Rosa's TurbineBenchmark

Percentage Accurate: 85.1% → 99.8%

Time: 19.5s

Alternatives: 19

Speedup: 1.3×

• 54.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 78.0%

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

   1. *-commutative 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. times-frac 78.0%

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

   4. +-commutative 78.0%

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

   5. fma-def 78.0%

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

   6. unswap-sqr 99.7%

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

   7. pow2 99.7%

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

5. Applied egg-rr 99.7%

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

   1. div-inv 99.7%

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

   2. pow2 99.7%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -4.5 + ((3.0 + (2.0 * pow(r, -2.0))) - ((0.125 * (3.0 + (-2.0 * v))) / ((1.0 / (r * w)) * ((1.0 - v) / (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 = (-4.5d0) + ((3.0d0 + (2.0d0 * (r ** (-2.0d0)))) - ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) / ((1.0d0 / (r * w)) * ((1.0d0 - v) / (r * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 78.0%

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

   1. *-un-lft-identity 78.0%

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

   2. add-sqr-sqrt 78.0%

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

   3. times-frac 78.0%

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

   4. unswap-sqr 78.0%

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

   5. sqrt-prod 45.9%

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

   6. add-sqr-sqrt 62.0%

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

   7. unswap-sqr 80.7%

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

   8. sqrt-prod 58.4%

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

   9. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. div-inv 99.7%

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

   2. pow2 99.7%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

7. Applied egg-rr 99.8%

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

8. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -4.5 + ((3.0 + (2.0 / (r * r))) - ((0.125 * (3.0 + (-2.0 * v))) / ((1.0 / (r * w)) * ((1.0 - v) / (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 = (-4.5d0) + ((3.0d0 + (2.0d0 / (r * r))) - ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) / ((1.0d0 / (r * w)) * ((1.0d0 - v) / (r * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 78.0%

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

   1. *-un-lft-identity 78.0%

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

   2. add-sqr-sqrt 78.0%

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

   3. times-frac 78.0%

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

   4. unswap-sqr 78.0%

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

   5. sqrt-prod 45.9%

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

   6. add-sqr-sqrt 62.0%

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

   7. unswap-sqr 80.7%

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

   8. sqrt-prod 58.4%

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

   9. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -4.5 + ((3.0 + ((2.0 / r) / r)) - ((0.125 * (3.0 + (-2.0 * v))) / ((1.0 / (r * w)) * ((1.0 - v) / (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 = (-4.5d0) + ((3.0d0 + ((2.0d0 / r) / r)) - ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) / ((1.0d0 / (r * w)) * ((1.0d0 - v) / (r * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 78.0%

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

   1. *-un-lft-identity 78.0%

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

   2. add-sqr-sqrt 78.0%

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

   3. times-frac 78.0%

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

   4. unswap-sqr 78.0%

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

   5. sqrt-prod 45.9%

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

   6. add-sqr-sqrt 62.0%

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

   7. unswap-sqr 80.7%

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

   8. sqrt-prod 58.4%

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

   9. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. div-inv 99.7%

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

   2. pow2 99.7%

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

   3. pow-flip 99.9%

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

   4. metadata-eval 99.9%

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

7. Applied egg-rr 99.8%

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

   1. metadata-eval 42.6%

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

   2. pow-flip 42.5%

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

   3. pow2 42.5%

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

   4. div-inv 42.5%

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

   5. associate-/r* 42.5%

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

9. Applied egg-rr 99.8%

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

10. Final simplification 99.8%

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

Alternative 5: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+220}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - 0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e+220)
     (+
      t_0
      (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* r (* w w))))))
     (+ -4.5 (- (+ 3.0 t_0) (* 0.375 (* w (* w (* r r)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+220) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((w * w) <= 2d+220) then
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (r * (w * w)))))
    else
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (0.375d0 * (w * (w * (r * r)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+220) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 2e+220:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	else:
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e+220)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(0.375 * Float64(w * Float64(w * Float64(r * r))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 2e+220)
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	else
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e+220], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(0.375 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2e220

   1. Initial program 88.8%

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

      1. associate--l- 88.8%

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

      2. +-commutative 88.8%

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

      3. associate--l+ 88.8%

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

      4. +-commutative 88.8%

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

      5. associate--r+ 88.8%

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

      6. metadata-eval 88.8%

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

      7. associate-*l/ 96.8%

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

      8. *-commutative 96.8%

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

      9. *-commutative 96.8%

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

      10. *-commutative 96.8%

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

   3. Simplified 96.8%

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

   if 2e220 < (*.f64 w w)

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in v around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow2 67.6%

