Rosa's TurbineBenchmark

Percentage Accurate: 84.6% → 99.8%

Time: 11.6s

Alternatives: 12

Speedup: 1.7×

• 52.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[v_, w_, r_] := N[(N[(N[Power[N[(N[(1.0 - v), $MachinePrecision] / N[(v * 0.25 + -0.375), $MachinePrecision]), $MachinePrecision], -1.0], $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 84.9%

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

   1. sub-neg 84.9%

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

   2. +-commutative 84.9%

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

   3. associate--l+ 84.9%

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

   4. associate-/l* 86.3%

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

   5. distribute-neg-frac 86.3%

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

   6. associate-/r/ 86.3%

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

   7. fma-def 86.3%

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

   8. sub-neg 86.3%

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

3. Simplified 78.3%

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

   1. fma-udef 78.3%

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

   2. unswap-sqr 99.8%

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

   3. pow2 99.8%

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

   4. div-inv 99.8%

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

   5. fma-def 99.8%

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

   6. pow2 99.8%

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

   7. pow-flip 99.9%

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

   8. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

   1. clear-num 99.9%

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

   2. inv-pow 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. sub-neg 84.9%

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

   2. +-commutative 84.9%

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

   3. associate--l+ 84.9%

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

   4. associate-/l* 86.3%

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

   5. distribute-neg-frac 86.3%

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

   6. associate-/r/ 86.3%

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

   7. fma-def 86.3%

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

   8. sub-neg 86.3%

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

3. Simplified 78.3%

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

   1. fma-udef 78.3%

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

   2. unswap-sqr 99.8%

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

   3. pow2 99.8%

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

   4. div-inv 99.8%

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

   5. fma-def 99.8%

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

   6. pow2 99.8%

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

   7. pow-flip 99.9%

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

   8. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.7% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) + ((-1.5d0) - (((r * w) ** 2.0d0) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v))))
end function

Java

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

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 - (math.pow((r * w), 2.0) * ((0.375 + (v * -0.25)) / (1.0 - v))))

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 - (((r * w) ^ 2.0) * ((0.375 + (v * -0.25)) / (1.0 - v))));
end

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. associate--l- 84.9%

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

   2. +-commutative 84.9%

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

   3. associate--l+ 84.9%

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

   4. +-commutative 84.9%

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

   5. associate--r+ 84.9%

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

   6. metadata-eval 84.9%

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

   7. associate-*l/ 86.3%

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

   8. *-commutative 86.3%

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

   9. *-commutative 86.3%

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

   10. *-commutative 86.3%

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

3. Simplified 86.3%

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

4. Taylor expanded in r around 0 78.3%

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

   1. *-commutative 78.3%

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

   2. unpow2 78.3%

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

   3. unpow2 78.3%

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

   4. swap-sqr 99.8%

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

   5. unpow2 99.8%

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

   6. *-commutative 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	double t_0 = 1.0 / (r * w);
	return (((2.0 / (r * r)) + 3.0) - ((0.125 * (3.0 + (v * -2.0))) / ((1.0 - v) * (t_0 * t_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
    real(8) :: t_0
    t_0 = 1.0d0 / (r * w)
    code = (((2.0d0 / (r * r)) + 3.0d0) - ((0.125d0 * (3.0d0 + (v * (-2.0d0)))) / ((1.0d0 - v) * (t_0 * t_0)))) + (-4.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. sub-neg 84.9%

