Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 99.3%

Time: 18.1s

Alternatives: 13

Speedup: 1.0×

• 53.6% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

   1. associate--l- 86.6%

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

   2. associate-*l* 82.3%

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

   3. sqr-neg 82.3%

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

   4. associate-*l* 86.6%

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

   5. associate-/l* 90.2%

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

   6. fma-define 90.2%

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

3. Simplified 90.2%

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

   1. associate-/l* 89.5%

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

   2. *-commutative 89.5%

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

   3. associate-*r/ 89.5%

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

   4. associate-*l* 96.8%

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

   5. associate-*r* 99.8%

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

   6. add-sqr-sqrt 50.3%

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

   7. sqrt-prod 78.0%

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

   8. add-sqr-sqrt 35.4%

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

   9. sqrt-prod 67.8%

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

   10. sqrt-prod 67.8%

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

   11. associate-*r* 73.6%

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

   12. *-commutative 73.6%

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

   13. sqrt-prod 37.4%

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

   14. *-commutative 37.4%

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

   15. sqrt-prod 37.4%

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

   16. sqrt-prod 24.9%

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

   17. add-sqr-sqrt 50.3%

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

   18. associate-*r* 50.3%

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

   19. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 70.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v}\right)\right) - 4.5\\ t_2 := t\_0 + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{if}\;r \leq 6.2 \cdot 10^{-77}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{elif}\;r \leq 6.2 \cdot 10^{+80}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;r \leq 4.5 \cdot 10^{+125}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;r \leq 4.6 \cdot 10^{+230}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;r \leq 5 \cdot 10^{+259}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r)))
        (t_1
         (- (+ 3.0 (* r (* (+ 0.375 (* v -0.25)) (/ (* w (* r w)) v)))) 4.5))
        (t_2 (+ t_0 (+ -1.5 (* 0.375 (* r (* (* w w) (/ r (+ v -1.0)))))))))
   (if (<= r 6.2e-77)
     (+ t_0 -1.5)
     (if (<= r 6.2e+80)
       t_2
       (if (<= r 4.5e+125)
         t_1
         (if (<= r 4.6e+230)
           t_2
           (if (<= r 5e+259)
             t_1
             (-
              3.0
              (+
               4.5
               (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* r w))))))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / v)))) - 4.5;
	double t_2 = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	double tmp;
	if (r <= 6.2e-77) {
		tmp = t_0 + -1.5;
	} else if (r <= 6.2e+80) {
		tmp = t_2;
	} else if (r <= 4.5e+125) {
		tmp = t_1;
	} else if (r <= 4.6e+230) {
		tmp = t_2;
	} else if (r <= 5e+259) {
		tmp = t_1;
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = (3.0d0 + (r * ((0.375d0 + (v * (-0.25d0))) * ((w * (r * w)) / v)))) - 4.5d0
    t_2 = t_0 + ((-1.5d0) + (0.375d0 * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    if (r <= 6.2d-77) then
        tmp = t_0 + (-1.5d0)
    else if (r <= 6.2d+80) then
        tmp = t_2
    else if (r <= 4.5d+125) then
        tmp = t_1
    else if (r <= 4.6d+230) then
        tmp = t_2
    else if (r <= 5d+259) then
        tmp = t_1
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / v)))) - 4.5;
	double t_2 = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	double tmp;
	if (r <= 6.2e-77) {
		tmp = t_0 + -1.5;
	} else if (r <= 6.2e+80) {
		tmp = t_2;
	} else if (r <= 4.5e+125) {
		tmp = t_1;
	} else if (r <= 4.6e+230) {
		tmp = t_2;
	} else if (r <= 5e+259) {
		tmp = t_1;
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / v)))) - 4.5
	t_2 = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))))
	tmp = 0
	if r <= 6.2e-77:
		tmp = t_0 + -1.5
	elif r <= 6.2e+80:
		tmp = t_2
	elif r <= 4.5e+125:
		tmp = t_1
	elif r <= 4.6e+230:
		tmp = t_2
	elif r <= 5e+259:
		tmp = t_1
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(Float64(3.0 + Float64(r * Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(w * Float64(r * w)) / v)))) - 4.5)
	t_2 = Float64(t_0 + Float64(-1.5 + Float64(0.375 * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))))
	tmp = 0.0
	if (r <= 6.2e-77)
		tmp = Float64(t_0 + -1.5);
	elseif (r <= 6.2e+80)
		tmp = t_2;
	elseif (r <= 4.5e+125)
		tmp = t_1;
	elseif (r <= 4.6e+230)
		tmp = t_2;
	elseif (r <= 5e+259)
		tmp = t_1;
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / v)))) - 4.5;
	t_2 = t_0 + (-1.5 + (0.375 * (r * ((w * w) * (r / (v + -1.0))))));
	tmp = 0.0;
	if (r <= 6.2e-77)
		tmp = t_0 + -1.5;
	elseif (r <= 6.2e+80)
		tmp = t_2;
	elseif (r <= 4.5e+125)
		tmp = t_1;
	elseif (r <= 4.6e+230)
		tmp = t_2;
	elseif (r <= 5e+259)
		tmp = t_1;
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(3.0 + N[(r * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(-1.5 + N[(0.375 * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 6.2e-77], N[(t$95$0 + -1.5), $MachinePrecision], If[LessEqual[r, 6.2e+80], t$95$2, If[LessEqual[r, 4.5e+125], t$95$1, If[LessEqual[r, 4.6e+230], t$95$2, If[LessEqual[r, 5e+259], t$95$1, N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if r < 6.20000000000000016e-77