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

      3. unpow2 67.6%

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

      4. *-commutative 67.6%

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

      5. swap-sqr 98.5%

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

      6. unpow2 98.5%

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

      7. *-commutative 98.5%

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

   6. Simplified 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

      3. unswap-sqr 67.6%

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

      4. associate-*l* 98.5%

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \end{array} \]

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+220}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - 0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e+220)
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))
     (+ -4.5 (- (+ 3.0 t_0) (* 0.375 (* w (* w (* r r)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+220) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((w * w) <= 2d+220) then
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * ((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v))))
    else
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (0.375d0 * (w * (w * (r * r)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+220) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 2e+220:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))))
	else:
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e+220)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(0.375 * Float64(w * Float64(w * Float64(r * r))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 2e+220)
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	else
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e+220], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(0.375 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2e220

   1. Initial program 88.8%

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

      1. associate--l- 88.8%

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

      2. +-commutative 88.8%

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

      3. associate--l+ 88.8%

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

      4. +-commutative 88.8%

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

      5. associate--r+ 88.8%

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

      6. metadata-eval 88.8%

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

      7. associate-*l/ 96.8%

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

      8. *-commutative 96.8%

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

      9. *-commutative 96.8%

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

      10. *-commutative 96.8%

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

   3. Simplified 96.8%

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

   4. Taylor expanded in r around 0 96.8%

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

      1. unpow2 96.8%

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

      2. *-commutative 96.8%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 2e220 < (*.f64 w w)

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in v around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow2 67.6%

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

      3. unpow2 67.6%

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

      4. *-commutative 67.6%

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

      5. swap-sqr 98.5%

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

      6. unpow2 98.5%

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

      7. *-commutative 98.5%

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

   6. Simplified 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

      3. unswap-sqr 67.6%

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

      4. associate-*l* 98.5%

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 97.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -2.45 \cdot 10^{+42}:\\ \;\;\;\;t_1 + \left(-1.5 - \frac{r}{\frac{4}{t_0}}\right)\\ \mathbf{elif}\;v \leq 0.13:\\ \;\;\;\;-4.5 + \left(\left(3 + t_1\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - \frac{r}{4} \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (<= v -2.45e+42)
     (+ t_1 (- -1.5 (/ r (/ 4.0 t_0))))
     (if (<= v 0.13)
       (+ -4.5 (- (+ 3.0 t_1) (* 0.375 (* (* r w) (* r w)))))
       (+ t_1 (- -1.5 (* (/ r 4.0) t_0)))))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -2.45e+42) {
		tmp = t_1 + (-1.5 - (r / (4.0 / t_0)));
	} else if (v <= 0.13) {
		tmp = -4.5 + ((3.0 + t_1) - (0.375 * ((r * w) * (r * w))));
	} else {
		tmp = t_1 + (-1.5 - ((r / 4.0) * 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 = w * (r * w)
    t_1 = 2.0d0 / (r * r)
    if (v <= (-2.45d+42)) then
        tmp = t_1 + ((-1.5d0) - (r / (4.0d0 / t_0)))
    else if (v <= 0.13d0) then
        tmp = (-4.5d0) + ((3.0d0 + t_1) - (0.375d0 * ((r * w) * (r * w))))
    else
        tmp = t_1 + ((-1.5d0) - ((r / 4.0d0) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -2.45e+42) {
		tmp = t_1 + (-1.5 - (r / (4.0 / t_0)));
	} else if (v <= 0.13) {
		tmp = -4.5 + ((3.0 + t_1) - (0.375 * ((r * w) * (r * w))));
	} else {
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= -2.45e+42:
		tmp = t_1 + (-1.5 - (r / (4.0 / t_0)))
	elif v <= 0.13:
		tmp = -4.5 + ((3.0 + t_1) - (0.375 * ((r * w) * (r * w))))
	else:
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -2.45e+42)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(r / Float64(4.0 / t_0))));
	elseif (v <= 0.13)
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_1) - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(r / 4.0) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -2.45e+42)
		tmp = t_1 + (-1.5 - (r / (4.0 / t_0)));
	elseif (v <= 0.13)
		tmp = -4.5 + ((3.0 + t_1) - (0.375 * ((r * w) * (r * w))));
	else
		tmp = t_1 + (-1.5 - ((r / 4.0) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -2.45e+42], N[(t$95$1 + N[(-1.5 - N[(r / N[(4.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 0.13], N[(-4.5 + N[(N[(3.0 + t$95$1), $MachinePrecision] - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(N[(r / 4.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.4500000000000001e42

