Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.8%

Time: 12.8s

Alternatives: 15

Speedup: 1.3×

• 52.7% 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, 0.1× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (* (/ (fma v 0.25 -0.375) (- 1.0 v)) (pow (* r w) 2.0))
  (fma 2.0 (pow r -2.0) -1.5)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 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. sub-neg 86.6%

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

   2. +-commutative 86.6%

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

   3. associate--l+ 86.6%

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

   4. associate-/l* 90.3%

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

   5. distribute-neg-frac 90.3%

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

   6. associate-/r/ 90.3%

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

   7. fma-def 90.3%

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

   8. sub-neg 90.3%

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

3. Simplified 82.4%

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

   1. fma-udef 82.4%

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

   2. unswap-sqr 99.7%

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

   3. pow2 99.7%

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

   4. div-inv 99.7%

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

   5. fma-def 99.7%

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

   6. pow2 99.7%

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

   7. pow-flip 99.8%

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

   8. metadata-eval 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return ((3.0 + (2.0 / (r * r))) - ((0.125 * (3.0 + (v * -2.0))) / pow((pow((r * w), 2.0) / (1.0 - v)), -1.0))) + -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 + (v * (-2.0d0)))) / ((((r * w) ** 2.0d0) / (1.0d0 - v)) ** (-1.0d0)))) + (-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 + (v * -2.0))) / Math.pow((Math.pow((r * w), 2.0) / (1.0 - v)), -1.0))) + -4.5;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. sub-neg 86.6%

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

   2. associate-/l* 90.3%

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

   3. cancel-sign-sub-inv 90.3%

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

   4. metadata-eval 90.3%

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

   5. *-commutative 90.3%

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

   6. *-commutative 90.3%

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

   7. metadata-eval 90.3%

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

3. Simplified 90.3%

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

   1. clear-num 90.3%

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

   2. inv-pow 90.3%

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

   3. associate-*r* 82.4%

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

   4. unswap-sqr 99.8%

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

   5. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. sub-neg 86.6%

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

   2. associate-/l* 90.3%

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

   3. cancel-sign-sub-inv 90.3%

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

   4. metadata-eval 90.3%

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

   5. *-commutative 90.3%

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

   6. *-commutative 90.3%

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

   7. metadata-eval 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in r around 0 80.6%

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

   1. associate-*r/ 80.6%

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

   2. associate-/l* 80.6%

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

   3. +-commutative 80.6%

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

   4. fma-udef 80.6%

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

   5. *-commutative 80.6%

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

   6. unpow2 80.6%

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

   7. unpow2 80.6%

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

   8. swap-sqr 95.0%

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

   9. unpow2 95.0%

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

   10. associate-/l/ 99.8%

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

   11. associate-/r/ 99.7%

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

   12. *-commutative 99.7%

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

   13. fma-udef 99.7%

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

   14. *-commutative 99.7%

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

   15. fma-def 99.7%

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

   16. unpow2 99.7%

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

   17. swap-sqr 82.4%

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

   18. unpow2 82.4%

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

   19. unpow2 82.4%

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

   20. *-commutative 82.4%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. sub-neg 86.6%

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

   2. associate-/l* 90.3%

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

   3. cancel-sign-sub-inv 90.3%

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

   4. metadata-eval 90.3%

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

   5. *-commutative 90.3%

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

   6. *-commutative 90.3%

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

   7. metadata-eval 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in r around 0 82.4%

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

   1. *-commutative 82.4%

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

   2. unpow2 82.4%

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

   3. unpow2 82.4%

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

   4. swap-sqr 99.7%

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

   5. unpow2 99.7%

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

   6. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;\left(3 + t_0\right) - \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{1 - v} \leq 3:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<=
        (-
         (+ 3.0 t_0)
         (/ (* (* 0.125 (- 3.0 (* v 2.0))) (* r (* r (* w w)))) (- 1.0 v)))
        3.0)
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))
     (+ -1.5 t_0))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (1.0 - v))) <= 3.0) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -1.5 + t_0;
	}
	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 (((3.0d0 + t_0) - (((0.125d0 * (3.0d0 - (v * 2.0d0))) * (r * (r * (w * w)))) / (1.0d0 - v))) <= 3.0d0) then
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v))))
    else
        tmp = (-1.5d0) + t_0
    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 (((3.0 + t_0) - (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (1.0 - v))) <= 3.0) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if ((3.0 + t_0) - (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (1.0 - v))) <= 3.0:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))))
	else:
		tmp = -1.5 + t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(Float64(3.0 + t_0) - Float64(Float64(Float64(0.125 * Float64(3.0 - Float64(v * 2.0))) * Float64(r * Float64(r * Float64(w * w)))) / Float64(1.0 - v))) <= 3.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	else
		tmp = Float64(-1.5 + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (((3.0 + t_0) - (((0.125 * (3.0 - (v * 2.0))) * (r * (r * (w * w)))) / (1.0 - v))) <= 3.0)
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	else
		tmp = -1.5 + t_0;
	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[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(N[(0.125 * N[(3.0 - N[(v * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) < 3

   1. Initial program 85.7%

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

      1. associate--l- 85.7%

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

      2. +-commutative 85.7%

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

      3. associate--l+ 85.7%

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

      4. +-commutative 85.7%

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

      5. associate--r+ 85.7%

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

      6. metadata-eval 85.7%

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

      7. associate-*l/ 92.3%

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

      8. *-commutative 92.3%

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

      9. *-commutative 92.3%

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

      10. *-commutative 92.3%

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

   3. Simplified 92.3%

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

   4. Taylor expanded in r around 0 92.3%

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

      1. unpow2 92.3%

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

      2. associate-*l* 97.7%

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

   6. Simplified 97.7%

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

   if 3 < (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))

   1. Initial program 87.8%

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

      1. sub-neg 87.8%

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

      2. +-commutative 87.8%

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

      3. associate--l+ 87.8%

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

      4. associate-/l* 87.8%

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

      5. distribute-neg-frac 87.8%

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

      6. associate-/r/ 87.8%

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

      7. fma-def 87.8%

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

      8. sub-neg 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in r around 0 99.9%