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

   2. associate-/l* 86.3%

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

   3. cancel-sign-sub-inv 86.3%

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

   4. metadata-eval 86.3%

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

   5. *-commutative 86.3%

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

   6. *-commutative 86.3%

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

   7. metadata-eval 86.3%

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

3. Simplified 86.3%

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

   1. div-inv 86.3%

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

   2. associate-*r* 78.3%

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

   3. unswap-sqr 99.8%

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

   4. pow2 99.8%

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

5. Applied egg-rr 99.8%

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

   1. add-sqr-sqrt 99.7%

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

   2. pow2 99.7%

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

   3. sqrt-div 99.8%

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

   4. metadata-eval 99.8%

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

   5. sqrt-prod 55.3%

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

   6. add-sqr-sqrt 77.9%

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

   7. pow2 77.9%

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

   8. sqrt-div 77.9%

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

   9. metadata-eval 77.9%

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

   10. sqrt-prod 55.3%

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

   11. add-sqr-sqrt 99.7%

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

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%

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

Alternative 5: 92.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 4 \cdot 10^{-91}:\\ \;\;\;\;t_0 + \left(-1.5 + -0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 4e-91)
     (+ t_0 (+ -1.5 (* -0.375 (* w (* w (* r r))))))
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* w (* r w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 4e-91) {
		tmp = t_0 + (-1.5 + (-0.375 * (w * (w * (r * r)))));
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 4d-91) then
        tmp = t_0 + ((-1.5d0) + ((-0.375d0) * (w * (w * (r * r)))))
    else
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (w * (r * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 4e-91) {
		tmp = t_0 + (-1.5 + (-0.375 * (w * (w * (r * r)))));
	} else {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 4e-91:
		tmp = t_0 + (-1.5 + (-0.375 * (w * (w * (r * r)))))
	else:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 4e-91)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(-0.375 * Float64(w * Float64(w * Float64(r * r))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(w * Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 4e-91)
		tmp = t_0 + (-1.5 + (-0.375 * (w * (w * (r * r)))));
	else
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (w * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 4e-91], N[(t$95$0 + N[(-1.5 + N[(-0.375 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4.00000000000000009e-91

   1. Initial program 82.2%

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

      1. sub-neg 82.2%

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

      2. +-commutative 82.2%

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

      3. associate--l+ 82.2%

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

      4. associate-/l* 83.7%

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

      5. distribute-neg-frac 83.7%

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

      6. associate-/r/ 83.7%

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

      7. fma-def 83.7%

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

      8. sub-neg 83.7%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in v around 0 76.3%

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

      1. associate--l+ 76.3%

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

      2. associate-*r/ 76.3%

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

      3. metadata-eval 76.3%

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

      4. unpow2 76.3%

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

      5. *-commutative 76.3%

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

      6. fma-neg 76.3%

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

      7. *-commutative 76.3%

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

      8. unpow2 76.3%

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

      9. unpow2 76.3%

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

      10. swap-sqr 95.7%

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

      11. unpow2 95.7%

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

      12. *-commutative 95.7%

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

      13. metadata-eval 95.7%

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

   6. Simplified 95.7%

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

      1. fma-udef 95.7%

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

      2. *-commutative 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. *-commutative 95.7%

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

      2. unpow2 95.7%

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

      3. unswap-sqr 76.3%

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

      4. associate-*l* 89.3%

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

   10. Applied egg-rr 89.3%

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

   if 4.00000000000000009e-91 < r

   1. Initial program 91.8%

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

      1. associate--l- 91.8%

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

      2. +-commutative 91.8%

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

      3. associate--l+ 91.8%

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

      4. +-commutative 91.8%

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

      5. associate--r+ 91.8%

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

      6. metadata-eval 91.8%

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

      7. associate-*l/ 93.1%

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

      8. *-commutative 93.1%

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

      9. *-commutative 93.1%

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

      10. *-commutative 93.1%

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

   3. Simplified 93.1%

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

   4. Taylor expanded in r around 0 93.1%

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

      1. *-commutative 93.1%

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

      2. unpow2 93.1%

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

      3. associate-*r* 99.8%

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

      4. *-commutative 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 89.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 6 \cdot 10^{+115}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{w}{\frac{-2.6666666666666665}{r \cdot \left(r \cdot w\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 6e+115)
   (+ (/ 2.0 (* r r)) (+ -1.5 (* -0.375 (* w (* w (* r r))))))
   (+ -1.5 (/ w (/ -2.6666666666666665 (* r (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 6e+115) {
		tmp = (2.0 / (r * r)) + (-1.5 + (-0.375 * (w * (w * (r * r)))));
	} else {
		tmp = -1.5 + (w / (-2.6666666666666665 / (r * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 6d+115) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + ((-0.375d0) * (w * (w * (r * r)))))
    else
        tmp = (-1.5d0) + (w / ((-2.6666666666666665d0) / (r * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 6e+115) {
		tmp = (2.0 / (r * r)) + (-1.5 + (-0.375 * (w * (w * (r * r)))));
	} else {
		tmp = -1.5 + (w / (-2.6666666666666665 / (r * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 6e+115:
		tmp = (2.0 / (r * r)) + (-1.5 + (-0.375 * (w * (w * (r * r)))))
	else:
		tmp = -1.5 + (w / (-2.6666666666666665 / (r * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 6e+115)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(-0.375 * Float64(w * Float64(w * Float64(r * r))))));
	else
		tmp = Float64(-1.5 + Float64(w / Float64(-2.6666666666666665 / Float64(r * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 6e+115)
		tmp = (2.0 / (r * r)) + (-1.5 + (-0.375 * (w * (w * (r * r)))));
	else
		tmp = -1.5 + (w / (-2.6666666666666665 / (r * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 6e+115], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(-0.375 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(w / N[(-2.6666666666666665 / N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 6.0000000000000001e115