   1. Initial program 80.5%

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

      1. associate--l- 80.5%

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

      2. +-commutative 80.5%

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

      3. associate--l+ 80.5%

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

      4. +-commutative 80.5%

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

      5. associate--r+ 80.5%

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

      6. metadata-eval 80.5%

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

      7. associate-*r* 80.5%

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

      8. *-commutative 80.5%

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

      9. associate-/l* 86.4%

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

      10. *-commutative 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in v around inf 89.2%

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

      1. unpow2 89.2%

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

      2. *-commutative 89.2%

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

      3. associate-*l* 96.9%

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

      4. *-commutative 96.9%

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

   6. Simplified 96.9%

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

   if -2.4500000000000001e42 < v < 0.13

   1. Initial program 84.3%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in v around 0 80.5%

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

      1. *-commutative 80.5%

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

      2. unpow2 80.5%

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

      3. unpow2 80.5%

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

      4. *-commutative 80.5%

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

      5. swap-sqr 99.0%

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

      6. unpow2 99.0%

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

      7. *-commutative 99.0%

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

   6. Simplified 99.0%

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

      1. *-commutative 99.0%

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

      2. unpow2 99.0%

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

   8. Applied egg-rr 99.0%

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

   if 0.13 < v

   1. Initial program 75.6%

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

      1. associate--l- 75.6%

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

      2. +-commutative 75.6%

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

      3. associate--l+ 75.6%

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

      4. +-commutative 75.6%

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

      5. associate--r+ 75.6%

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

      6. metadata-eval 75.6%

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

      7. associate-*r* 74.1%

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

      8. *-commutative 74.1%

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

      9. associate-/l* 82.7%

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

      10. *-commutative 82.7%

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

   3. Simplified 82.7%

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

   4. Taylor expanded in v around inf 86.3%

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

      1. unpow2 86.3%

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

      2. *-commutative 86.3%

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

      3. associate-*l* 93.9%

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

      4. *-commutative 93.9%

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

   6. Simplified 93.9%

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

      1. associate-/r/ 94.0%

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

      2. *-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.45 \cdot 10^{+42}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{elif}\;v \leq 0.13:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 8: 93.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+220}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - 0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 2e+220)
     (+ t_0 (- -1.5 (/ r (/ 4.0 (* w (* r w))))))
     (+ -4.5 (- (+ 3.0 t_0) (* 0.375 (* w (* w (* r r)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+220) {
		tmp = t_0 + (-1.5 - (r / (4.0 / (w * (r * w)))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((w * w) <= 2d+220) then
        tmp = t_0 + ((-1.5d0) - (r / (4.0d0 / (w * (r * w)))))
    else
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (0.375d0 * (w * (w * (r * r)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 2e+220) {
		tmp = t_0 + (-1.5 - (r / (4.0 / (w * (r * w)))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 2e+220:
		tmp = t_0 + (-1.5 - (r / (4.0 / (w * (r * w)))))
	else:
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 2e+220)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r / Float64(4.0 / Float64(w * Float64(r * w))))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(0.375 * Float64(w * Float64(w * Float64(r * r))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 2e+220)
		tmp = t_0 + (-1.5 - (r / (4.0 / (w * (r * w)))));
	else
		tmp = -4.5 + ((3.0 + t_0) - (0.375 * (w * (w * (r * r)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 2e+220], N[(t$95$0 + N[(-1.5 - N[(r / N[(4.0 / N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(0.375 * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2e220

   1. Initial program 88.8%

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

      1. associate--l- 88.8%

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

      2. +-commutative 88.8%

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

      3. associate--l+ 88.8%

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

      4. +-commutative 88.8%

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

      5. associate--r+ 88.8%

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

      6. metadata-eval 88.8%

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

      7. associate-*r* 88.8%

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

      8. *-commutative 88.8%

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

      9. associate-/l* 94.5%

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

      10. *-commutative 94.5%

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

   3. Simplified 94.5%

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

   4. Taylor expanded in v around inf 92.4%

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

      1. unpow2 92.4%

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

      2. *-commutative 92.4%

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

      3. associate-*l* 93.7%

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

      4. *-commutative 93.7%

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

   6. Simplified 93.7%

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

   if 2e220 < (*.f64 w w)

   1. Initial program 67.6%

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

   3. Simplified 67.6%

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

   4. Taylor expanded in v around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. unpow2 67.6%

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

      3. unpow2 67.6%

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

      4. *-commutative 67.6%

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

      5. swap-sqr 98.5%

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

      6. unpow2 98.5%

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

      7. *-commutative 98.5%

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

   6. Simplified 98.5%

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

      1. *-commutative 98.5%

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

      2. unpow2 98.5%

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

      3. unswap-sqr 67.6%

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

      4. associate-*l* 98.5%

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

   8. Applied egg-rr 98.5%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - 0.375 \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\right)\\ \end{array} \]