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

      1. sub-neg 99.9%

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

      2. associate-*r/ 99.9%

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

      3. metadata-eval 99.9%

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

      4. unpow2 99.9%

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

      5. metadata-eval 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - v \cdot 2\right)\right) \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}{1 - v} \leq 3:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]

Alternative 6: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{r \cdot w}\\ -4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + v \cdot -2\right)}{\left(1 - v\right) \cdot \left(t_0 \cdot t_0\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* r w))))
   (+
    -4.5
    (-
     (+ 3.0 (/ 2.0 (* r r)))
     (/ (* 0.125 (+ 3.0 (* v -2.0))) (* (- 1.0 v) (* t_0 t_0)))))))

C

double code(double v, double w, double r) {
	double t_0 = 1.0 / (r * w);
	return -4.5 + ((3.0 + (2.0 / (r * r))) - ((0.125 * (3.0 + (v * -2.0))) / ((1.0 - v) * (t_0 * t_0))));
}

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
    t_0 = 1.0d0 / (r * w)
    code = (-4.5d0) + ((3.0d0 + (2.0d0 / (r * r))) - ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) / ((1.0d0 - v) * (t_0 * t_0))))
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 1.0 / (r * w);
	return -4.5 + ((3.0 + (2.0 / (r * r))) - ((0.125 * (3.0 + (v * -2.0))) / ((1.0 - v) * (t_0 * t_0))));
}

Python

def code(v, w, r):
	t_0 = 1.0 / (r * w)
	return -4.5 + ((3.0 + (2.0 / (r * r))) - ((0.125 * (3.0 + (v * -2.0))) / ((1.0 - v) * (t_0 * t_0))))

Julia

function code(v, w, r)
	t_0 = Float64(1.0 / Float64(r * w))
	return Float64(-4.5 + Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(0.125 * Float64(3.0 + Float64(v * -2.0))) / Float64(Float64(1.0 - v) * Float64(t_0 * t_0)))))
end

MATLAB

function tmp = code(v, w, r)
	t_0 = 1.0 / (r * w);
	tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((0.125 * (3.0 + (v * -2.0))) / ((1.0 - v) * (t_0 * t_0))));
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]}, N[(-4.5 + N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.125 * N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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. sub-neg 86.6%

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

   2. associate-/l* 90.3%

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

   3. cancel-sign-sub-inv 90.3%

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

   4. metadata-eval 90.3%

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

   5. *-commutative 90.3%

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

   6. *-commutative 90.3%

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

   7. metadata-eval 90.3%

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

3. Simplified 90.3%

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

   1. div-inv 90.3%

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

   2. associate-*r* 82.4%

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

   3. unswap-sqr 99.7%

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

   4. pow2 99.7%

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

5. Applied egg-rr 99.7%

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

   1. metadata-eval 99.7%

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

   2. unpow2 99.7%

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

   3. frac-times 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 7: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq -2.02 \cdot 10^{-48} \lor \neg \left(r \leq 1.4 \cdot 10^{-121}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + -4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= r -2.02e-48) (not (<= r 1.4e-121)))
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* r (* w w))))))
     (+ t_0 -4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((r <= -2.02e-48) || !(r <= 1.4e-121)) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = t_0 + -4.5;
	}
	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 <= (-2.02d-48)) .or. (.not. (r <= 1.4d-121))) then
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (r * (w * w)))))
    else
        tmp = t_0 + (-4.5d0)
    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 <= -2.02e-48) || !(r <= 1.4e-121)) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = t_0 + -4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (r <= -2.02e-48) or not (r <= 1.4e-121):
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	else:
		tmp = t_0 + -4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((r <= -2.02e-48) || !(r <= 1.4e-121))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(t_0 + -4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((r <= -2.02e-48) || ~((r <= 1.4e-121)))
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	else
		tmp = t_0 + -4.5;
	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[r, -2.02e-48], N[Not[LessEqual[r, 1.4e-121]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < -2.0199999999999999e-48 or 1.4000000000000001e-121 < r

   1. Initial program 90.0%

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

      1. associate--l- 90.0%

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

      2. +-commutative 90.0%

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

      3. associate--l+ 90.0%

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

      4. +-commutative 90.0%

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

      5. associate--r+ 90.0%

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

      6. metadata-eval 90.0%

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

      7. associate-*l/ 95.7%

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

      8. *-commutative 95.7%

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

      9. *-commutative 95.7%

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

      10. *-commutative 95.7%

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

   3. Simplified 95.7%

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

   if -2.0199999999999999e-48 < r < 1.4000000000000001e-121

   1. Initial program 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-/l* 81.0%

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

      3. cancel-sign-sub-inv 81.0%

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

      4. metadata-eval 81.0%

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

      5. *-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in v around 0 81.0%

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

      1. unpow2 81.0%

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

      2. unpow2 81.0%

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

   6. Simplified 81.0%

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

      1. add-sqr-sqrt 81.0%

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

      2. unswap-sqr 81.0%

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

      3. unpow2 81.0%

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

      4. sqrt-div 81.0%

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

      5. metadata-eval 81.0%

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

      6. unpow2 81.0%

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

      7. sqrt-prod 58.9%

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

      8. add-sqr-sqrt 81.1%

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

      9. *-commutative 81.1%

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

      10. unswap-sqr 97.8%

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

      11. unpow2 97.8%

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

      12. sqrt-div 97.8%

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

      13. metadata-eval 97.8%

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

      14. unpow2 97.8%

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

      15. sqrt-prod 65.2%

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

      16. add-sqr-sqrt 97.8%

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

      17. *-commutative 97.8%

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

   8. Applied egg-rr 97.8%

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

   9. Taylor expanded in r around 0 99.9%

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + -4.5 \]
   10. Step-by-step derivation