   1. Initial program 83.7%

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

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 73.8%

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

   if 6.20000000000000016e-77 < r < 6.19999999999999976e80 or 4.5e125 < r < 4.5999999999999996e230

   1. Initial program 97.9%

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

   3. Simplified 98.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 88.1%

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

   if 6.19999999999999976e80 < r < 4.5e125 or 4.5999999999999996e230 < r < 5.00000000000000033e259

   1. Initial program 82.0%

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

   3. Taylor expanded in r around inf 82.0%

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

      1. associate-/l* 99.1%

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

      2. *-commutative 99.1%

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

      3. *-commutative 99.1%

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

      4. associate-/l* 94.1%

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

      5. *-commutative 94.1%

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

      6. associate-*r/ 94.0%

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

      7. *-commutative 94.0%

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

      8. associate-*r* 93.9%

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

   5. Applied egg-rr 94.0%

      \[\leadsto \left(3 - \color{blue}{\left(\left(0.375 + -0.25 \cdot v\right) \cdot \frac{{w}^{2} \cdot r}{1 - v}\right) \cdot r}\right) - 4.5 \]
   6. Step-by-step derivation

      1. pow2 94.0%

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

      2. associate-*r* 94.0%

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

      3. *-commutative 94.0%

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

      4. *-un-lft-identity 94.0%

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

      5. times-frac 90.9%

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

   7. Applied egg-rr 90.9%

      \[\leadsto \left(3 - \left(\left(0.375 + -0.25 \cdot v\right) \cdot \color{blue}{\left(\frac{w \cdot r}{1} \cdot \frac{w}{1 - v}\right)}\right) \cdot r\right) - 4.5 \]
   8. Step-by-step derivation

      1. /-rgt-identity 90.9%

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

      2. associate-*r/ 94.0%

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

      3. *-commutative 94.0%

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

   9. Simplified 94.0%

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

   10. Taylor expanded in v around inf 84.8%

       \[\leadsto \left(3 - \left(\left(0.375 + -0.25 \cdot v\right) \cdot \frac{\left(r \cdot w\right) \cdot w}{\color{blue}{-1 \cdot v}}\right) \cdot r\right) - 4.5 \]
   11. Step-by-step derivation

