Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.7%

Time: 12.2s

Alternatives: 14

Speedup: 1.4×

• 53.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.9% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (-
   (+ 3.0 (/ 2.0 (* r r)))
   (/ (* 0.125 (+ 3.0 (* -2.0 v))) (/ 1.0 (/ (* (* r w) (* r w)) (- 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))) / (1.0 / (((r * w) * (r * w)) / (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))) / (1.0d0 / (((r * w) * (r * w)) / (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))) / (1.0 / (((r * w) * (r * w)) / (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))) / (1.0 / (((r * w) * (r * w)) / (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(0.125 * Float64(3.0 + Float64(-2.0 * v))) / Float64(1.0 / Float64(Float64(Float64(r * w) * Float64(r * w)) / 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))) / (1.0 / (((r * w) * (r * w)) / (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[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 84.4%

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

3. Simplified 79.5%

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

   1. clear-num 79.5%

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

   2. inv-pow 79.5%

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

   3. unswap-sqr 99.4%

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

   4. pow2 99.4%

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

5. Applied egg-rr 99.4%

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

   1. unpow-1 99.4%

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

   2. *-commutative 99.4%

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

7. Simplified 99.4%

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

   1. unpow2 99.4%

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 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 84.4%

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

3. Simplified 79.5%

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

   1. *-un-lft-identity 79.5%

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

   2. add-sqr-sqrt 79.5%

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

   3. times-frac 79.5%

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

   4. unswap-sqr 79.5%

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

   5. sqrt-prod 45.0%

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

   6. add-sqr-sqrt 63.7%

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

   7. unswap-sqr 78.0%

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

   8. sqrt-prod 54.6%

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

   9. add-sqr-sqrt 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := w \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -118000000:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot \frac{r}{4}\right)\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \frac{0.375}{\frac{1}{r \cdot w} \cdot \frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r}{\frac{4}{t_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* w (* r w))))
   (if (<= v -118000000.0)
     (+ t_0 (- -1.5 (* t_1 (/ r 4.0))))
     (if (<= v 1.65e-25)
       (+
        -4.5
        (- (+ 3.0 t_0) (/ 0.375 (* (/ 1.0 (* r w)) (/ (- 1.0 v) (* r w))))))
       (+ t_0 (- -1.5 (/ r (/ 4.0 t_1))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = w * (r * w);
	double tmp;
	if (v <= -118000000.0) {
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	} else if (v <= 1.65e-25) {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))));
	} else {
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = w * (r * w)
    if (v <= (-118000000.0d0)) then
        tmp = t_0 + ((-1.5d0) - (t_1 * (r / 4.0d0)))
    else if (v <= 1.65d-25) then
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (0.375d0 / ((1.0d0 / (r * w)) * ((1.0d0 - v) / (r * w)))))
    else
        tmp = t_0 + ((-1.5d0) - (r / (4.0d0 / t_1)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = w * (r * w);
	double tmp;
	if (v <= -118000000.0) {
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	} else if (v <= 1.65e-25) {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))));
	} else {
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = w * (r * w)
	tmp = 0
	if v <= -118000000.0:
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)))
	elif v <= 1.65e-25:
		tmp = -4.5 + ((3.0 + t_0) - (0.375 / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))))
	else:
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(w * Float64(r * w))
	tmp = 0.0
	if (v <= -118000000.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * Float64(r / 4.0))));
	elseif (v <= 1.65e-25)
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(0.375 / Float64(Float64(1.0 / Float64(r * w)) * Float64(Float64(1.0 - v) / Float64(r * w))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r / Float64(4.0 / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = w * (r * w);
	tmp = 0.0;
	if (v <= -118000000.0)
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	elseif (v <= 1.65e-25)
		tmp = -4.5 + ((3.0 + t_0) - (0.375 / ((1.0 / (r * w)) * ((1.0 - v) / (r * w)))));
	else
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -118000000.0], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * N[(r / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.65e-25], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(0.375 / N[(N[(1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - v), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(r / N[(4.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1.18e8

   1. Initial program 80.8%

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

      1. associate--l- 80.8%

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

      2. +-commutative 80.8%

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

      3. associate--l+ 80.8%

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

      4. +-commutative 80.8%

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

      5. associate--r+ 80.8%

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

      6. metadata-eval 80.8%

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

      7. associate-*r* 80.8%

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

      8. *-commutative 80.8%

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

      9. associate-/l* 82.3%

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

      10. *-commutative 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in v around inf 83.7%