   1. Initial program 84.7%

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

      1. sub-neg 84.7%

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

      2. +-commutative 84.7%

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

      3. associate--l+ 84.7%

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

      4. associate-/l* 86.0%

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

      5. distribute-neg-frac 86.0%

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

      6. associate-/r/ 86.0%

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

      7. fma-def 86.0%

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

      8. sub-neg 86.0%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in v around 0 79.2%

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

      1. associate--l+ 79.2%

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

      2. associate-*r/ 79.2%

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

      3. metadata-eval 79.2%

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

      4. unpow2 79.2%

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

      5. *-commutative 79.2%

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

      6. fma-neg 79.2%

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

      7. *-commutative 79.2%

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

      8. unpow2 79.2%

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

      9. unpow2 79.2%

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

      10. swap-sqr 95.9%

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

      11. unpow2 95.9%

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

      12. *-commutative 95.9%

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

      13. metadata-eval 95.9%

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

   6. Simplified 95.9%

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

      1. fma-udef 95.9%

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

      2. *-commutative 95.9%

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

   8. Applied egg-rr 95.9%

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

      1. *-commutative 95.9%

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

      2. unpow2 95.9%

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

      3. unswap-sqr 79.2%

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

      4. associate-*l* 90.4%

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

   10. Applied egg-rr 90.4%

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

   if 6.0000000000000001e115 < r

   1. Initial program 85.8%

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

      1. sub-neg 85.8%

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

      2. +-commutative 85.8%

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

      3. associate--l+ 85.8%

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

      4. associate-/l* 88.0%

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

      5. distribute-neg-frac 88.0%

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

      6. associate-/r/ 88.1%

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

      7. fma-def 88.1%

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

      8. sub-neg 88.1%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in r around inf 63.2%

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

      1. sub-neg 63.2%

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

      2. associate-/l* 63.2%

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

      3. unpow2 63.2%

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

      4. unpow2 63.2%

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

      5. *-commutative 63.2%

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

      6. fma-neg 63.2%

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

      7. metadata-eval 63.2%

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

      8. metadata-eval 63.2%

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

   6. Simplified 63.2%

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

      1. clear-num 63.2%

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

      2. inv-pow 63.2%

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

      3. associate-*l* 63.2%

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

   8. Applied egg-rr 63.2%

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

      1. unpow-1 63.2%

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

      2. clear-num 63.2%

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

      3. associate-/l* 64.1%

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

      4. associate-/l/ 64.1%

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

      5. associate-*r* 64.1%

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

      6. *-commutative 64.1%

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

   10. Applied egg-rr 64.1%

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

   11. Taylor expanded in v around 0 64.5%

       \[\leadsto \frac{w}{\color{blue}{\frac{-2.6666666666666665}{w \cdot {r}^{2}}}} + -1.5 \]
   12. Step-by-step derivation

       1. *-commutative 64.5%

          \[\leadsto \frac{w}{\frac{-2.6666666666666665}{\color{blue}{{r}^{2} \cdot w}}} + -1.5 \]

       2. unpow2 64.5%

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

       3. associate-*r* 82.2%

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

       4. *-commutative 82.2%

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

   13. Simplified 82.2%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6 \cdot 10^{+115}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{w}{\frac{-2.6666666666666665}{r \cdot \left(r \cdot w\right)}}\\ \end{array} \]

Alternative 7: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 6 \cdot 10^{-38}:\\ \;\;\;\;t_0 + -4.5\\ \mathbf{elif}\;r \leq 1.45 \cdot 10^{-24} \lor \neg \left(r \leq 0.42\right):\\ \;\;\;\;-1.5 + \frac{w}{\frac{-4}{r \cdot \left(r \cdot w\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 6e-38)
     (+ t_0 -4.5)
     (if (or (<= r 1.45e-24) (not (<= r 0.42)))
       (+ -1.5 (/ w (/ -4.0 (* r (* r w)))))
       (+ -1.5 t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6e-38) {
		tmp = t_0 + -4.5;
	} else if ((r <= 1.45e-24) || !(r <= 0.42)) {
		tmp = -1.5 + (w / (-4.0 / (r * (r * w))));
	} 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 (r <= 6d-38) then
        tmp = t_0 + (-4.5d0)
    else if ((r <= 1.45d-24) .or. (.not. (r <= 0.42d0))) then
        tmp = (-1.5d0) + (w / ((-4.0d0) / (r * (r * w))))
    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 (r <= 6e-38) {
		tmp = t_0 + -4.5;
	} else if ((r <= 1.45e-24) || !(r <= 0.42)) {
		tmp = -1.5 + (w / (-4.0 / (r * (r * w))));
	} else {
		tmp = -1.5 + t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 6e-38:
		tmp = t_0 + -4.5
	elif (r <= 1.45e-24) or not (r <= 0.42):
		tmp = -1.5 + (w / (-4.0 / (r * (r * w))))
	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 (r <= 6e-38)
		tmp = Float64(t_0 + -4.5);
	elseif ((r <= 1.45e-24) || !(r <= 0.42))
		tmp = Float64(-1.5 + Float64(w / Float64(-4.0 / Float64(r * Float64(r * w)))));
	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 (r <= 6e-38)
		tmp = t_0 + -4.5;
	elseif ((r <= 1.45e-24) || ~((r <= 0.42)))
		tmp = -1.5 + (w / (-4.0 / (r * (r * w))));
	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[r, 6e-38], N[(t$95$0 + -4.5), $MachinePrecision], If[Or[LessEqual[r, 1.45e-24], N[Not[LessEqual[r, 0.42]], $MachinePrecision]], N[(-1.5 + N[(w / N[(-4.0 / N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.99999999999999977e-38

