   12. Simplified 58.4%

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

   if 2.44999999999999994e-57 < r < 4.5000000000000001e-40 or 1.80000000000000008e103 < 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. associate--l- 88.2%

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

      2. +-commutative 88.2%

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

      3. associate--l+ 88.2%

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

      4. +-commutative 88.2%

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

      5. associate--r+ 88.2%

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

      6. metadata-eval 88.2%

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

      7. associate-*r* 88.2%

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

      8. *-commutative 88.2%

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

      9. associate-/l* 91.7%

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

      10. *-commutative 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in v around inf 87.3%

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

      1. unpow2 87.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 87.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 89.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 89.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 89.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

   7. Taylor expanded in w around 0 87.3%

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

      1. *-commutative 87.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

      2. unpow2 87.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      3. associate-*r* 89.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. associate-/r* 89.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      5. *-commutative 89.8%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

   9. Simplified 89.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

   10. Taylor expanded in r around inf 64.0%

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

       1. *-commutative 64.0%

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

       2. *-commutative 64.0%

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

       3. associate-*l* 64.0%

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

       4. unpow2 64.0%

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

       5. unpow2 64.0%

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

   12. Simplified 64.0%

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

   if 4.5000000000000001e-40 < r < 1.80000000000000008e103

   1. Initial program 96.9%

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

      1. associate--l- 96.9%

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

      2. +-commutative 96.9%

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

      3. associate--l+ 96.9%

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

      4. +-commutative 96.9%

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

      5. associate--r+ 96.9%

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

      6. metadata-eval 96.9%

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

      7. associate-*r* 96.8%

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

      8. *-commutative 96.8%

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

      9. associate-/l* 96.7%

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

      10. *-commutative 96.7%

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

   3. Simplified 96.7%

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

   4. Taylor expanded in v around inf 97.3%

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

      1. unpow2 97.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

      2. *-commutative 97.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

      3. associate-*l* 97.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. *-commutative 97.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

   6. Simplified 97.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

   7. Taylor expanded in w around 0 97.3%

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

      1. *-commutative 97.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

      2. unpow2 97.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

      3. associate-*r* 97.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

      4. associate-/r* 97.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

      5. *-commutative 97.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

   9. Simplified 97.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

   10. Taylor expanded in r around 0 68.6%

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

       1. sub-neg 68.6%

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

       2. associate-*r/ 68.6%

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

       3. metadata-eval 68.6%

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

       4. unpow2 68.6%

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

       5. metadata-eval 68.6%

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

   12. Simplified 68.6%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.45 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 4.5 \cdot 10^{-40} \lor \neg \left(r \leq 1.8 \cdot 10^{+103}\right):\\ \;\;\;\;\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]

Alternative 10: 57.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (+ (/ 2.0 (* r r)) -1.5))

C

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

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)) + (-1.5d0)
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + -1.5;
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + -1.5

Julia

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

MATLAB

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

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

   1. associate--l- 84.5%

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

   2. +-commutative 84.5%

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

   3. associate--l+ 84.5%

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

   4. +-commutative 84.5%

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

   5. associate--r+ 84.5%

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

   6. metadata-eval 84.5%

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

   7. associate-*r* 83.7%

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

   8. *-commutative 83.7%

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

   9. associate-/l* 84.8%

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

   10. *-commutative 84.8%

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

3. Simplified 84.8%

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

4. Taylor expanded in v around inf 82.9%

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

   1. unpow2 82.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   2. *-commutative 82.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

   3. associate-*l* 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

   4. *-commutative 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

6. Simplified 90.5%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

7. Taylor expanded in w around 0 82.9%

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

   1. *-commutative 82.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

   2. unpow2 82.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

   3. associate-*r* 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

   4. associate-/r* 91.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

   5. *-commutative 91.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

9. Simplified 91.4%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

10. Taylor expanded in r around 0 59.5%

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

    1. sub-neg 59.5%

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

    2. associate-*r/ 59.5%

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

    3. metadata-eval 59.5%

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

    4. unpow2 59.5%

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

    5. metadata-eval 59.5%

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

12. Simplified 59.5%

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

13. Final simplification 59.5%

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

Alternative 11: 44.1% 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 84.5%

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

   1. associate--l- 84.5%

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

   2. +-commutative 84.5%

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

   3. associate--l+ 84.5%

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

   4. +-commutative 84.5%

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

   5. associate--r+ 84.5%

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

   6. metadata-eval 84.5%

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

   7. associate-*r* 83.7%

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

   8. *-commutative 83.7%

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

   9. associate-/l* 84.8%

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

   10. *-commutative 84.8%

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

3. Simplified 84.8%

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

4. Taylor expanded in v around inf 82.9%

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

   1. unpow2 82.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{r \cdot \color{blue}{\left(w \cdot w\right)}}}\right) \]

   2. *-commutative 82.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right) \cdot r}}}\right) \]

   3. associate-*l* 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

   4. *-commutative 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \color{blue}{\left(r \cdot w\right)}}}\right) \]

6. Simplified 90.5%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{4}{w \cdot \left(r \cdot w\right)}}}\right) \]

7. Taylor expanded in w around 0 82.9%

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

   1. *-commutative 82.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{{w}^{2} \cdot r}}}\right) \]

   2. unpow2 82.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot r}}\right) \]

   3. associate-*r* 90.5%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{\color{blue}{w \cdot \left(w \cdot r\right)}}}\right) \]

   4. associate-/r* 91.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{w \cdot r}}}\right) \]

   5. *-commutative 91.4%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{4}{w}}{\color{blue}{r \cdot w}}}\right) \]

9. Simplified 91.4%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{4}{w}}{r \cdot w}}}\right) \]

10. Taylor expanded in r around 0 42.9%

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

    1. unpow2 42.9%

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

12. Simplified 42.9%

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

13. Final simplification 42.9%

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

Reproduce

herbie shell --seed 2023271 
(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))