       1. unpow2 99.9%

          \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + -4.5 \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.02 \cdot 10^{-48} \lor \neg \left(r \leq 1.4 \cdot 10^{-121}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -4.5\\ \end{array} \]

Alternative 8: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq -2.02 \cdot 10^{-48} \lor \neg \left(r \leq 1.8 \cdot 10^{-120}\right):\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - r \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + -4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= r -2.02e-48) (not (<= r 1.8e-120)))
     (+ -4.5 (- (+ 3.0 t_0) (* r (* (* r (* w w)) 0.375))))
     (+ t_0 -4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((r <= -2.02e-48) || !(r <= 1.8e-120)) {
		tmp = -4.5 + ((3.0 + t_0) - (r * ((r * (w * w)) * 0.375)));
	} else {
		tmp = t_0 + -4.5;
	}
	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 <= (-2.02d-48)) .or. (.not. (r <= 1.8d-120))) then
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (r * ((r * (w * w)) * 0.375d0)))
    else
        tmp = t_0 + (-4.5d0)
    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 <= -2.02e-48) || !(r <= 1.8e-120)) {
		tmp = -4.5 + ((3.0 + t_0) - (r * ((r * (w * w)) * 0.375)));
	} else {
		tmp = t_0 + -4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (r <= -2.02e-48) or not (r <= 1.8e-120):
		tmp = -4.5 + ((3.0 + t_0) - (r * ((r * (w * w)) * 0.375)))
	else:
		tmp = t_0 + -4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((r <= -2.02e-48) || !(r <= 1.8e-120))
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(r * Float64(Float64(r * Float64(w * w)) * 0.375))));
	else
		tmp = Float64(t_0 + -4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((r <= -2.02e-48) || ~((r <= 1.8e-120)))
		tmp = -4.5 + ((3.0 + t_0) - (r * ((r * (w * w)) * 0.375)));
	else
		tmp = t_0 + -4.5;
	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[r, -2.02e-48], N[Not[LessEqual[r, 1.8e-120]], $MachinePrecision]], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(r * N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < -2.0199999999999999e-48 or 1.8000000000000001e-120 < r

   1. Initial program 90.0%

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

      1. sub-neg 90.0%

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

      2. associate-/l* 95.7%

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

      3. cancel-sign-sub-inv 95.7%

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

      4. metadata-eval 95.7%

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

      5. *-commutative 95.7%

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

      6. *-commutative 95.7%

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

      7. metadata-eval 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in v around 0 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

   6. Simplified 65.4%

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

      1. unswap-sqr 73.9%

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

      2. unpow2 73.9%

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

      3. associate-/r/ 73.9%

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

      4. /-rgt-identity 73.9%

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

      5. unpow2 73.9%

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

      6. unswap-sqr 65.4%

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

      7. associate-*r* 73.9%

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

      8. associate-*r* 73.4%

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

      9. distribute-lft-in 73.4%

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

      10. metadata-eval 73.4%

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

      11. associate-*r* 73.4%

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

      12. metadata-eval 73.4%

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

   8. Applied egg-rr 73.4%

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

   9. Taylor expanded in v around 0 87.3%

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

       1. unpow2 87.3%

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

   11. Simplified 87.3%

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

   if -2.0199999999999999e-48 < r < 1.8000000000000001e-120

   1. Initial program 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-/l* 81.0%

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

      3. cancel-sign-sub-inv 81.0%

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

      4. metadata-eval 81.0%

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

      5. *-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in v around 0 81.0%

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

      1. unpow2 81.0%

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

      2. unpow2 81.0%

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

   6. Simplified 81.0%

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

      1. add-sqr-sqrt 81.0%

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

      2. unswap-sqr 81.0%

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

      3. unpow2 81.0%

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

      4. sqrt-div 81.0%

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

      5. metadata-eval 81.0%

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

      6. unpow2 81.0%

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

      7. sqrt-prod 58.9%

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

      8. add-sqr-sqrt 81.1%

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

      9. *-commutative 81.1%

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

      10. unswap-sqr 97.8%

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

      11. unpow2 97.8%

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

      12. sqrt-div 97.8%

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

      13. metadata-eval 97.8%

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

      14. unpow2 97.8%

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

      15. sqrt-prod 65.2%

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

      16. add-sqr-sqrt 97.8%

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

      17. *-commutative 97.8%

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

   8. Applied egg-rr 97.8%

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

   9. Taylor expanded in r around 0 99.9%

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + -4.5 \]
   10. Step-by-step derivation

       1. unpow2 99.9%

          \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + -4.5 \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.02 \cdot 10^{-48} \lor \neg \left(r \leq 1.8 \cdot 10^{-120}\right):\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - r \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot 0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -4.5\\ \end{array} \]