   1. Initial program 83.0%

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

      1. sub-neg 83.0%

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

      2. associate-/l* 84.4%

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

      3. cancel-sign-sub-inv 84.4%

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

      4. metadata-eval 84.4%

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

      5. *-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. metadata-eval 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in v around 0 70.0%

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

      1. unpow2 70.0%

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

      2. unpow2 70.0%

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

   6. Simplified 70.0%

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

   7. Taylor expanded in r around 0 58.2%

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

      1. unpow2 58.2%

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

   9. Simplified 58.2%

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

   if 5.99999999999999977e-38 < r < 1.4499999999999999e-24 or 0.419999999999999984 < r

   1. Initial program 90.5%

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

      1. sub-neg 90.5%

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

      2. +-commutative 90.5%

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

      3. associate--l+ 90.5%

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

      4. associate-/l* 92.0%

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

      5. distribute-neg-frac 92.0%

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

      6. associate-/r/ 92.0%

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

      7. fma-def 92.0%

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

      8. sub-neg 92.0%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in r around inf 70.4%

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

      1. sub-neg 70.4%

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

      2. associate-/l* 70.4%

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

      3. unpow2 70.4%

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

      4. unpow2 70.4%

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

      5. *-commutative 70.4%

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

      6. fma-neg 70.4%

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

      7. metadata-eval 70.4%

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

      8. metadata-eval 70.4%

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

   6. Simplified 70.4%

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

      1. clear-num 70.5%

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

      2. inv-pow 70.5%

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

      3. associate-*l* 70.5%

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

   8. Applied egg-rr 70.5%

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

      1. unpow-1 70.5%

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

      2. clear-num 70.4%

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

      3. associate-/l* 71.1%

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

      4. associate-/l/ 71.1%

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

      5. associate-*r* 71.1%

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

      6. *-commutative 71.1%

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

   10. Applied egg-rr 71.1%

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

   11. Taylor expanded in v around inf 65.4%

       \[\leadsto \frac{w}{\color{blue}{\frac{-4}{w \cdot {r}^{2}}}} + -1.5 \]
   12. Step-by-step derivation

       1. *-commutative 65.4%

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

       2. unpow2 65.4%

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

       3. associate-*r* 75.8%

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

       4. *-commutative 75.8%

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

   13. Simplified 75.8%

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

   if 1.4499999999999999e-24 < r < 0.419999999999999984

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.2%

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

      4. associate-/l* 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. associate-/r/ 99.2%

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

      7. fma-def 99.2%

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

      8. sub-neg 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in r around 0 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. unpow2 99.2%

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

      5. metadata-eval 99.2%

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

   6. Simplified 99.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{r \cdot r} + -4.5\\ \mathbf{elif}\;r \leq 1.45 \cdot 10^{-24} \lor \neg \left(r \leq 0.42\right):\\ \;\;\;\;-1.5 + \frac{w}{\frac{-4}{r \cdot \left(r \cdot w\right)}}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \end{array} \]

Alternative 8: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.9 \cdot 10^{-38}:\\ \;\;\;\;t_1 + -4.5\\ \mathbf{elif}\;r \leq 1.45 \cdot 10^{-24}:\\ \;\;\;\;-1.5 + \frac{w}{\frac{-4}{t_0}}\\ \mathbf{elif}\;r \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;-1.5 + t_1\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{w}{\frac{-2.6666666666666665}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (<= r 2.9e-38)
     (+ t_1 -4.5)
     (if (<= r 1.45e-24)
       (+ -1.5 (/ w (/ -4.0 t_0)))
       (if (<= r 9.5e-6)
         (+ -1.5 t_1)
         (+ -1.5 (/ w (/ -2.6666666666666665 t_0))))))))