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

      1. unpow2 83.7%

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

   6. Simplified 83.7%

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

      1. associate-/r/ 83.7%

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

      2. associate-*r* 96.1%

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

      3. *-commutative 96.1%

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

      4. *-commutative 96.1%

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

   8. Applied egg-rr 96.1%

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

   if -1.18e8 < v < 1.6499999999999999e-25

   1. Initial program 86.5%

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

   3. Simplified 79.1%

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

      1. *-un-lft-identity 79.1%

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

      2. add-sqr-sqrt 79.0%

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

      3. times-frac 79.0%

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

      4. unswap-sqr 79.0%

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

      5. sqrt-prod 44.1%

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

      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 79.5%

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

      8. sqrt-prod 55.1%

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

      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. Taylor expanded in v around 0 99.7%

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

   if 1.6499999999999999e-25 < v

   1. Initial program 84.1%

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

      1. associate--l- 84.1%

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

      2. +-commutative 84.1%

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

      3. associate--l+ 84.1%

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

      4. +-commutative 84.1%

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

      5. associate--r+ 84.1%

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

      6. metadata-eval 84.1%

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

      7. associate-*r* 84.0%

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

      8. *-commutative 84.0%

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

      9. associate-/l* 85.5%

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

      10. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in v around inf 88.4%

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

      1. unpow2 88.4%

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

      2. associate-*r* 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-*l* 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -118000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{4}\right)\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375}{\frac{1}{r \cdot w} \cdot \frac{1 - v}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := w \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -100000000:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot \frac{r}{4}\right)\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \frac{0.375}{\frac{1}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r}{\frac{4}{t_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* w (* r w))))
   (if (<= v -100000000.0)
     (+ t_0 (- -1.5 (* t_1 (/ r 4.0))))
     (if (<= v 1.65e-25)
       (+
        -4.5
        (- (+ 3.0 t_0) (/ 0.375 (/ 1.0 (/ (* (* r w) (* r w)) (- 1.0 v))))))
       (+ t_0 (- -1.5 (/ r (/ 4.0 t_1))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = w * (r * w);
	double tmp;
	if (v <= -100000000.0) {
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	} else if (v <= 1.65e-25) {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	} else {
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = w * (r * w)
    if (v <= (-100000000.0d0)) then
        tmp = t_0 + ((-1.5d0) - (t_1 * (r / 4.0d0)))
    else if (v <= 1.65d-25) then
        tmp = (-4.5d0) + ((3.0d0 + t_0) - (0.375d0 / (1.0d0 / (((r * w) * (r * w)) / (1.0d0 - v)))))
    else
        tmp = t_0 + ((-1.5d0) - (r / (4.0d0 / t_1)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = w * (r * w);
	double tmp;
	if (v <= -100000000.0) {
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	} else if (v <= 1.65e-25) {
		tmp = -4.5 + ((3.0 + t_0) - (0.375 / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	} else {
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = w * (r * w)
	tmp = 0
	if v <= -100000000.0:
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)))
	elif v <= 1.65e-25:
		tmp = -4.5 + ((3.0 + t_0) - (0.375 / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))))
	else:
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(w * Float64(r * w))
	tmp = 0.0
	if (v <= -100000000.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * Float64(r / 4.0))));
	elseif (v <= 1.65e-25)
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(0.375 / Float64(1.0 / Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r / Float64(4.0 / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = w * (r * w);
	tmp = 0.0;
	if (v <= -100000000.0)
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	elseif (v <= 1.65e-25)
		tmp = -4.5 + ((3.0 + t_0) - (0.375 / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	else
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -100000000.0], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * N[(r / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.65e-25], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(0.375 / N[(1.0 / N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(r / N[(4.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -1e8

   1. Initial program 80.8%

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

      1. associate--l- 80.8%

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

      2. +-commutative 80.8%

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

      3. associate--l+ 80.8%

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

      4. +-commutative 80.8%

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

      5. associate--r+ 80.8%

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

      6. metadata-eval 80.8%

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

      7. associate-*r* 80.8%

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

      8. *-commutative 80.8%

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

      9. associate-/l* 82.3%

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

      10. *-commutative 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in v around inf 83.7%