Alternative 9: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := 3 + t_0\\ \mathbf{if}\;r \leq -2.02 \cdot 10^{-48}:\\ \;\;\;\;-4.5 + \left(t_1 - r \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot 0.375\right)\right)\\ \mathbf{elif}\;r \leq 7 \cdot 10^{-118}:\\ \;\;\;\;t_0 + -4.5\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(t_1 - r \cdot \left(r \cdot \left(w \cdot \left(w \cdot 0.375\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (+ 3.0 t_0)))
   (if (<= r -2.02e-48)
     (+ -4.5 (- t_1 (* r (* (* r (* w w)) 0.375))))
     (if (<= r 7e-118)
       (+ t_0 -4.5)
       (+ -4.5 (- t_1 (* r (* r (* w (* w 0.375))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double tmp;
	if (r <= -2.02e-48) {
		tmp = -4.5 + (t_1 - (r * ((r * (w * w)) * 0.375)));
	} else if (r <= 7e-118) {
		tmp = t_0 + -4.5;
	} else {
		tmp = -4.5 + (t_1 - (r * (r * (w * (w * 0.375)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = 3.0d0 + t_0
    if (r <= (-2.02d-48)) then
        tmp = (-4.5d0) + (t_1 - (r * ((r * (w * w)) * 0.375d0)))
    else if (r <= 7d-118) then
        tmp = t_0 + (-4.5d0)
    else
        tmp = (-4.5d0) + (t_1 - (r * (r * (w * (w * 0.375d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = 3.0 + t_0;
	double tmp;
	if (r <= -2.02e-48) {
		tmp = -4.5 + (t_1 - (r * ((r * (w * w)) * 0.375)));
	} else if (r <= 7e-118) {
		tmp = t_0 + -4.5;
	} else {
		tmp = -4.5 + (t_1 - (r * (r * (w * (w * 0.375)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = 3.0 + t_0
	tmp = 0
	if r <= -2.02e-48:
		tmp = -4.5 + (t_1 - (r * ((r * (w * w)) * 0.375)))
	elif r <= 7e-118:
		tmp = t_0 + -4.5
	else:
		tmp = -4.5 + (t_1 - (r * (r * (w * (w * 0.375)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(3.0 + t_0)
	tmp = 0.0
	if (r <= -2.02e-48)
		tmp = Float64(-4.5 + Float64(t_1 - Float64(r * Float64(Float64(r * Float64(w * w)) * 0.375))));
	elseif (r <= 7e-118)
		tmp = Float64(t_0 + -4.5);
	else
		tmp = Float64(-4.5 + Float64(t_1 - Float64(r * Float64(r * Float64(w * Float64(w * 0.375))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = 3.0 + t_0;
	tmp = 0.0;
	if (r <= -2.02e-48)
		tmp = -4.5 + (t_1 - (r * ((r * (w * w)) * 0.375)));
	elseif (r <= 7e-118)
		tmp = t_0 + -4.5;
	else
		tmp = -4.5 + (t_1 - (r * (r * (w * (w * 0.375)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 + t$95$0), $MachinePrecision]}, If[LessEqual[r, -2.02e-48], N[(-4.5 + N[(t$95$1 - N[(r * N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 7e-118], N[(t$95$0 + -4.5), $MachinePrecision], N[(-4.5 + N[(t$95$1 - N[(r * N[(r * N[(w * N[(w * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < -2.0199999999999999e-48

   1. Initial program 87.0%

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

      1. sub-neg 87.0%

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

      2. associate-/l* 97.1%

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

      3. cancel-sign-sub-inv 97.1%

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

      4. metadata-eval 97.1%

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

      5. *-commutative 97.1%

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

      6. *-commutative 97.1%

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

      7. metadata-eval 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in v around 0 56.3%

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

      1. unpow2 56.3%

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

      2. unpow2 56.3%

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

   6. Simplified 56.3%

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

      1. unswap-sqr 68.0%

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

      2. unpow2 68.0%

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

      3. associate-/r/ 68.0%

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

      4. /-rgt-identity 68.0%

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

      5. unpow2 68.0%

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

      6. unswap-sqr 56.3%

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

      7. associate-*r* 68.1%

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

      8. associate-*r* 68.1%

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

      9. distribute-lft-in 68.1%

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

      10. metadata-eval 68.1%

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

      11. associate-*r* 68.1%

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

      12. metadata-eval 68.1%

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

   8. Applied egg-rr 68.1%

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

   9. Taylor expanded in v around 0 85.3%

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

       1. unpow2 85.3%

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

   11. Simplified 85.3%

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

   if -2.0199999999999999e-48 < r < 7e-118

   1. Initial program 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-/l* 81.0%

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

      3. cancel-sign-sub-inv 81.0%

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

      4. metadata-eval 81.0%

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

      5. *-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in v around 0 81.0%

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

      1. unpow2 81.0%

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

      2. unpow2 81.0%

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

   6. Simplified 81.0%

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

      1. add-sqr-sqrt 81.0%

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

      2. unswap-sqr 81.0%

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

      3. unpow2 81.0%

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

      4. sqrt-div 81.0%

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

      5. metadata-eval 81.0%

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

      6. unpow2 81.0%

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

      7. sqrt-prod 58.9%

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

      8. add-sqr-sqrt 81.1%

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

      9. *-commutative 81.1%

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

      10. unswap-sqr 97.8%

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

      11. unpow2 97.8%

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

      12. sqrt-div 97.8%

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

      13. metadata-eval 97.8%

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

      14. unpow2 97.8%

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

      15. sqrt-prod 65.2%

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

      16. add-sqr-sqrt 97.8%

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

      17. *-commutative 97.8%

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

   8. Applied egg-rr 97.8%

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

   9. Taylor expanded in r around 0 99.9%

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + -4.5 \]
   10. Step-by-step derivation

       1. unpow2 99.9%

          \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + -4.5 \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + -4.5 \]

   if 7e-118 < r

   1. Initial program 92.5%

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

      1. sub-neg 92.5%

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

      2. associate-/l* 94.6%

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

      3. cancel-sign-sub-inv 94.6%

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

      4. metadata-eval 94.6%

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

      5. *-commutative 94.6%

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

      6. *-commutative 94.6%

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

      7. metadata-eval 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in v around 0 73.0%