C

double code(double v, double w, double r) {
	double t_0 = r * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.9e-38) {
		tmp = t_1 + -4.5;
	} else if (r <= 1.45e-24) {
		tmp = -1.5 + (w / (-4.0 / t_0));
	} else if (r <= 9.5e-6) {
		tmp = -1.5 + t_1;
	} else {
		tmp = -1.5 + (w / (-2.6666666666666665 / 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) :: t_1
    real(8) :: tmp
    t_0 = r * (r * w)
    t_1 = 2.0d0 / (r * r)
    if (r <= 2.9d-38) then
        tmp = t_1 + (-4.5d0)
    else if (r <= 1.45d-24) then
        tmp = (-1.5d0) + (w / ((-4.0d0) / t_0))
    else if (r <= 9.5d-6) then
        tmp = (-1.5d0) + t_1
    else
        tmp = (-1.5d0) + (w / ((-2.6666666666666665d0) / 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);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.9e-38) {
		tmp = t_1 + -4.5;
	} else if (r <= 1.45e-24) {
		tmp = -1.5 + (w / (-4.0 / t_0));
	} else if (r <= 9.5e-6) {
		tmp = -1.5 + t_1;
	} else {
		tmp = -1.5 + (w / (-2.6666666666666665 / t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.9e-38:
		tmp = t_1 + -4.5
	elif r <= 1.45e-24:
		tmp = -1.5 + (w / (-4.0 / t_0))
	elif r <= 9.5e-6:
		tmp = -1.5 + t_1
	else:
		tmp = -1.5 + (w / (-2.6666666666666665 / t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.9e-38)
		tmp = Float64(t_1 + -4.5);
	elseif (r <= 1.45e-24)
		tmp = Float64(-1.5 + Float64(w / Float64(-4.0 / t_0)));
	elseif (r <= 9.5e-6)
		tmp = Float64(-1.5 + t_1);
	else
		tmp = Float64(-1.5 + Float64(w / Float64(-2.6666666666666665 / t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.9e-38)
		tmp = t_1 + -4.5;
	elseif (r <= 1.45e-24)
		tmp = -1.5 + (w / (-4.0 / t_0));
	elseif (r <= 9.5e-6)
		tmp = -1.5 + t_1;
	else
		tmp = -1.5 + (w / (-2.6666666666666665 / t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.9e-38], N[(t$95$1 + -4.5), $MachinePrecision], If[LessEqual[r, 1.45e-24], N[(-1.5 + N[(w / N[(-4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 9.5e-6], N[(-1.5 + t$95$1), $MachinePrecision], N[(-1.5 + N[(w / N[(-2.6666666666666665 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if r < 2.89999999999999994e-38

   1. Initial program 83.0%

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

      1. sub-neg 83.0%

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

      2. associate-/l* 84.4%

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

      3. cancel-sign-sub-inv 84.4%

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

      4. metadata-eval 84.4%

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

      5. *-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. metadata-eval 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in v around 0 70.0%

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

      1. unpow2 70.0%

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

      2. unpow2 70.0%

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

   6. Simplified 70.0%

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

   7. Taylor expanded in r around 0 58.2%

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

      1. unpow2 58.2%

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

   9. Simplified 58.2%

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

   if 2.89999999999999994e-38 < r < 1.4499999999999999e-24

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate--l+ 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. associate-/r/ 100.0%

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

      7. fma-def 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in r around inf 75.0%

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

      1. sub-neg 75.0%

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

      2. associate-/l* 75.0%

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

      3. unpow2 75.0%

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

      4. unpow2 75.0%

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

      5. *-commutative 75.0%

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

      6. fma-neg 75.0%

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

      7. metadata-eval 75.0%

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

      8. metadata-eval 75.0%

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

   6. Simplified 75.0%

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

      1. clear-num 75.0%

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

      2. inv-pow 75.0%

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

      3. associate-*l* 75.0%

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

   8. Applied egg-rr 75.0%

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

      1. unpow-1 75.0%

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

      2. clear-num 75.0%

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

      3. associate-/l* 75.0%

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

      4. associate-/l/ 75.0%

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

      5. associate-*r* 75.0%

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

      6. *-commutative 75.0%

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

   10. Applied egg-rr 75.0%

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

   11. Taylor expanded in v around inf 55.3%

       \[\leadsto \frac{w}{\color{blue}{\frac{-4}{w \cdot {r}^{2}}}} + -1.5 \]
   12. Step-by-step derivation

       1. *-commutative 55.3%

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

       2. unpow2 55.3%

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

       3. associate-*r* 55.3%

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

       4. *-commutative 55.3%

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

   13. Simplified 55.3%

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

   if 1.4499999999999999e-24 < r < 9.5000000000000005e-6

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.2%

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

      4. associate-/l* 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. associate-/r/ 99.2%