Alternative 9: 68.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.75 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.75e-141)
   (/ (/ 2.0 r) r)
   (+ (/ 2.0 (* r r)) (- -1.5 (* (* w (* r w)) (/ r 2.6666666666666665))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.75e-141) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.75d-141) then
        tmp = (2.0d0 / r) / r
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((w * (r * w)) * (r / 2.6666666666666665d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.75e-141) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.75e-141:
		tmp = (2.0 / r) / r
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.75e-141)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(w * Float64(r * w)) * Float64(r / 2.6666666666666665))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.75e-141)
		tmp = (2.0 / r) / r;
	else
		tmp = (2.0 / (r * r)) + (-1.5 - ((w * (r * w)) * (r / 2.6666666666666665)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.75e-141], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r / 2.6666666666666665), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.7500000000000001e-141

   1. Initial program 77.6%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in v around inf 73.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. unpow2 73.5%

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

      4. *-commutative 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in r around 0 53.9%

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

      1. unpow2 53.9%

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

   9. Simplified 53.9%

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

       1. div-inv 99.8%

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

       2. pow2 99.8%

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

       3. pow-flip 99.9%

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

       4. metadata-eval 99.9%

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

   11. Applied egg-rr 54.0%

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

       1. metadata-eval 54.0%

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

       2. pow-flip 53.9%

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

       3. pow2 53.9%

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

       4. div-inv 53.9%

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

       5. associate-/r* 53.9%

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

   13. Applied egg-rr 53.9%

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

   if 1.7500000000000001e-141 < r

   1. Initial program 88.6%

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

      1. associate--l- 88.6%

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

      2. +-commutative 88.6%

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

      3. associate--l+ 88.6%

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

      4. +-commutative 88.6%

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

      5. associate--r+ 88.6%

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

      6. metadata-eval 88.6%

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

      7. associate-*r* 88.6%

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

      8. *-commutative 88.6%

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

      9. associate-/l* 91.8%

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

      10. *-commutative 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in v around 0 85.4%

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

      1. unpow2 85.4%

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

      2. *-commutative 85.4%

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

      3. associate-*l* 88.6%

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

      4. *-commutative 88.6%

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

   6. Simplified 88.6%

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

      1. associate-/r/ 88.7%

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

      2. *-commutative 88.7%

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

   8. Applied egg-rr 88.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.75 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{2.6666666666666665}\right)\\ \end{array} \]

Alternative 10: 67.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 10^{-142}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot w\right)\right) \cdot \frac{r}{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1e-142)
   (/ (/ 2.0 r) r)
   (+ (/ 2.0 (* r r)) (- -1.5 (* (* r (* w w)) (/ r 4.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1e-142) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * w)) * (r / 4.0)));
	}
	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) :: tmp
    if (r <= 1d-142) then
        tmp = (2.0d0 / r) / r
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((r * (w * w)) * (r / 4.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1e-142) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * w)) * (r / 4.0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1e-142:
		tmp = (2.0 / r) / r
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * w)) * (r / 4.0)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1e-142)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(r * Float64(w * w)) * Float64(r / 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1e-142)
		tmp = (2.0 / r) / r;
	else
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * w)) * (r / 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1e-142], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision] * N[(r / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1e-142

   1. Initial program 77.6%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in v around inf 73.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. unpow2 73.5%

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

      4. *-commutative 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in r around 0 53.9%

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

      1. unpow2 53.9%

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

   9. Simplified 53.9%

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

       1. div-inv 99.8%

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

       2. pow2 99.8%

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

       3. pow-flip 99.9%

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

       4. metadata-eval 99.9%

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

   11. Applied egg-rr 54.0%

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

       1. metadata-eval 54.0%

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

       2. pow-flip 53.9%

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

       3. pow2 53.9%

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

       4. div-inv 53.9%

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

       5. associate-/r* 53.9%

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

   13. Applied egg-rr 53.9%

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

   if 1e-142 < r

   1. Initial program 88.6%

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

      1. associate--l- 88.6%

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

      2. +-commutative 88.6%

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

      3. associate--l+ 88.6%

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

      4. +-commutative 88.6%

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

      5. associate--r+ 88.6%

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

      6. metadata-eval 88.6%

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

      7. associate-*r* 88.6%

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

      8. *-commutative 88.6%

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

      9. associate-/l* 91.8%

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

      10. *-commutative 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in v around inf 89.1%