       1. neg-mul-1 84.8%

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

   12. Simplified 84.8%

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

   if 5.00000000000000033e259 < r

   1. Initial program 92.0%

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

      1. associate--l- 92.0%

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

      2. associate-*l* 83.9%

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

      3. sqr-neg 83.9%

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

      4. associate-*l* 92.0%

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

      5. associate-/l* 92.0%

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

      6. fma-define 92.0%

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

   3. Simplified 92.0%

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

      1. associate-/l* 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r/ 92.0%

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

      4. associate-*l* 99.9%

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

      5. associate-*r* 99.9%

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

      6. add-sqr-sqrt 99.7%

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

      7. sqrt-prod 84.2%

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

      8. add-sqr-sqrt 59.2%

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

      9. sqrt-prod 58.9%

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

      10. sqrt-prod 58.9%

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

      11. associate-*r* 67.0%

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

      12. *-commutative 67.0%

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

      13. sqrt-prod 67.0%

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

      14. *-commutative 67.0%

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

      15. sqrt-prod 67.1%

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

      16. sqrt-prod 74.7%

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

      17. add-sqr-sqrt 99.9%

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

      18. associate-*r* 99.7%

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

      19. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.5%

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

   8. Taylor expanded in r around inf 91.5%

      \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(w \cdot r\right) \cdot \left(r \cdot w\right)\right) + 4.5\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 6.2 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{elif}\;r \leq 4.5 \cdot 10^{+125}:\\ \;\;\;\;\left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v}\right)\right) - 4.5\\ \mathbf{elif}\;r \leq 4.6 \cdot 10^{+230}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{elif}\;r \leq 5 \cdot 10^{+259}:\\ \;\;\;\;\left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 9 \cdot 10^{-90}:\\ \;\;\;\;t\_1 + -1.5\\ \mathbf{elif}\;r \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;t\_1 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot t\_0\right)\\ \mathbf{elif}\;r \leq 82000:\\ \;\;\;\;t\_1 + \left(-1.5 + 0.375 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* r (* (* w w) (/ r (+ v -1.0))))) (t_1 (/ 2.0 (* r r))))
   (if (<= r 9e-90)
     (+ t_1 -1.5)
     (if (<= r 2.2e-5)
       (+ t_1 (+ -1.5 (* (* v -0.25) t_0)))
       (if (<= r 82000.0)
         (+ t_1 (+ -1.5 (* 0.375 t_0)))
         (+
          3.0
          (-
           (* (* 0.125 (+ 3.0 (* -2.0 v))) (/ (* r (* w (* r w))) (+ v -1.0)))
           4.5)))))))

C

double code(double v, double w, double r) {
	double t_0 = r * ((w * w) * (r / (v + -1.0)));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 9e-90) {
		tmp = t_1 + -1.5;
	} else if (r <= 2.2e-5) {
		tmp = t_1 + (-1.5 + ((v * -0.25) * t_0));
	} else if (r <= 82000.0) {
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * (w * (r * w))) / (v + -1.0))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r * ((w * w) * (r / (v + (-1.0d0))))
    t_1 = 2.0d0 / (r * r)
    if (r <= 9d-90) then
        tmp = t_1 + (-1.5d0)
    else if (r <= 2.2d-5) then
        tmp = t_1 + ((-1.5d0) + ((v * (-0.25d0)) * t_0))
    else if (r <= 82000.0d0) then
        tmp = t_1 + ((-1.5d0) + (0.375d0 * t_0))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * (w * (r * w))) / (v + (-1.0d0)))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = r * ((w * w) * (r / (v + -1.0)));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (r <= 9e-90) {
		tmp = t_1 + -1.5;
	} else if (r <= 2.2e-5) {
		tmp = t_1 + (-1.5 + ((v * -0.25) * t_0));
	} else if (r <= 82000.0) {
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * (w * (r * w))) / (v + -1.0))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = r * ((w * w) * (r / (v + -1.0)))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if r <= 9e-90:
		tmp = t_1 + -1.5
	elif r <= 2.2e-5:
		tmp = t_1 + (-1.5 + ((v * -0.25) * t_0))
	elif r <= 82000.0:
		tmp = t_1 + (-1.5 + (0.375 * t_0))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * (w * (r * w))) / (v + -1.0))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0))))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 9e-90)
		tmp = Float64(t_1 + -1.5);
	elseif (r <= 2.2e-5)
		tmp = Float64(t_1 + Float64(-1.5 + Float64(Float64(v * -0.25) * t_0)));
	elseif (r <= 82000.0)
		tmp = Float64(t_1 + Float64(-1.5 + Float64(0.375 * t_0)));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * Float64(w * Float64(r * w))) / Float64(v + -1.0))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = r * ((w * w) * (r / (v + -1.0)));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 9e-90)
		tmp = t_1 + -1.5;
	elseif (r <= 2.2e-5)
		tmp = t_1 + (-1.5 + ((v * -0.25) * t_0));
	elseif (r <= 82000.0)
		tmp = t_1 + (-1.5 + (0.375 * t_0));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * (w * (r * w))) / (v + -1.0))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 9e-90], N[(t$95$1 + -1.5), $MachinePrecision], If[LessEqual[r, 2.2e-5], N[(t$95$1 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 82000.0], N[(t$95$1 + N[(-1.5 + N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if r < 9.00000000000000017e-90