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

      1. unpow2 73.0%

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

      2. unpow2 73.0%

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

   6. Simplified 73.0%

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

      1. unswap-sqr 78.8%

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

      2. unpow2 78.8%

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

      3. associate-/r/ 78.8%

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

      4. /-rgt-identity 78.8%

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

      5. unpow2 78.8%

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

      6. unswap-sqr 73.0%

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

      7. associate-*r* 78.8%

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

      8. associate-*r* 77.7%

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

      9. distribute-lft-in 77.7%

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

      10. metadata-eval 77.7%

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

      11. associate-*r* 77.7%

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

      12. metadata-eval 77.7%

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

   8. Applied egg-rr 77.7%

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

   9. Taylor expanded in v around 0 89.0%

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

       1. associate-*r* 89.0%

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

       2. *-commutative 89.0%

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

       3. unpow2 89.0%

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

       4. associate-*r* 89.0%

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

   11. Simplified 89.0%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.02 \cdot 10^{-48}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - r \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot 0.375\right)\right)\\ \mathbf{elif}\;r \leq 7 \cdot 10^{-118}:\\ \;\;\;\;\frac{2}{r \cdot r} + -4.5\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - r \cdot \left(r \cdot \left(w \cdot \left(w \cdot 0.375\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq -2.02 \cdot 10^{-48} \lor \neg \left(r \leq 10^{-119}\right):\\ \;\;\;\;t_0 + \left(-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + -4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (or (<= r -2.02e-48) (not (<= r 1e-119)))
     (+ t_0 (- (* -0.375 (* (* r r) (* w w))) 1.5))
     (+ t_0 -4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((r <= -2.02e-48) || !(r <= 1e-119)) {
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5);
	} else {
		tmp = t_0 + -4.5;
	}
	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 <= (-2.02d-48)) .or. (.not. (r <= 1d-119))) then
        tmp = t_0 + (((-0.375d0) * ((r * r) * (w * w))) - 1.5d0)
    else
        tmp = t_0 + (-4.5d0)
    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 <= -2.02e-48) || !(r <= 1e-119)) {
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5);
	} else {
		tmp = t_0 + -4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (r <= -2.02e-48) or not (r <= 1e-119):
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5)
	else:
		tmp = t_0 + -4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if ((r <= -2.02e-48) || !(r <= 1e-119))
		tmp = Float64(t_0 + Float64(Float64(-0.375 * Float64(Float64(r * r) * Float64(w * w))) - 1.5));
	else
		tmp = Float64(t_0 + -4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((r <= -2.02e-48) || ~((r <= 1e-119)))
		tmp = t_0 + ((-0.375 * ((r * r) * (w * w))) - 1.5);
	else
		tmp = t_0 + -4.5;
	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[r, -2.02e-48], N[Not[LessEqual[r, 1e-119]], $MachinePrecision]], N[(t$95$0 + N[(N[(-0.375 * N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + -4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < -2.0199999999999999e-48 or 1.00000000000000001e-119 < r

   1. Initial program 90.0%

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

      1. sub-neg 90.0%

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

      2. +-commutative 90.0%

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

      3. associate--l+ 90.0%

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

      4. associate-/l* 95.7%

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

      5. distribute-neg-frac 95.7%

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

      6. associate-/r/ 95.7%

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

      7. fma-def 95.8%

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

      8. sub-neg 95.8%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in v around 0 78.2%

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

      1. associate--l+ 78.2%

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

      2. associate-*r/ 78.2%

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

      3. metadata-eval 78.2%

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

      4. unpow2 78.2%

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

      5. *-commutative 78.2%

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

      6. unpow2 78.2%

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

      7. unpow2 78.2%

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

   6. Simplified 78.2%

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

   if -2.0199999999999999e-48 < r < 1.00000000000000001e-119

   1. Initial program 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-/l* 81.0%

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

      3. cancel-sign-sub-inv 81.0%

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

      4. metadata-eval 81.0%

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

      5. *-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in v around 0 81.0%

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

      1. unpow2 81.0%

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

      2. unpow2 81.0%

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

   6. Simplified 81.0%

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

      1. add-sqr-sqrt 81.0%

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

      2. unswap-sqr 81.0%

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

      3. unpow2 81.0%

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

      4. sqrt-div 81.0%

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

      5. metadata-eval 81.0%

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

      6. unpow2 81.0%

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

      7. sqrt-prod 58.9%

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

      8. add-sqr-sqrt 81.1%

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

      9. *-commutative 81.1%

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

      10. unswap-sqr 97.8%

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

      11. unpow2 97.8%

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

      12. sqrt-div 97.8%

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

      13. metadata-eval 97.8%

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

      14. unpow2 97.8%

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

      15. sqrt-prod 65.2%

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

      16. add-sqr-sqrt 97.8%

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

      17. *-commutative 97.8%

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

   8. Applied egg-rr 97.8%

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

   9. Taylor expanded in r around 0 99.9%

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + -4.5 \]
   10. Step-by-step derivation

       1. unpow2 99.9%

          \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + -4.5 \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + -4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.02 \cdot 10^{-48} \lor \neg \left(r \leq 10^{-119}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -4.5\\ \end{array} \]

Alternative 11: 84.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(r \cdot r\right) \cdot \left(w \cdot w\right)\\ \mathbf{if}\;r \leq -2.02 \cdot 10^{-48}:\\ \;\;\;\;t_0 + \left(-0.25 \cdot t_1 - 1.5\right)\\ \mathbf{elif}\;r \leq 10^{-120}:\\ \;\;\;\;t_0 + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-0.375 \cdot t_1 - 1.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* (* r r) (* w w))))
   (if (<= r -2.02e-48)
     (+ t_0 (- (* -0.25 t_1) 1.5))
     (if (<= r 1e-120) (+ t_0 -4.5) (+ t_0 (- (* -0.375 t_1) 1.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (r * r) * (w * w);
	double tmp;
	if (r <= -2.02e-48) {
		tmp = t_0 + ((-0.25 * t_1) - 1.5);
	} else if (r <= 1e-120) {
		tmp = t_0 + -4.5;
	} else {
		tmp = t_0 + ((-0.375 * t_1) - 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) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (r * r) * (w * w)
    if (r <= (-2.02d-48)) then
        tmp = t_0 + (((-0.25d0) * t_1) - 1.5d0)
    else if (r <= 1d-120) then
        tmp = t_0 + (-4.5d0)
    else
        tmp = t_0 + (((-0.375d0) * t_1) - 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 t_1 = (r * r) * (w * w);
	double tmp;
	if (r <= -2.02e-48) {
		tmp = t_0 + ((-0.25 * t_1) - 1.5);
	} else if (r <= 1e-120) {
		tmp = t_0 + -4.5;
	} else {
		tmp = t_0 + ((-0.375 * t_1) - 1.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (r * r) * (w * w)
	tmp = 0
	if r <= -2.02e-48:
		tmp = t_0 + ((-0.25 * t_1) - 1.5)
	elif r <= 1e-120:
		tmp = t_0 + -4.5
	else:
		tmp = t_0 + ((-0.375 * t_1) - 1.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(r * r) * Float64(w * w))
	tmp = 0.0
	if (r <= -2.02e-48)
		tmp = Float64(t_0 + Float64(Float64(-0.25 * t_1) - 1.5));
	elseif (r <= 1e-120)
		tmp = Float64(t_0 + -4.5);
	else
		tmp = Float64(t_0 + Float64(Float64(-0.375 * t_1) - 1.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (r * r) * (w * w);
	tmp = 0.0;
	if (r <= -2.02e-48)
		tmp = t_0 + ((-0.25 * t_1) - 1.5);
	elseif (r <= 1e-120)
		tmp = t_0 + -4.5;
	else
		tmp = t_0 + ((-0.375 * t_1) - 1.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, -2.02e-48], N[(t$95$0 + N[(N[(-0.25 * t$95$1), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1e-120], N[(t$95$0 + -4.5), $MachinePrecision], N[(t$95$0 + N[(N[(-0.375 * t$95$1), $MachinePrecision] - 1.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < -2.0199999999999999e-48