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

      7. fma-def 99.2%

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

      8. sub-neg 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in r around 0 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. unpow2 99.2%

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

      5. metadata-eval 99.2%

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

   6. Simplified 99.2%

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

   if 9.5000000000000005e-6 < r

   1. Initial program 89.8%

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

      1. sub-neg 89.8%

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

      2. +-commutative 89.8%

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

      3. associate--l+ 89.8%

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

      4. associate-/l* 91.4%

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

      5. distribute-neg-frac 91.4%

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

      6. associate-/r/ 91.5%

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

      7. fma-def 91.4%

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

      8. sub-neg 91.4%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in r around inf 70.1%

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

      1. sub-neg 70.1%

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

      2. associate-/l* 70.1%

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

      3. unpow2 70.1%

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

      4. unpow2 70.1%

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

      5. *-commutative 70.1%

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

      6. fma-neg 70.1%

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

      7. metadata-eval 70.1%

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

      8. metadata-eval 70.1%

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

   6. Simplified 70.1%

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

      1. clear-num 70.1%

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

      2. inv-pow 70.1%

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

      3. associate-*l* 70.2%

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

   8. Applied egg-rr 70.2%

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

      1. unpow-1 70.2%

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

      2. clear-num 70.1%

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

      3. associate-/l* 70.9%

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

      4. associate-/l/ 70.8%

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

      5. associate-*r* 70.8%

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

      6. *-commutative 70.8%

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

   10. Applied egg-rr 70.8%

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

   11. Taylor expanded in v around 0 73.2%

       \[\leadsto \frac{w}{\color{blue}{\frac{-2.6666666666666665}{w \cdot {r}^{2}}}} + -1.5 \]
   12. Step-by-step derivation

       1. *-commutative 73.2%

          \[\leadsto \frac{w}{\frac{-2.6666666666666665}{\color{blue}{{r}^{2} \cdot w}}} + -1.5 \]

       2. unpow2 73.2%

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

       3. associate-*r* 85.8%

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

       4. *-commutative 85.8%

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

   13. Simplified 85.8%

       \[\leadsto \frac{w}{\color{blue}{\frac{-2.6666666666666665}{r \cdot \left(w \cdot r\right)}}} + -1.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.9 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{r \cdot r} + -4.5\\ \mathbf{elif}\;r \leq 1.45 \cdot 10^{-24}:\\ \;\;\;\;-1.5 + \frac{w}{\frac{-4}{r \cdot \left(r \cdot w\right)}}\\ \mathbf{elif}\;r \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{w}{\frac{-2.6666666666666665}{r \cdot \left(r \cdot w\right)}}\\ \end{array} \]

Alternative 9: 66.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 5.8 \cdot 10^{-38}:\\ \;\;\;\;t_0 + -4.5\\ \mathbf{elif}\;r \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;-1.5 + \frac{w \cdot w}{\frac{-4}{r \cdot r}}\\ \mathbf{elif}\;r \leq 1.04 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + t_0\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{w}{\frac{-2.6666666666666665}{r \cdot \left(r \cdot w\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 5.8e-38)
     (+ t_0 -4.5)
     (if (<= r 2.3e-24)
       (+ -1.5 (/ (* w w) (/ -4.0 (* r r))))
       (if (<= r 1.04e-5)
         (+ -1.5 t_0)
         (+ -1.5 (/ w (/ -2.6666666666666665 (* r (* r w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 5.8e-38) {
		tmp = t_0 + -4.5;
	} else if (r <= 2.3e-24) {
		tmp = -1.5 + ((w * w) / (-4.0 / (r * r)));
	} else if (r <= 1.04e-5) {
		tmp = -1.5 + t_0;
	} else {
		tmp = -1.5 + (w / (-2.6666666666666665 / (r * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 5.8d-38) then
        tmp = t_0 + (-4.5d0)
    else if (r <= 2.3d-24) then
        tmp = (-1.5d0) + ((w * w) / ((-4.0d0) / (r * r)))
    else if (r <= 1.04d-5) then
        tmp = (-1.5d0) + t_0
    else
        tmp = (-1.5d0) + (w / ((-2.6666666666666665d0) / (r * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 5.8e-38) {
		tmp = t_0 + -4.5;
	} else if (r <= 2.3e-24) {
		tmp = -1.5 + ((w * w) / (-4.0 / (r * r)));
	} else if (r <= 1.04e-5) {
		tmp = -1.5 + t_0;
	} else {
		tmp = -1.5 + (w / (-2.6666666666666665 / (r * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 5.8e-38:
		tmp = t_0 + -4.5
	elif r <= 2.3e-24:
		tmp = -1.5 + ((w * w) / (-4.0 / (r * r)))
	elif r <= 1.04e-5:
		tmp = -1.5 + t_0
	else:
		tmp = -1.5 + (w / (-2.6666666666666665 / (r * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 5.8e-38)
		tmp = Float64(t_0 + -4.5);
	elseif (r <= 2.3e-24)
		tmp = Float64(-1.5 + Float64(Float64(w * w) / Float64(-4.0 / Float64(r * r))));
	elseif (r <= 1.04e-5)
		tmp = Float64(-1.5 + t_0);
	else
		tmp = Float64(-1.5 + Float64(w / Float64(-2.6666666666666665 / Float64(r * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 5.8e-38)
		tmp = t_0 + -4.5;
	elseif (r <= 2.3e-24)
		tmp = -1.5 + ((w * w) / (-4.0 / (r * r)));
	elseif (r <= 1.04e-5)
		tmp = -1.5 + t_0;
	else
		tmp = -1.5 + (w / (-2.6666666666666665 / (r * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 5.8e-38], N[(t$95$0 + -4.5), $MachinePrecision], If[LessEqual[r, 2.3e-24], N[(-1.5 + N[(N[(w * w), $MachinePrecision] / N[(-4.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 1.04e-5], N[(-1.5 + t$95$0), $MachinePrecision], N[(-1.5 + N[(w / N[(-2.6666666666666665 / N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if r < 5.79999999999999988e-38