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

      1. unpow2 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*l* 91.6%

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

      4. *-commutative 91.6%

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

   6. Simplified 91.6%

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

      1. associate-/r/ 91.5%

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

      2. *-commutative 91.5%

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

   8. Applied egg-rr 91.5%

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

   9. Taylor expanded in w around 0 89.1%

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

       1. unpow2 89.1%

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

   11. Simplified 89.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10^{-142}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot w\right)\right) \cdot \frac{r}{4}\right)\\ \end{array} \]

Alternative 11: 68.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2e-141)
   (/ (/ 2.0 r) r)
   (+ (/ 2.0 (* r r)) (- -1.5 (* (/ r 4.0) (* w (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-141) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r / 4.0) * (w * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2d-141) then
        tmp = (2.0d0 / r) / r
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((r / 4.0d0) * (w * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2e-141) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r / 4.0) * (w * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2e-141:
		tmp = (2.0 / r) / r
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - ((r / 4.0) * (w * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2e-141)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(r / 4.0) * Float64(w * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2e-141)
		tmp = (2.0 / r) / r;
	else
		tmp = (2.0 / (r * r)) + (-1.5 - ((r / 4.0) * (w * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2e-141], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(r / 4.0), $MachinePrecision] * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.0000000000000001e-141

   1. Initial program 77.6%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in v around inf 73.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. unpow2 73.5%

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

      4. *-commutative 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in r around 0 53.9%

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

      1. unpow2 53.9%

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

   9. Simplified 53.9%

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

       1. div-inv 99.8%

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

       2. pow2 99.8%

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

       3. pow-flip 99.9%

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

       4. metadata-eval 99.9%

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

   11. Applied egg-rr 54.0%

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

       1. metadata-eval 54.0%

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

       2. pow-flip 53.9%

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

       3. pow2 53.9%

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

       4. div-inv 53.9%

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

       5. associate-/r* 53.9%

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

   13. Applied egg-rr 53.9%

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

   if 2.0000000000000001e-141 < r

   1. Initial program 88.6%

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

      1. associate--l- 88.6%

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

      2. +-commutative 88.6%

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

      3. associate--l+ 88.6%

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

      4. +-commutative 88.6%

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

      5. associate--r+ 88.6%

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

      6. metadata-eval 88.6%

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

      7. associate-*r* 88.6%

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

      8. *-commutative 88.6%

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

      9. associate-/l* 91.8%

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

      10. *-commutative 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in v around inf 89.1%

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

      1. unpow2 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*l* 91.6%

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

      4. *-commutative 91.6%

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

   6. Simplified 91.6%

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

      1. associate-/r/ 91.5%

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

      2. *-commutative 91.5%

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

   8. Applied egg-rr 91.5%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{4} \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 12: 68.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.5e-141)
   (/ (/ 2.0 r) r)
   (+ (/ 2.0 (* r r)) (- -1.5 (/ r (/ 4.0 (* w (* r w))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.5e-141) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2.5d-141) then
        tmp = (2.0d0 / r) / r
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - (r / (4.0d0 / (w * (r * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.5e-141) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.5e-141:
		tmp = (2.0 / r) / r
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - (r / (4.0 / (w * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.5e-141)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(r / Float64(4.0 / Float64(w * Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.5e-141)
		tmp = (2.0 / r) / r;
	else
		tmp = (2.0 / (r * r)) + (-1.5 - (r / (4.0 / (w * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.5e-141], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(r / N[(4.0 / N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.5e-141

   1. Initial program 77.6%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in v around inf 73.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. unpow2 73.5%

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

      4. *-commutative 73.5%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in r around 0 53.9%