   1. Initial program 87.0%

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

      1. sub-neg 87.0%

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

      2. +-commutative 87.0%

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

      3. associate--l+ 87.0%

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

      4. associate-/l* 97.1%

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

      5. distribute-neg-frac 97.1%

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

      6. associate-/r/ 97.1%

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

      7. fma-def 97.2%

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

      8. sub-neg 97.2%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in v around inf 72.6%

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

      1. associate--l+ 72.6%

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

      2. associate-*r/ 72.6%

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

      3. metadata-eval 72.6%

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

      4. unpow2 72.6%

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

      5. *-commutative 72.6%

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

      6. unpow2 72.6%

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

      7. unpow2 72.6%

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

   6. Simplified 72.6%

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

   if -2.0199999999999999e-48 < r < 9.99999999999999979e-121

   1. Initial program 81.0%

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

      1. sub-neg 81.0%

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

      2. associate-/l* 81.0%

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

      3. cancel-sign-sub-inv 81.0%

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

      4. metadata-eval 81.0%

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

      5. *-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. metadata-eval 81.0%

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

   3. Simplified 81.0%

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

   4. Taylor expanded in v around 0 81.0%

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

      1. unpow2 81.0%

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

      2. unpow2 81.0%

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

   6. Simplified 81.0%

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

      1. add-sqr-sqrt 81.0%

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

      2. unswap-sqr 81.0%

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

      3. unpow2 81.0%

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

      4. sqrt-div 81.0%

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

      5. metadata-eval 81.0%

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

      6. unpow2 81.0%

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

      7. sqrt-prod 58.9%

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

      8. add-sqr-sqrt 81.1%

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

      9. *-commutative 81.1%

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

      10. unswap-sqr 97.8%

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

      11. unpow2 97.8%

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

      12. sqrt-div 97.8%

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

      13. metadata-eval 97.8%

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

      14. unpow2 97.8%

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

      15. sqrt-prod 65.2%

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

      16. add-sqr-sqrt 97.8%

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

      17. *-commutative 97.8%

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

   8. Applied egg-rr 97.8%

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

   9. Taylor expanded in r around 0 99.9%

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + -4.5 \]
   10. Step-by-step derivation

       1. unpow2 99.9%

          \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + -4.5 \]

   11. Simplified 99.9%

       \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + -4.5 \]

   if 9.99999999999999979e-121 < r

   1. Initial program 92.5%

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

      1. sub-neg 92.5%

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

      2. +-commutative 92.5%

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

      3. associate--l+ 92.5%

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

      4. associate-/l* 94.6%

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

      5. distribute-neg-frac 94.6%

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

      6. associate-/r/ 94.6%

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

      7. fma-def 94.6%

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

      8. sub-neg 94.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in v around 0 82.9%

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

      1. associate--l+ 82.9%

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

      2. associate-*r/ 82.9%

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

      3. metadata-eval 82.9%

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

      4. unpow2 82.9%

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

      5. *-commutative 82.9%

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

      6. unpow2 82.9%

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

      7. unpow2 82.9%

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

   6. Simplified 82.9%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.02 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{elif}\;r \leq 10^{-120}:\\ \;\;\;\;\frac{2}{r \cdot r} + -4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \end{array} \]

Alternative 12: 81.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq -5.3 \lor \neg \left(r \leq 5.6\right):\\ \;\;\;\;-1.5 + -0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= r -5.3) (not (<= r 5.6)))
   (+ -1.5 (* -0.375 (* (* r r) (* w w))))
   (+ -1.5 (/ 2.0 (* r r)))))

C

double code(double v, double w, double r) {
	double tmp;
	if ((r <= -5.3) || !(r <= 5.6)) {
		tmp = -1.5 + (-0.375 * ((r * r) * (w * w)));
	} else {
		tmp = -1.5 + (2.0 / (r * r));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if ((r <= (-5.3d0)) .or. (.not. (r <= 5.6d0))) then
        tmp = (-1.5d0) + ((-0.375d0) * ((r * r) * (w * w)))
    else
        tmp = (-1.5d0) + (2.0d0 / (r * r))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((r <= -5.3) || !(r <= 5.6)) {
		tmp = -1.5 + (-0.375 * ((r * r) * (w * w)));
	} else {
		tmp = -1.5 + (2.0 / (r * r));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (r <= -5.3) or not (r <= 5.6):
		tmp = -1.5 + (-0.375 * ((r * r) * (w * w)))
	else:
		tmp = -1.5 + (2.0 / (r * r))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((r <= -5.3) || !(r <= 5.6))
		tmp = Float64(-1.5 + Float64(-0.375 * Float64(Float64(r * r) * Float64(w * w))));
	else
		tmp = Float64(-1.5 + Float64(2.0 / Float64(r * r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((r <= -5.3) || ~((r <= 5.6)))
		tmp = -1.5 + (-0.375 * ((r * r) * (w * w)));
	else
		tmp = -1.5 + (2.0 / (r * r));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[r, -5.3], N[Not[LessEqual[r, 5.6]], $MachinePrecision]], N[(-1.5 + N[(-0.375 * N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < -5.29999999999999982 or 5.5999999999999996 < r