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

      1. unpow2 83.7%

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

   6. Simplified 83.7%

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

      1. associate-/r/ 83.7%

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

      2. associate-*r* 96.1%

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

      3. *-commutative 96.1%

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

      4. *-commutative 96.1%

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

   8. Applied egg-rr 96.1%

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

   if -1e8 < v < 1.6499999999999999e-25

   1. Initial program 86.5%

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

   3. Simplified 79.1%

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

      1. clear-num 79.1%

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

      2. inv-pow 79.1%

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

      3. unswap-sqr 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in v around 0 99.7%

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

   if 1.6499999999999999e-25 < v

   1. Initial program 84.1%

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

      1. associate--l- 84.1%

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

      2. +-commutative 84.1%

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

      3. associate--l+ 84.1%

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

      4. +-commutative 84.1%

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

      5. associate--r+ 84.1%

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

      6. metadata-eval 84.1%

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

      7. associate-*r* 84.0%

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

      8. *-commutative 84.0%

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

      9. associate-/l* 85.5%

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

      10. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in v around inf 88.4%

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

      1. unpow2 88.4%

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

      2. associate-*r* 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-*l* 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -100000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{4}\right)\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.375}{\frac{1}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \end{array} \]

Alternative 5: 91.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.26 \cdot 10^{-50}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{v \cdot -0.25}{\frac{1}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{v \cdot -0.25 + 0.375}{1 - v}\right) + \frac{2}{r} \cdot \frac{1}{r}\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.26e-50)
   (+
    -4.5
    (-
     (+ 3.0 (/ 2.0 (* r r)))
     (/ (* v -0.25) (/ 1.0 (/ (* (* r w) (* r w)) (- 1.0 v))))))
   (+
    (- -1.5 (* (* r (* w (* r w))) (/ (+ (* v -0.25) 0.375) (- 1.0 v))))
    (* (/ 2.0 r) (/ 1.0 r)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.26e-50) {
		tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((v * -0.25) / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	} else {
		tmp = (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v)))) + ((2.0 / r) * (1.0 / r));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.26d-50) then
        tmp = (-4.5d0) + ((3.0d0 + (2.0d0 / (r * r))) - ((v * (-0.25d0)) / (1.0d0 / (((r * w) * (r * w)) / (1.0d0 - v)))))
    else
        tmp = ((-1.5d0) - ((r * (w * (r * w))) * (((v * (-0.25d0)) + 0.375d0) / (1.0d0 - v)))) + ((2.0d0 / r) * (1.0d0 / r))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.26e-50) {
		tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((v * -0.25) / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	} else {
		tmp = (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v)))) + ((2.0 / r) * (1.0 / r));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.26e-50:
		tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((v * -0.25) / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))))
	else:
		tmp = (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v)))) + ((2.0 / r) * (1.0 / r))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.26e-50)
		tmp = Float64(-4.5 + Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(v * -0.25) / Float64(1.0 / Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v))))));
	else
		tmp = Float64(Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(1.0 - v)))) + Float64(Float64(2.0 / r) * Float64(1.0 / r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.26e-50)
		tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((v * -0.25) / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	else
		tmp = (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v)))) + ((2.0 / r) * (1.0 / r));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.26e-50], N[(-4.5 + N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(v * -0.25), $MachinePrecision] / N[(1.0 / N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / r), $MachinePrecision] * N[(1.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.26e-50

   1. Initial program 82.1%

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

   3. Simplified 78.1%

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

      1. clear-num 78.1%

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

      2. inv-pow 78.1%

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

      3. unswap-sqr 99.8%

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

      4. pow2 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

      2. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

   10. Taylor expanded in v around inf 90.4%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{-0.25 \cdot v}}{\frac{1}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}}}\right) + -4.5 \]
   11. Step-by-step derivation

       1. *-commutative 90.4%

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

   12. Simplified 90.4%

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

   if 1.26e-50 < r

   1. Initial program 91.9%

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

      1. associate--l- 91.9%

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

      2. +-commutative 91.9%

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

      3. associate--l+ 91.9%

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

      4. +-commutative 91.9%

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

      5. associate--r+ 91.9%

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

      6. metadata-eval 91.9%

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

      7. associate-*l/ 95.1%

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

      8. *-commutative 95.1%

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

      9. *-commutative 95.1%

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

      10. *-commutative 95.1%

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

   3. Simplified 95.1%

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

      1. clear-num 13.1%

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

      2. inv-pow 13.1%

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

   5. Applied egg-rr 95.1%

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

      1. unpow-1 13.1%

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

      2. associate-/l* 13.1%

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

   7. Simplified 95.2%

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

   8. Taylor expanded in r around 0 95.2%

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

      1. unpow2 95.1%

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

      2. associate-*r* 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l* 98.3%

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

   10. Simplified 98.3%

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

       1. associate-/r/ 98.3%

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

   12. Applied egg-rr 98.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.26 \cdot 10^{-50}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{v \cdot -0.25}{\frac{1}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{v \cdot -0.25 + 0.375}{1 - v}\right) + \frac{2}{r} \cdot \frac{1}{r}\\ \end{array} \]