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

      1. unpow2 53.9%

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

   9. Simplified 53.9%

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

       1. div-inv 99.8%

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

       2. pow2 99.8%

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

       3. pow-flip 99.9%

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

       4. metadata-eval 99.9%

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

   11. Applied egg-rr 54.0%

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

       1. metadata-eval 54.0%

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

       2. pow-flip 53.9%

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

       3. pow2 53.9%

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

       4. div-inv 53.9%

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

       5. associate-/r* 53.9%

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

   13. Applied egg-rr 53.9%

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

   if 2.5e-141 < r

   1. Initial program 88.6%

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

      1. associate--l- 88.6%

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

      2. +-commutative 88.6%

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

      3. associate--l+ 88.6%

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

      4. +-commutative 88.6%

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

      5. associate--r+ 88.6%

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

      6. metadata-eval 88.6%

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

      7. associate-*r* 88.6%

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

      8. *-commutative 88.6%

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

      9. associate-/l* 91.8%

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

      10. *-commutative 91.8%

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

   3. Simplified 91.8%

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

   4. Taylor expanded in v around inf 89.1%

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

      1. unpow2 89.1%

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

      2. *-commutative 89.1%

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

      3. associate-*l* 91.6%

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

      4. *-commutative 91.6%

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

   6. Simplified 91.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \end{array} \]

Alternative 13: 58.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.15 \cdot 10^{+75} \lor \neg \left(r \leq 9.2 \cdot 10^{+95}\right):\\ \;\;\;\;\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.02e-7)
   (/ (/ 2.0 r) r)
   (if (or (<= r 1.15e+75) (not (<= r 9.2e+95)))
     (* (* (* r r) (* w w)) -0.375)
     (+ (/ 2.0 (* r r)) -1.5))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.02e-7) {
		tmp = (2.0 / r) / r;
	} else if ((r <= 1.15e+75) || !(r <= 9.2e+95)) {
		tmp = ((r * r) * (w * w)) * -0.375;
	} else {
		tmp = (2.0 / (r * r)) + -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.02d-7) then
        tmp = (2.0d0 / r) / r
    else if ((r <= 1.15d+75) .or. (.not. (r <= 9.2d+95))) then
        tmp = ((r * r) * (w * w)) * (-0.375d0)
    else
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.02e-7) {
		tmp = (2.0 / r) / r;
	} else if ((r <= 1.15e+75) || !(r <= 9.2e+95)) {
		tmp = ((r * r) * (w * w)) * -0.375;
	} else {
		tmp = (2.0 / (r * r)) + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.02e-7:
		tmp = (2.0 / r) / r
	elif (r <= 1.15e+75) or not (r <= 9.2e+95):
		tmp = ((r * r) * (w * w)) * -0.375
	else:
		tmp = (2.0 / (r * r)) + -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.02e-7)
		tmp = Float64(Float64(2.0 / r) / r);
	elseif ((r <= 1.15e+75) || !(r <= 9.2e+95))
		tmp = Float64(Float64(Float64(r * r) * Float64(w * w)) * -0.375);
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.02e-7)
		tmp = (2.0 / r) / r;
	elseif ((r <= 1.15e+75) || ~((r <= 9.2e+95)))
		tmp = ((r * r) * (w * w)) * -0.375;
	else
		tmp = (2.0 / (r * r)) + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.02e-7], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], If[Or[LessEqual[r, 1.15e+75], N[Not[LessEqual[r, 9.2e+95]], $MachinePrecision]], N[(N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision] * -0.375), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.02e-7

   1. Initial program 79.2%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in v around inf 75.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 75.6%

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

      2. unpow2 75.6%

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

      3. unpow2 75.6%

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

      4. *-commutative 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in r around 0 56.3%

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

      1. unpow2 56.3%

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

   9. Simplified 56.3%

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

       1. div-inv 99.7%

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

       2. pow2 99.7%

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

       3. pow-flip 99.9%

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

       4. metadata-eval 99.9%

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

   11. Applied egg-rr 56.5%

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

       1. metadata-eval 56.5%

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

       2. pow-flip 56.3%

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

       3. pow2 56.3%

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

       4. div-inv 56.3%

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

       5. associate-/r* 56.4%

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

   13. Applied egg-rr 56.4%

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

   if 1.02e-7 < r < 1.1499999999999999e75 or 9.19999999999999989e95 < r

   1. Initial program 86.7%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in v around 0 73.8%

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

      1. *-commutative 73.8%

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

      2. unpow2 73.8%

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

      3. unpow2 73.8%

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

      4. *-commutative 73.8%

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

      5. swap-sqr 88.0%

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

      6. unpow2 88.0%

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

      7. *-commutative 88.0%

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

   6. Simplified 88.0%

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

      1. *-commutative 88.0%

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

      2. unpow2 88.0%

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

   8. Applied egg-rr 88.0%

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

   9. Taylor expanded in r around inf 66.4%

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

       1. *-commutative 66.4%

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

       2. unpow2 66.4%

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

       3. *-commutative 66.4%

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

       4. unpow2 66.4%

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

   11. Simplified 66.4%

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

   if 1.1499999999999999e75 < r < 9.19999999999999989e95

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in v around inf 100.0%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. unpow2 100.0%

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

      4. *-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in r around 0 100.0%

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

      1. sub-neg 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. unpow2 100.0%

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

      5. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.02 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.15 \cdot 10^{+75} \lor \neg \left(r \leq 9.2 \cdot 10^{+95}\right):\\ \;\;\;\;\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]