   1. Initial program 90.2%

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

      1. sub-neg 90.2%

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

      2. +-commutative 90.2%

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

      3. associate--l+ 90.2%

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

      4. associate-/l* 97.6%

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

      5. distribute-neg-frac 97.6%

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

      6. associate-/r/ 97.6%

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

      7. fma-def 97.6%

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

      8. sub-neg 97.6%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in r around inf 69.4%

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

      1. sub-neg 69.4%

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

      2. associate-*r* 77.2%

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

      3. associate-/l* 80.9%

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

      4. *-commutative 80.9%

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

      5. unpow2 80.9%

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

      6. unpow2 80.9%

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

      7. swap-sqr 99.2%

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

      8. unpow2 99.2%

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

      9. *-commutative 99.2%

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

      10. *-commutative 99.2%

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

      11. fma-neg 99.2%

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

      12. metadata-eval 99.2%

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

      13. metadata-eval 99.2%

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

   6. Simplified 99.2%

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

      1. unpow2 99.2%

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

      2. unswap-sqr 80.9%

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

      3. div-inv 80.9%

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

      4. times-frac 70.3%

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

   8. Applied egg-rr 70.3%

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

   9. Taylor expanded in v around 0 49.8%

      \[\leadsto \frac{w \cdot w}{1 - v} \cdot \frac{r \cdot r}{\color{blue}{-1.7777777777777777 \cdot v - 2.6666666666666665}} + -1.5 \]

   10. Taylor expanded in v around 0 75.1%

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

       1. *-commutative 75.1%

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

       2. unpow2 75.1%

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

       3. *-commutative 75.1%

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

       4. unpow2 75.1%

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

   12. Simplified 75.1%

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

   if -5.29999999999999982 < r < 5.5999999999999996

   1. Initial program 83.3%

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

      1. sub-neg 83.3%

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

      2. +-commutative 83.3%

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

      3. associate--l+ 83.3%

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

      4. associate-/l* 83.3%

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

      5. distribute-neg-frac 83.3%

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

      6. associate-/r/ 83.3%

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

      7. fma-def 83.3%

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

      8. sub-neg 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in r around 0 91.5%

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

      1. sub-neg 91.5%

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

      2. associate-*r/ 91.5%

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

      3. metadata-eval 91.5%

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

      4. unpow2 91.5%

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

      5. metadata-eval 91.5%

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

   6. Simplified 91.5%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -5.3 \lor \neg \left(r \leq 5.6\right):\\ \;\;\;\;-1.5 + -0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]

Alternative 13: 81.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot r\right) \cdot \left(w \cdot w\right)\\ \mathbf{if}\;r \leq -8:\\ \;\;\;\;-1.5 + -0.25 \cdot t_0\\ \mathbf{elif}\;r \leq 120:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + -0.375 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r r) (* w w))))
   (if (<= r -8.0)
     (+ -1.5 (* -0.25 t_0))
     (if (<= r 120.0) (+ -1.5 (/ 2.0 (* r r))) (+ -1.5 (* -0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * r) * (w * w);
	double tmp;
	if (r <= -8.0) {
		tmp = -1.5 + (-0.25 * t_0);
	} else if (r <= 120.0) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -1.5 + (-0.375 * t_0);
	}
	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 = (r * r) * (w * w)
    if (r <= (-8.0d0)) then
        tmp = (-1.5d0) + ((-0.25d0) * t_0)
    else if (r <= 120.0d0) then
        tmp = (-1.5d0) + (2.0d0 / (r * r))
    else
        tmp = (-1.5d0) + ((-0.375d0) * t_0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * r) * (w * w);
	double tmp;
	if (r <= -8.0) {
		tmp = -1.5 + (-0.25 * t_0);
	} else if (r <= 120.0) {
		tmp = -1.5 + (2.0 / (r * r));
	} else {
		tmp = -1.5 + (-0.375 * t_0);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * r) * (w * w)
	tmp = 0
	if r <= -8.0:
		tmp = -1.5 + (-0.25 * t_0)
	elif r <= 120.0:
		tmp = -1.5 + (2.0 / (r * r))
	else:
		tmp = -1.5 + (-0.375 * t_0)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * r) * Float64(w * w))
	tmp = 0.0
	if (r <= -8.0)
		tmp = Float64(-1.5 + Float64(-0.25 * t_0));
	elseif (r <= 120.0)
		tmp = Float64(-1.5 + Float64(2.0 / Float64(r * r)));
	else
		tmp = Float64(-1.5 + Float64(-0.375 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * r) * (w * w);
	tmp = 0.0;
	if (r <= -8.0)
		tmp = -1.5 + (-0.25 * t_0);
	elseif (r <= 120.0)
		tmp = -1.5 + (2.0 / (r * r));
	else
		tmp = -1.5 + (-0.375 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, -8.0], N[(-1.5 + N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 120.0], N[(-1.5 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(-0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < -8

   1. Initial program 87.4%

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

      1. sub-neg 87.4%

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

      2. +-commutative 87.4%

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

      3. associate--l+ 87.4%

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

      4. associate-/l* 98.3%

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

      5. distribute-neg-frac 98.3%

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

      6. associate-/r/ 98.3%

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

      7. fma-def 98.3%

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

      8. sub-neg 98.3%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in r around inf 69.6%