Alternative 6: 91.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{v \cdot -0.25}{\frac{1}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{v \cdot -0.25 + 0.375}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.05e-35)
   (+
    -4.5
    (-
     (+ 3.0 (/ 2.0 (* r r)))
     (/ (* v -0.25) (/ 1.0 (/ (* (* r w) (* r w)) (- 1.0 v))))))
   (+
    (/ 1.0 (/ r (/ 2.0 r)))
    (- -1.5 (* (* r (* w (* r w))) (/ (+ (* v -0.25) 0.375) (- 1.0 v)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.05e-35) {
		tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((v * -0.25) / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	} else {
		tmp = (1.0 / (r / (2.0 / r))) + (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2.05d-35) then
        tmp = (-4.5d0) + ((3.0d0 + (2.0d0 / (r * r))) - ((v * (-0.25d0)) / (1.0d0 / (((r * w) * (r * w)) / (1.0d0 - v)))))
    else
        tmp = (1.0d0 / (r / (2.0d0 / r))) + ((-1.5d0) - ((r * (w * (r * w))) * (((v * (-0.25d0)) + 0.375d0) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.05e-35) {
		tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((v * -0.25) / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	} else {
		tmp = (1.0 / (r / (2.0 / r))) + (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.05e-35:
		tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((v * -0.25) / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))))
	else:
		tmp = (1.0 / (r / (2.0 / r))) + (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.05e-35)
		tmp = Float64(-4.5 + Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) - Float64(Float64(v * -0.25) / Float64(1.0 / Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(1.0 - v))))));
	else
		tmp = Float64(Float64(1.0 / Float64(r / Float64(2.0 / r))) + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.05e-35)
		tmp = -4.5 + ((3.0 + (2.0 / (r * r))) - ((v * -0.25) / (1.0 / (((r * w) * (r * w)) / (1.0 - v)))));
	else
		tmp = (1.0 / (r / (2.0 / r))) + (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.05e-35], N[(-4.5 + N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(v * -0.25), $MachinePrecision] / N[(1.0 / N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(r / N[(2.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.05000000000000013e-35

   1. Initial program 82.4%

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

   3. Simplified 78.5%

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

      1. clear-num 78.5%

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

      2. inv-pow 78.5%

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

      3. unswap-sqr 99.7%

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

      4. pow2 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. unpow2 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in v around inf 90.5%

       \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{-0.25 \cdot v}}{\frac{1}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}}}\right) + -4.5 \]
   11. Step-by-step derivation

       1. *-commutative 90.5%

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

   12. Simplified 90.5%

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

   if 2.05000000000000013e-35 < r

   1. Initial program 91.5%

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

      1. associate--l- 91.5%

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

      2. +-commutative 91.5%

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

      3. associate--l+ 91.5%

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

      4. +-commutative 91.5%

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

      5. associate--r+ 91.5%

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

      6. metadata-eval 91.5%

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

      7. associate-*l/ 94.8%

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

      8. *-commutative 94.8%

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

      9. *-commutative 94.8%

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

      10. *-commutative 94.8%

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

   3. Simplified 94.9%

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

      1. clear-num 8.6%

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

      2. inv-pow 8.6%

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

   5. Applied egg-rr 94.9%

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

      1. unpow-1 8.6%

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

      2. associate-/l* 8.6%

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

   7. Simplified 94.9%

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

   8. Taylor expanded in r around 0 94.9%

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

      1. unpow2 94.9%

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

      2. associate-*r* 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-*l* 98.3%

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

   10. Simplified 98.3%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.05 \cdot 10^{-35}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{v \cdot -0.25}{\frac{1}{\frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{1 - v}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{v \cdot -0.25 + 0.375}{1 - v}\right)\\ \end{array} \]