Alternative 14: 58.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4e-6) (/ (/ 2.0 r) r) (* -0.25 (* (* r r) (* w w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4e-6) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -0.25 * ((r * r) * (w * w));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 4d-6) then
        tmp = (2.0d0 / r) / r
    else
        tmp = (-0.25d0) * ((r * r) * (w * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 4e-6) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -0.25 * ((r * r) * (w * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 4e-6:
		tmp = (2.0 / r) / r
	else:
		tmp = -0.25 * ((r * r) * (w * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4e-6)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(-0.25 * Float64(Float64(r * r) * Float64(w * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 4e-6)
		tmp = (2.0 / r) / r;
	else
		tmp = -0.25 * ((r * r) * (w * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4e-6], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(-0.25 * N[(N[(r * r), $MachinePrecision] * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.99999999999999982e-6

   1. Initial program 79.2%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in v around inf 75.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 75.6%

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

      2. unpow2 75.6%

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

      3. unpow2 75.6%

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

      4. *-commutative 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in r around 0 56.3%

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

      1. unpow2 56.3%

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

   9. Simplified 56.3%

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

       1. div-inv 99.7%

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

       2. pow2 99.7%

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

       3. pow-flip 99.9%

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

       4. metadata-eval 99.9%

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

   11. Applied egg-rr 56.5%

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

       1. metadata-eval 56.5%

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

       2. pow-flip 56.3%

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

       3. pow2 56.3%

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

       4. div-inv 56.3%

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

       5. associate-/r* 56.4%

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

   13. Applied egg-rr 56.4%

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

   if 3.99999999999999982e-6 < r

   1. Initial program 86.9%

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

      1. associate--l- 86.9%

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

      2. +-commutative 86.9%

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

      3. associate--l+ 86.9%

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

      4. +-commutative 86.9%

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

      5. associate--r+ 86.9%

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

      6. metadata-eval 86.9%

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

      7. associate-*r* 86.9%

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

      8. *-commutative 86.9%

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

      9. associate-/l* 91.1%

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

      10. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in v around inf 87.5%

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

      1. unpow2 87.5%

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

      2. *-commutative 87.5%

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

      3. associate-*l* 90.7%

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

      4. *-commutative 90.7%

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

   6. Simplified 90.7%

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

      1. associate-/r/ 90.7%

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

      2. *-commutative 90.7%

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

   8. Applied egg-rr 90.7%

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

   9. Taylor expanded in r around inf 68.2%

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

       1. *-commutative 68.2%

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

       2. unpow2 68.2%

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

       3. *-commutative 68.2%

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

       4. unpow2 68.2%

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

   11. Simplified 68.2%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \end{array} \]

Alternative 15: 45.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.00031:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.00031) (/ 2.0 (* r r)) -4.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00031) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.00031d0) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00031) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.00031:
		tmp = 2.0 / (r * r)
	else:
		tmp = -4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.00031)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -4.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.00031)
		tmp = 2.0 / (r * r);
	else
		tmp = -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.00031], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -4.5]

Derivation

1. Split input into 2 regimes

2. if r < 3.1e-4

   1. Initial program 79.2%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in v around inf 75.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 75.6%

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

      2. unpow2 75.6%

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

      3. unpow2 75.6%

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

      4. *-commutative 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in r around 0 56.3%

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

      1. unpow2 56.3%

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

   9. Simplified 56.3%

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

   if 3.1e-4 < r

   1. Initial program 86.9%

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

   3. Simplified 79.1%

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

      1. *-commutative 79.1%

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

      2. *-un-lft-identity 79.1%

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

      3. times-frac 79.1%

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

      4. +-commutative 79.1%

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

      5. fma-def 79.1%

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

      6. unswap-sqr 99.8%

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

      7. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 59.7%

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

      1. associate-*r/ 59.7%

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

      2. *-commutative 59.7%

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

      3. unpow2 59.7%

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

      4. unpow2 59.7%

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

      5. swap-sqr 70.2%

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

      6. unpow2 70.2%

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

      7. associate-/l* 70.2%

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

      8. *-commutative 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in r around 0 6.0%

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

       1. unpow2 6.0%

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

   11. Simplified 6.0%

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

   12. Taylor expanded in r around inf 6.0%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.00031:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-4.5\\ \end{array} \]

Alternative 16: 45.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.00031:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 0.00031) (/ (/ 2.0 r) r) -4.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00031) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.00031d0) then
        tmp = (2.0d0 / r) / r
    else
        tmp = -4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.00031) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.00031:
		tmp = (2.0 / r) / r
	else:
		tmp = -4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.00031)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = -4.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.00031)
		tmp = (2.0 / r) / r;
	else
		tmp = -4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.00031], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], -4.5]

Derivation

1. Split input into 2 regimes

2. if r < 3.1e-4

   1. Initial program 79.2%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in v around inf 75.6%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
   5. Step-by-step derivation