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

      1. sub-neg 69.6%

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

      2. associate-*r* 72.5%

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

      3. associate-/l* 78.0%

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

      4. *-commutative 78.0%

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

      5. unpow2 78.0%

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

      6. unpow2 78.0%

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

      7. swap-sqr 98.7%

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

      8. unpow2 98.7%

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

      9. *-commutative 98.7%

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

      10. *-commutative 98.7%

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

      11. fma-neg 98.7%

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

      12. metadata-eval 98.7%

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

      13. metadata-eval 98.7%

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

   6. Simplified 98.7%

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

      1. unpow2 98.7%

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

      2. unswap-sqr 78.0%

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

      3. div-inv 78.0%

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

      4. times-frac 70.4%

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

   8. Applied egg-rr 70.4%

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

   9. Taylor expanded in v around inf 71.9%

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

       1. *-commutative 71.9%

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

       2. unpow2 71.9%

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

       3. unpow2 71.9%

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

   11. Simplified 71.9%

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

   if -8 < r < 120

   1. Initial program 83.3%

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

      1. sub-neg 83.3%

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

      2. +-commutative 83.3%

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

      3. associate--l+ 83.3%

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

      4. associate-/l* 83.3%

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

      5. distribute-neg-frac 83.3%

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

      6. associate-/r/ 83.3%

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

      7. fma-def 83.3%

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

      8. sub-neg 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in r around 0 91.5%

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

      1. sub-neg 91.5%

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

      2. associate-*r/ 91.5%

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

      3. metadata-eval 91.5%

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

      4. unpow2 91.5%

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

      5. metadata-eval 91.5%

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

   6. Simplified 91.5%

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

   if 120 < r

   1. Initial program 93.5%

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

      1. sub-neg 93.5%

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

      2. +-commutative 93.5%

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

      3. associate--l+ 93.5%

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

      4. associate-/l* 96.7%

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

      5. distribute-neg-frac 96.7%

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

      6. associate-/r/ 96.6%

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

      7. fma-def 96.7%

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

      8. sub-neg 96.7%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in r around inf 69.1%

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

      1. sub-neg 69.1%

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

      2. associate-*r* 82.9%

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

      3. associate-/l* 84.5%

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

      4. *-commutative 84.5%

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

      5. unpow2 84.5%

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

      6. unpow2 84.5%

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

      7. swap-sqr 99.8%

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

      8. unpow2 99.8%

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

      9. *-commutative 99.8%

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

      10. *-commutative 99.8%

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

      11. fma-neg 99.8%

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

      12. metadata-eval 99.8%

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

      13. metadata-eval 99.8%

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

   6. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. unswap-sqr 84.5%

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

      3. div-inv 84.4%

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

      4. times-frac 70.3%

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

   8. Applied egg-rr 70.3%

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

   9. Taylor expanded in v around 0 60.0%

      \[\leadsto \frac{w \cdot w}{1 - v} \cdot \frac{r \cdot r}{\color{blue}{-1.7777777777777777 \cdot v - 2.6666666666666665}} + -1.5 \]

   10. Taylor expanded in v around 0 80.1%

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

       1. *-commutative 80.1%

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

       2. unpow2 80.1%

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

       3. *-commutative 80.1%

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

       4. unpow2 80.1%

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

   12. Simplified 80.1%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -8:\\ \;\;\;\;-1.5 + -0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{elif}\;r \leq 120:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + -0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \end{array} \]

Alternative 14: 46.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. sub-neg 86.6%

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

   2. associate-/l* 90.3%

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

   3. cancel-sign-sub-inv 90.3%

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

   4. metadata-eval 90.3%

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

   5. *-commutative 90.3%

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

   6. *-commutative 90.3%

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

   7. metadata-eval 90.3%

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

3. Simplified 90.3%

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

4. Taylor expanded in v around 0 71.2%

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

   1. unpow2 71.2%

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

   2. unpow2 71.2%

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

6. Simplified 71.2%

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

   1. add-sqr-sqrt 71.2%

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

   2. unswap-sqr 71.2%

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

   3. unpow2 71.2%

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

   4. sqrt-div 71.2%

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

   5. metadata-eval 71.2%

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

   6. unpow2 71.2%

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

   7. sqrt-prod 40.1%

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

   8. add-sqr-sqrt 58.8%

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

   9. *-commutative 58.8%

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

   10. unswap-sqr 68.9%

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

   11. unpow2 68.9%

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

   12. sqrt-div 69.0%

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

   13. metadata-eval 69.0%

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

   14. unpow2 69.0%

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

   15. sqrt-prod 45.7%

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

   16. add-sqr-sqrt 82.8%

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

   17. *-commutative 82.8%

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

8. Applied egg-rr 82.8%

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

9. Taylor expanded in r around 0 49.8%

   \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} + -4.5 \]
10. Step-by-step derivation

    1. unpow2 49.8%

       \[\leadsto \frac{2}{\color{blue}{r \cdot r}} + -4.5 \]

11. Simplified 49.8%

    \[\leadsto \color{blue}{\frac{2}{r \cdot r}} + -4.5 \]

12. Final simplification 49.8%

    \[\leadsto \frac{2}{r \cdot r} + -4.5 \]

Alternative 15: 56.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (-1.5d0) + (2.0d0 / (r * r))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. sub-neg 86.6%

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

   2. +-commutative 86.6%

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

   3. associate--l+ 86.6%

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

   4. associate-/l* 90.3%

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

   5. distribute-neg-frac 90.3%

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

   6. associate-/r/ 90.3%

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

   7. fma-def 90.3%

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

   8. sub-neg 90.3%

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

3. Simplified 82.4%

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

4. Taylor expanded in r around 0 60.4%

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

   1. sub-neg 60.4%

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

   2. associate-*r/ 60.4%

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

   3. metadata-eval 60.4%

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

   4. unpow2 60.4%

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

   5. metadata-eval 60.4%

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

6. Simplified 60.4%

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

7. Final simplification 60.4%

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

Reproduce

herbie shell --seed 2023182 
(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))