Alternative 7: 70.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{v \cdot -0.25 + 0.375}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4e-147)
   (/ 1.0 (/ r (/ 2.0 r)))
   (+
    (/ 2.0 (* r r))
    (- -1.5 (* (* r (* w (* r w))) (/ (+ (* v -0.25) 0.375) (- 1.0 v)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4e-147) {
		tmp = 1.0 / (r / (2.0 / r));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 4d-147) then
        tmp = 1.0d0 / (r / (2.0d0 / r))
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((r * (w * (r * w))) * (((v * (-0.25d0)) + 0.375d0) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 4e-147) {
		tmp = 1.0 / (r / (2.0 / r));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 4e-147:
		tmp = 1.0 / (r / (2.0 / r))
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4e-147)
		tmp = Float64(1.0 / Float64(r / Float64(2.0 / r)));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(Float64(v * -0.25) + 0.375) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 4e-147)
		tmp = 1.0 / (r / (2.0 / r));
	else
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * (w * (r * w))) * (((v * -0.25) + 0.375) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4e-147], N[(1.0 / N[(r / N[(2.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.9999999999999999e-147

   1. Initial program 80.4%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in v around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. unpow2 74.0%

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

      4. *-commutative 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in r around 0 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. clear-num 53.9%

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

       2. inv-pow 53.9%

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

   11. Applied egg-rr 53.9%

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

       1. unpow-1 53.9%

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

       2. associate-/l* 53.9%

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

   13. Simplified 53.9%

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

   if 3.9999999999999999e-147 < r

   1. Initial program 92.8%

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

      1. associate--l- 92.8%

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

      2. +-commutative 92.8%

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

      3. associate--l+ 92.8%

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

      4. +-commutative 92.8%

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

      5. associate--r+ 92.8%

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

      6. metadata-eval 92.8%

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

      7. associate-*l/ 95.1%

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

      8. *-commutative 95.1%

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

      9. *-commutative 95.1%

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

      10. *-commutative 95.1%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in r around 0 95.2%

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

      1. unpow2 95.2%

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

      2. associate-*r* 98.7%

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

      3. *-commutative 98.7%

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

      4. associate-*l* 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{v \cdot -0.25 + 0.375}{1 - v}\right)\\ \end{array} \]

Alternative 8: 96.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := w \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -23000000:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot \frac{r}{4}\right)\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r}{\frac{\frac{2.6666666666666665}{r \cdot w}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{r}{\frac{4}{t_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (* w (* r w))))
   (if (<= v -23000000.0)
     (+ t_0 (- -1.5 (* t_1 (/ r 4.0))))
     (if (<= v 1.65e-25)
       (+ t_0 (- -1.5 (/ r (/ (/ 2.6666666666666665 (* r w)) w))))
       (+ t_0 (- -1.5 (/ r (/ 4.0 t_1))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = w * (r * w);
	double tmp;
	if (v <= -23000000.0) {
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	} else if (v <= 1.65e-25) {
		tmp = t_0 + (-1.5 - (r / ((2.6666666666666665 / (r * w)) / w)));
	} else {
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = w * (r * w)
    if (v <= (-23000000.0d0)) then
        tmp = t_0 + ((-1.5d0) - (t_1 * (r / 4.0d0)))
    else if (v <= 1.65d-25) then
        tmp = t_0 + ((-1.5d0) - (r / ((2.6666666666666665d0 / (r * w)) / w)))
    else
        tmp = t_0 + ((-1.5d0) - (r / (4.0d0 / t_1)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = w * (r * w);
	double tmp;
	if (v <= -23000000.0) {
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	} else if (v <= 1.65e-25) {
		tmp = t_0 + (-1.5 - (r / ((2.6666666666666665 / (r * w)) / w)));
	} else {
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = w * (r * w)
	tmp = 0
	if v <= -23000000.0:
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)))
	elif v <= 1.65e-25:
		tmp = t_0 + (-1.5 - (r / ((2.6666666666666665 / (r * w)) / w)))
	else:
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(w * Float64(r * w))
	tmp = 0.0
	if (v <= -23000000.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * Float64(r / 4.0))));
	elseif (v <= 1.65e-25)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r / Float64(Float64(2.6666666666666665 / Float64(r * w)) / w))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(r / Float64(4.0 / t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = w * (r * w);
	tmp = 0.0;
	if (v <= -23000000.0)
		tmp = t_0 + (-1.5 - (t_1 * (r / 4.0)));
	elseif (v <= 1.65e-25)
		tmp = t_0 + (-1.5 - (r / ((2.6666666666666665 / (r * w)) / w)));
	else
		tmp = t_0 + (-1.5 - (r / (4.0 / t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -23000000.0], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * N[(r / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1.65e-25], N[(t$95$0 + N[(-1.5 - N[(r / N[(N[(2.6666666666666665 / N[(r * w), $MachinePrecision]), $MachinePrecision] / w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(r / N[(4.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.3e7