   1. Initial program 83.0%

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

      1. sub-neg 83.0%

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

      2. associate-/l* 84.4%

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

      3. cancel-sign-sub-inv 84.4%

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

      4. metadata-eval 84.4%

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

      5. *-commutative 84.4%

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

      6. *-commutative 84.4%

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

      7. metadata-eval 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in v around 0 70.0%

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

      1. unpow2 70.0%

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

      2. unpow2 70.0%

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

   6. Simplified 70.0%

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

   7. Taylor expanded in r around 0 58.2%

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

      1. unpow2 58.2%

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

   9. Simplified 58.2%

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

   if 5.79999999999999988e-38 < r < 2.3000000000000001e-24

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate--l+ 100.0%

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

      4. associate-/l* 100.0%

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

      5. distribute-neg-frac 100.0%

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

      6. associate-/r/ 100.0%

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

      7. fma-def 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in r around inf 75.0%

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

      1. sub-neg 75.0%

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

      2. associate-/l* 75.0%

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

      3. unpow2 75.0%

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

      4. unpow2 75.0%

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

      5. *-commutative 75.0%

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

      6. fma-neg 75.0%

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

      7. metadata-eval 75.0%

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

      8. metadata-eval 75.0%

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

   6. Simplified 75.0%

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

   7. Taylor expanded in v around inf 55.3%

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

      1. unpow2 55.3%

         \[\leadsto \frac{w \cdot w}{\frac{-4}{\color{blue}{r \cdot r}}} + -1.5 \]

   9. Simplified 55.3%

      \[\leadsto \frac{w \cdot w}{\color{blue}{\frac{-4}{r \cdot r}}} + -1.5 \]

   if 2.3000000000000001e-24 < r < 1.04000000000000004e-5

   1. Initial program 99.2%

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

      1. sub-neg 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.2%

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

      4. associate-/l* 99.2%

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

      5. distribute-neg-frac 99.2%

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

      6. associate-/r/ 99.2%

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

      7. fma-def 99.2%

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

      8. sub-neg 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in r around 0 99.2%

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

      1. sub-neg 99.2%

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

      2. associate-*r/ 99.2%

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

      3. metadata-eval 99.2%

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

      4. unpow2 99.2%

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

      5. metadata-eval 99.2%

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

   6. Simplified 99.2%

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

   if 1.04000000000000004e-5 < r

   1. Initial program 89.8%

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

      1. sub-neg 89.8%

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

      2. +-commutative 89.8%

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

      3. associate--l+ 89.8%

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

      4. associate-/l* 91.4%

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

      5. distribute-neg-frac 91.4%

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

      6. associate-/r/ 91.5%

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

      7. fma-def 91.4%

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

      8. sub-neg 91.4%

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

   3. Simplified 75.3%

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

   4. Taylor expanded in r around inf 70.1%

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

      1. sub-neg 70.1%

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

      2. associate-/l* 70.1%

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

      3. unpow2 70.1%

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

      4. unpow2 70.1%

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

      5. *-commutative 70.1%

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

      6. fma-neg 70.1%

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

      7. metadata-eval 70.1%

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

      8. metadata-eval 70.1%

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

   6. Simplified 70.1%

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

      1. clear-num 70.1%

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

      2. inv-pow 70.1%

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

      3. associate-*l* 70.2%

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

   8. Applied egg-rr 70.2%

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

      1. unpow-1 70.2%

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

      2. clear-num 70.1%

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

      3. associate-/l* 70.9%

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

      4. associate-/l/ 70.8%

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

      5. associate-*r* 70.8%

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

      6. *-commutative 70.8%

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

   10. Applied egg-rr 70.8%

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

   11. Taylor expanded in v around 0 73.2%

       \[\leadsto \frac{w}{\color{blue}{\frac{-2.6666666666666665}{w \cdot {r}^{2}}}} + -1.5 \]
   12. Step-by-step derivation