      1. *-commutative 75.6%

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

      2. unpow2 75.6%

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

      3. unpow2 75.6%

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

      4. *-commutative 75.6%

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

   6. Simplified 75.6%

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

   7. Taylor expanded in r around 0 56.3%

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

      1. unpow2 56.3%

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

   9. Simplified 56.3%

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

       1. div-inv 99.7%

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

       2. pow2 99.7%

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

       3. pow-flip 99.9%

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

       4. metadata-eval 99.9%

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

   11. Applied egg-rr 56.5%

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

       1. metadata-eval 56.5%

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

       2. pow-flip 56.3%

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

       3. pow2 56.3%

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

       4. div-inv 56.3%

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

       5. associate-/r* 56.4%

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

   13. Applied egg-rr 56.4%

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

   if 3.1e-4 < r

   1. Initial program 86.9%

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

   3. Simplified 79.1%

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

      1. *-commutative 79.1%

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

      2. *-un-lft-identity 79.1%

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

      3. times-frac 79.1%

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

      4. +-commutative 79.1%

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

      5. fma-def 79.1%

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

      6. unswap-sqr 99.8%

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

      7. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in v around inf 59.7%

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

      1. associate-*r/ 59.7%

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

      2. *-commutative 59.7%

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

      3. unpow2 59.7%

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

      4. unpow2 59.7%

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

      5. swap-sqr 70.2%

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

      6. unpow2 70.2%

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

      7. associate-/l* 70.2%

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

      8. *-commutative 70.2%

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

   8. Simplified 70.2%

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

   9. Taylor expanded in r around 0 6.0%

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

       1. unpow2 6.0%

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

   11. Simplified 6.0%

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

   12. Taylor expanded in r around inf 6.0%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.00031:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-4.5\\ \end{array} \]

Alternative 17: 46.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 78.0%

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

   1. *-commutative 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. times-frac 78.0%

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

   4. +-commutative 78.0%

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

   5. fma-def 78.0%

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

   6. unswap-sqr 99.7%

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

   7. pow2 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in v around inf 66.2%

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

   1. associate-*r/ 66.2%

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

   2. *-commutative 66.2%

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

   3. unpow2 66.2%

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

   4. unpow2 66.2%

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

   5. swap-sqr 83.1%

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

   6. unpow2 83.1%

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

   7. associate-/l* 83.1%

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

   8. *-commutative 83.1%

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

8. Simplified 83.1%

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

9. Taylor expanded in r around 0 45.2%

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

    1. unpow2 45.2%

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

11. Simplified 45.2%

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

12. Final simplification 45.2%

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

Alternative 18: 57.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.2%

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

3. Simplified 78.0%

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

4. Taylor expanded in v around inf 75.7%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({r}^{2} \cdot {w}^{2}\right)}\right) + -4.5 \]
5. Step-by-step derivation

   1. *-commutative 75.7%

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

   2. unpow2 75.7%

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

   3. unpow2 75.7%

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

   4. *-commutative 75.7%

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

6. Simplified 75.7%

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

7. Taylor expanded in r around 0 54.3%

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

   1. sub-neg 54.3%

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

   2. associate-*r/ 54.3%

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

   3. metadata-eval 54.3%

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

   4. unpow2 54.3%

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

   5. metadata-eval 54.3%

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

9. Simplified 54.3%

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

10. Final simplification 54.3%

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

Alternative 19: 4.3% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -4.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -4.5)

C

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

Java

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

Python

def code(v, w, r):
	return -4.5

Julia

function code(v, w, r)
	return -4.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -4.5;
end

Wolfram

code[v_, w_, r_] := -4.5

Derivation

1. Initial program 81.2%

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

3. Simplified 78.0%

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

   1. *-commutative 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. times-frac 78.0%

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

   4. +-commutative 78.0%

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

   5. fma-def 78.0%

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

   6. unswap-sqr 99.7%

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

   7. pow2 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in v around inf 66.2%

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

   1. associate-*r/ 66.2%

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

   2. *-commutative 66.2%

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

   3. unpow2 66.2%

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

   4. unpow2 66.2%

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

   5. swap-sqr 83.1%

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

   6. unpow2 83.1%

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

   7. associate-/l* 83.1%

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

   8. *-commutative 83.1%

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

8. Simplified 83.1%

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

9. Taylor expanded in r around 0 45.2%

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

    1. unpow2 45.2%

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

11. Simplified 45.2%

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

12. Taylor expanded in r around inf 4.2%

    \[\leadsto \color{blue}{-4.5} \]

13. Final simplification 4.2%

    \[\leadsto -4.5 \]

Reproduce

herbie shell --seed 2023285 
(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))