   1. Initial program 80.8%

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

      1. associate--l- 80.8%

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

      2. +-commutative 80.8%

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

      3. associate--l+ 80.8%

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

      4. +-commutative 80.8%

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

      5. associate--r+ 80.8%

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

      6. metadata-eval 80.8%

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

      7. associate-*r* 80.8%

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

      8. *-commutative 80.8%

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

      9. associate-/l* 82.3%

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

      10. *-commutative 82.3%

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

   3. Simplified 82.3%

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

   4. Taylor expanded in v around inf 83.7%

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

      1. unpow2 83.7%

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

   6. Simplified 83.7%

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

      1. associate-/r/ 83.7%

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

      2. associate-*r* 96.1%

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

      3. *-commutative 96.1%

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

      4. *-commutative 96.1%

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

   8. Applied egg-rr 96.1%

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

   if -2.3e7 < v < 1.6499999999999999e-25

   1. Initial program 86.5%

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

      1. associate--l- 86.5%

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

      2. +-commutative 86.5%

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

      3. associate--l+ 86.5%

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

      4. +-commutative 86.5%

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

      5. associate--r+ 86.5%

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

      6. metadata-eval 86.5%

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

      7. associate-*r* 86.6%

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

      8. *-commutative 86.6%

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

      9. associate-/l* 86.6%

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

      10. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in v around 0 86.5%

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

      1. unpow2 86.5%

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

   6. Simplified 86.5%

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

   7. Taylor expanded in r around 0 86.5%

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

      1. unpow2 86.5%

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

      2. associate-*r* 92.5%

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

   9. Simplified 92.5%

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

       1. associate-/r* 95.4%

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

       2. div-inv 95.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot w} \cdot \frac{1}{w}}}\right) \]

   11. Applied egg-rr 95.4%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{2.6666666666666665}{r \cdot w} \cdot \frac{1}{w}}}\right) \]
   12. Step-by-step derivation

       1. associate-*r/ 95.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\color{blue}{\frac{\frac{2.6666666666666665}{r \cdot w} \cdot 1}{w}}}\right) \]

       2. *-rgt-identity 95.4%

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

   13. Simplified 95.4%

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

   if 1.6499999999999999e-25 < v

   1. Initial program 84.1%

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

      1. associate--l- 84.1%

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

      2. +-commutative 84.1%

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

      3. associate--l+ 84.1%

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

      4. +-commutative 84.1%

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

      5. associate--r+ 84.1%

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

      6. metadata-eval 84.1%

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

      7. associate-*r* 84.0%

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

      8. *-commutative 84.0%

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

      9. associate-/l* 85.5%

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

      10. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in v around inf 88.4%

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

      1. unpow2 88.4%

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

      2. associate-*r* 97.1%

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

      3. *-commutative 97.1%

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

      4. associate-*l* 97.1%

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

   6. Simplified 97.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -23000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{4}\right)\\ \mathbf{elif}\;v \leq 1.65 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{\frac{2.6666666666666665}{r \cdot w}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \end{array} \]

Alternative 9: 66.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.6 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.6e-145)
   (/ 1.0 (/ r (/ 2.0 r)))
   (+ (/ 2.0 (* r r)) (- -1.5 (* (* r 0.375) (* r (* w w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.6e-145) {
		tmp = 1.0 / (r / (2.0 / r));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * 0.375) * (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 <= 1.6d-145) then
        tmp = 1.0d0 / (r / (2.0d0 / r))
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((r * 0.375d0) * (r * (w * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.6e-145) {
		tmp = 1.0 / (r / (2.0 / r));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * 0.375) * (r * (w * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.6e-145:
		tmp = 1.0 / (r / (2.0 / r))
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * 0.375) * (r * (w * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.6e-145)
		tmp = Float64(1.0 / Float64(r / Float64(2.0 / r)));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(r * 0.375) * Float64(r * Float64(w * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.6e-145)
		tmp = 1.0 / (r / (2.0 / r));
	else
		tmp = (2.0 / (r * r)) + (-1.5 - ((r * 0.375) * (r * (w * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.6e-145], N[(1.0 / N[(r / N[(2.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(r * 0.375), $MachinePrecision] * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.60000000000000004e-145

   1. Initial program 80.4%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in v around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. unpow2 74.0%

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

      4. *-commutative 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in r around 0 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. clear-num 53.9%