       1. *-commutative 73.2%

          \[\leadsto \frac{w}{\frac{-2.6666666666666665}{\color{blue}{{r}^{2} \cdot w}}} + -1.5 \]

       2. unpow2 73.2%

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

       3. associate-*r* 85.8%

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

       4. *-commutative 85.8%

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

   13. Simplified 85.8%

       \[\leadsto \frac{w}{\color{blue}{\frac{-2.6666666666666665}{r \cdot \left(w \cdot r\right)}}} + -1.5 \]
3. Recombined 4 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.8 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{r \cdot r} + -4.5\\ \mathbf{elif}\;r \leq 2.3 \cdot 10^{-24}:\\ \;\;\;\;-1.5 + \frac{w \cdot w}{\frac{-4}{r \cdot r}}\\ \mathbf{elif}\;r \leq 1.04 \cdot 10^{-5}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{w}{\frac{-2.6666666666666665}{r \cdot \left(r \cdot w\right)}}\\ \end{array} \]

Alternative 10: 93.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + -0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) + ((-1.5d0) + ((-0.375d0) * ((r * w) * (r * w))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + (-0.375 * ((r * w) * (r * w))));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + (-0.375 * ((r * w) * (r * w))))

Julia

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

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + (-0.375 * ((r * w) * (r * w))));
end

Wolfram

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

Derivation

1. Initial program 84.9%

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

   1. sub-neg 84.9%

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

   2. +-commutative 84.9%

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

   3. associate--l+ 84.9%

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

   4. associate-/l* 86.3%

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

   5. distribute-neg-frac 86.3%

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

   6. associate-/r/ 86.3%

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

   7. fma-def 86.3%

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

   8. sub-neg 86.3%

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

3. Simplified 78.3%

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

4. Taylor expanded in v around 0 76.8%

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

   1. associate--l+ 76.8%

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

   2. associate-*r/ 76.8%

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

   3. metadata-eval 76.8%

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

   4. unpow2 76.8%

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

   5. *-commutative 76.8%

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

   6. fma-neg 76.8%

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

   7. *-commutative 76.8%

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

   8. unpow2 76.8%

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

   9. unpow2 76.8%

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

   10. swap-sqr 94.6%

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

   11. unpow2 94.6%

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

   12. *-commutative 94.6%

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

   13. metadata-eval 94.6%

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

6. Simplified 94.6%

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

   1. fma-udef 94.6%

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

   2. *-commutative 94.6%

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

8. Applied egg-rr 94.6%

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

   1. unpow2 94.6%

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

10. Applied egg-rr 94.6%

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

11. Final simplification 94.6%

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

Alternative 11: 46.8% 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 84.9%

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

   1. sub-neg 84.9%

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

   2. associate-/l* 86.3%

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

   3. cancel-sign-sub-inv 86.3%

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

   4. metadata-eval 86.3%

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

   5. *-commutative 86.3%

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

   6. *-commutative 86.3%

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

   7. metadata-eval 86.3%

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

3. Simplified 86.3%

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

4. Taylor expanded in v around 0 68.3%

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

   1. unpow2 68.3%

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

   2. unpow2 68.3%

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

6. Simplified 68.3%

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

7. Taylor expanded in r around 0 46.6%

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

   1. unpow2 46.6%

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

9. Simplified 46.6%

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

10. Final simplification 46.6%

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

Alternative 12: 57.0% 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 84.9%

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

   1. sub-neg 84.9%

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

   2. +-commutative 84.9%

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

   3. associate--l+ 84.9%

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

   4. associate-/l* 86.3%

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

   5. distribute-neg-frac 86.3%

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

   6. associate-/r/ 86.3%

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

   7. fma-def 86.3%

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

   8. sub-neg 86.3%

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

3. Simplified 78.3%

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

4. Taylor expanded in r around 0 52.2%

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

   1. sub-neg 52.2%

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

   2. associate-*r/ 52.2%

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

   3. metadata-eval 52.2%

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

   4. unpow2 52.2%

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

   5. metadata-eval 52.2%

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

6. Simplified 52.2%

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

7. Final simplification 52.2%

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

Reproduce

herbie shell --seed 2023240 
(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))