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

       2. inv-pow 53.9%

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

   11. Applied egg-rr 53.9%

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

       1. unpow-1 53.9%

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

       2. associate-/l* 53.9%

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

   13. Simplified 53.9%

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

   if 1.60000000000000004e-145 < r

   1. Initial program 92.8%

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

      1. associate--l- 92.8%

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

      2. +-commutative 92.8%

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

      3. associate--l+ 92.8%

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

      4. +-commutative 92.8%

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

      5. associate--r+ 92.8%

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

      6. metadata-eval 92.8%

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

      7. associate-*r* 92.9%

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

      8. *-commutative 92.9%

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

      9. associate-/l* 94.0%

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

      10. *-commutative 94.0%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in v around 0 90.4%

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

      1. unpow2 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in r around 0 90.4%

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

      1. unpow2 90.4%

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

      2. associate-*r* 92.0%

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

   9. Simplified 92.0%

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

       1. associate-/r/ 92.0%

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

       2. div-inv 92.0%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \frac{1}{2.6666666666666665}\right)} \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \]

       3. metadata-eval 92.0%

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

       4. associate-*l* 90.4%

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

   11. Applied egg-rr 90.4%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.6 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot 0.375\right) \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 10: 67.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{4}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 3.8e-146)
   (/ 1.0 (/ r (/ 2.0 r)))
   (+ (/ 2.0 (* r r)) (- -1.5 (* (* w (* r w)) (/ r 4.0))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 3.8e-146) {
		tmp = 1.0 / (r / (2.0 / r));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((w * (r * 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 <= 3.8d-146) then
        tmp = 1.0d0 / (r / (2.0d0 / r))
    else
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) - ((w * (r * 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 <= 3.8e-146) {
		tmp = 1.0 / (r / (2.0 / r));
	} else {
		tmp = (2.0 / (r * r)) + (-1.5 - ((w * (r * w)) * (r / 4.0)));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 3.8e-146:
		tmp = 1.0 / (r / (2.0 / r))
	else:
		tmp = (2.0 / (r * r)) + (-1.5 - ((w * (r * w)) * (r / 4.0)))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 3.8e-146)
		tmp = Float64(1.0 / Float64(r / Float64(2.0 / r)));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(Float64(w * Float64(r * w)) * Float64(r / 4.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 3.8e-146)
		tmp = 1.0 / (r / (2.0 / r));
	else
		tmp = (2.0 / (r * r)) + (-1.5 - ((w * (r * w)) * (r / 4.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 3.8e-146], N[(1.0 / N[(r / N[(2.0 / r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 3.79999999999999994e-146

   1. Initial program 80.4%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in v around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. unpow2 74.0%

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

      4. *-commutative 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in r around 0 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. clear-num 53.9%

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

       2. inv-pow 53.9%

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

   11. Applied egg-rr 53.9%

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

       1. unpow-1 53.9%

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

       2. associate-/l* 53.9%

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

   13. Simplified 53.9%

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

   if 3.79999999999999994e-146 < r

   1. Initial program 92.8%

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

      1. associate--l- 92.8%

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

      2. +-commutative 92.8%

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

      3. associate--l+ 92.8%

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

      4. +-commutative 92.8%

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

      5. associate--r+ 92.8%

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

      6. metadata-eval 92.8%

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

      7. associate-*r* 92.9%

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

      8. *-commutative 92.9%

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

      9. associate-/l* 94.0%

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

      10. *-commutative 94.0%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in v around inf 88.5%

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

      1. unpow2 88.5%

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

   6. Simplified 88.5%

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

      1. associate-/r/ 88.5%

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

      2. associate-*r* 92.0%

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

      3. *-commutative 92.0%

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

      4. *-commutative 92.0%

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

   8. Applied egg-rr 92.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 3.8 \cdot 10^{-146}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{4}\right)\\ \end{array} \]

Alternative 11: 67.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4e-147)
   (/ 1.0 (/ r (/ 2.0 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 <= 4e-147) {
		tmp = 1.0 / (r / (2.0 / 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 <= 4d-147) then
        tmp = 1.0d0 / (r / (2.0d0 / 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 <= 4e-147) {
		tmp = 1.0 / (r / (2.0 / 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 <= 4e-147:
		tmp = 1.0 / (r / (2.0 / 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 <= 4e-147)
		tmp = Float64(1.0 / Float64(r / Float64(2.0 / 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 <= 4e-147)
		tmp = 1.0 / (r / (2.0 / 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, 4e-147], N[(1.0 / N[(r / N[(2.0 / r), $MachinePrecision]), $MachinePrecision]), $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 < 3.9999999999999999e-147

   1. Initial program 80.4%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in v around inf 74.0%

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

      1. *-commutative 74.0%

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

      2. unpow2 74.0%

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

      3. unpow2 74.0%

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

      4. *-commutative 74.0%

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

   6. Simplified 74.0%

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

   7. Taylor expanded in r around 0 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. clear-num 53.9%

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

       2. inv-pow 53.9%

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

   11. Applied egg-rr 53.9%

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

       1. unpow-1 53.9%

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

       2. associate-/l* 53.9%

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

   13. Simplified 53.9%

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

   if 3.9999999999999999e-147 < r

   1. Initial program 92.8%

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

      1. associate--l- 92.8%

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

      2. +-commutative 92.8%

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

      3. associate--l+ 92.8%

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

      4. +-commutative 92.8%

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

      5. associate--r+ 92.8%

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

      6. metadata-eval 92.8%

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

      7. associate-*r* 92.9%

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

      8. *-commutative 92.9%

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

      9. associate-/l* 94.0%

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

      10. *-commutative 94.0%

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

   3. Simplified 94.1%

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

   4. Taylor expanded in v around inf 88.5%

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

      1. unpow2 88.5%

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

      2. associate-*r* 92.0%

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

      3. *-commutative 92.0%

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

      4. associate-*l* 92.0%

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

   6. Simplified 92.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{-147}:\\ \;\;\;\;\frac{1}{\frac{r}{\frac{2}{r}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{r}{\frac{4}{w \cdot \left(r \cdot w\right)}}\right)\\ \end{array} \]

Alternative 12: 49.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 8e-6) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 8e-6) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -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 <= 8d-6) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 8e-6) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 8e-6:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 8e-6)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 8e-6)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 8e-6], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 7.99999999999999964e-6

   1. Initial program 82.6%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in v around inf 77.0%

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

      1. *-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. unpow2 77.0%

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

      4. *-commutative 77.0%

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

   6. Simplified 77.0%

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

   7. Taylor expanded in r around 0 58.5%

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

      1. unpow2 58.5%

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

   9. Simplified 58.5%

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

   if 7.99999999999999964e-6 < r

   1. Initial program 92.0%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in v around inf 77.3%

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

      1. *-commutative 77.3%

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

      2. unpow2 77.3%

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

      3. unpow2 77.3%

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

      4. *-commutative 77.3%

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

   6. Simplified 77.3%

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

   7. Taylor expanded in r around 0 29.2%

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

      1. sub-neg 29.2%

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

      2. associate-*r/ 29.2%

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

      3. metadata-eval 29.2%

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

      4. unpow2 29.2%

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

      5. metadata-eval 29.2%

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

   9. Simplified 29.2%

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

   10. Taylor expanded in r around inf 29.2%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]

Alternative 13: 56.6% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.4%

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

3. Simplified 79.5%

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

4. Taylor expanded in v around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. unpow2 77.0%

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

   3. unpow2 77.0%

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

   4. *-commutative 77.0%

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

6. Simplified 77.0%

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

7. Taylor expanded in r around 0 58.6%

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

   1. sub-neg 58.6%

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

   2. associate-*r/ 58.6%

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

   3. metadata-eval 58.6%

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

   4. unpow2 58.6%

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

   5. metadata-eval 58.6%

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

9. Simplified 58.6%

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

10. Final simplification 58.6%

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

Alternative 14: 13.6% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -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 = -1.5d0
end function

Java

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

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 84.4%

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

3. Simplified 79.5%

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

4. Taylor expanded in v around inf 77.0%

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

   1. *-commutative 77.0%

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

   2. unpow2 77.0%

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

   3. unpow2 77.0%

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

   4. *-commutative 77.0%

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

6. Simplified 77.0%

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

7. Taylor expanded in r around 0 58.6%

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

   1. sub-neg 58.6%

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

   2. associate-*r/ 58.6%

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

   3. metadata-eval 58.6%

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

   4. unpow2 58.6%

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

   5. metadata-eval 58.6%

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

9. Simplified 58.6%

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

10. Taylor expanded in r around inf 12.2%

    \[\leadsto \color{blue}{-1.5} \]

11. Final simplification 12.2%

    \[\leadsto -1.5 \]

Reproduce

herbie shell --seed 2023293 
(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))