   1. Initial program 83.3%

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

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 73.8%

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

   if 9.00000000000000017e-90 < r < 2.1999999999999999e-5

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 99.8%

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

      1. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   if 2.1999999999999999e-5 < r < 82000

   1. Initial program 86.6%

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

   3. Simplified 89.5%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around 0 77.8%

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

   if 82000 < r

   1. Initial program 92.0%

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

      1. associate--l- 92.0%

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

      2. associate-*l* 86.5%

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

      3. sqr-neg 86.5%

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

      4. associate-*l* 92.0%

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

      5. associate-/l* 98.3%

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

      6. fma-define 98.3%

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

   3. Simplified 98.3%

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

      1. add-sqr-sqrt 98.2%

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

      2. pow2 98.2%

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

      3. *-commutative 98.2%

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

      4. sqrt-prod 98.2%

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

      5. sqrt-prod 52.6%

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

      6. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-prod-down 98.3%

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

      2. pow2 98.3%

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

      3. pow2 98.3%

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

      4. add-sqr-sqrt 98.3%

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

      5. associate-*l* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in r around inf 99.8%

      \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}{1 - v} + 4.5\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 2.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{elif}\;r \leq 82000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + 0.375 \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \frac{r}{v + -1}\\ \mathbf{if}\;r \leq 6.4 \cdot 10^{-90}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{elif}\;r \leq 24:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot t\_1\right)\right) - 4.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))) (t_1 (/ r (+ v -1.0))))
   (if (<= r 6.4e-90)
     (+ t_0 -1.5)
     (if (<= r 24.0)
       (+ t_0 (+ -1.5 (* (* v -0.25) (* r (* (* w w) t_1)))))
       (+
        3.0
        (- (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* w t_1))) 4.5))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r / (v + -1.0);
	double tmp;
	if (r <= 6.4e-90) {
		tmp = t_0 + -1.5;
	} else if (r <= 24.0) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * t_1))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * t_1))) - 4.5);
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    t_1 = r / (v + (-1.0d0))
    if (r <= 6.4d-90) then
        tmp = t_0 + (-1.5d0)
    else if (r <= 24.0d0) then
        tmp = t_0 + ((-1.5d0) + ((v * (-0.25d0)) * (r * ((w * w) * t_1))))
    else
        tmp = 3.0d0 + (((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (w * t_1))) - 4.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double t_1 = r / (v + -1.0);
	double tmp;
	if (r <= 6.4e-90) {
		tmp = t_0 + -1.5;
	} else if (r <= 24.0) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * t_1))));
	} else {
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * t_1))) - 4.5);
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	t_1 = r / (v + -1.0)
	tmp = 0
	if r <= 6.4e-90:
		tmp = t_0 + -1.5
	elif r <= 24.0:
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * t_1))))
	else:
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * t_1))) - 4.5)
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	t_1 = Float64(r / Float64(v + -1.0))
	tmp = 0.0
	if (r <= 6.4e-90)
		tmp = Float64(t_0 + -1.5);
	elseif (r <= 24.0)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(v * -0.25) * Float64(r * Float64(Float64(w * w) * t_1)))));
	else
		tmp = Float64(3.0 + Float64(Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(w * t_1))) - 4.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	t_1 = r / (v + -1.0);
	tmp = 0.0;
	if (r <= 6.4e-90)
		tmp = t_0 + -1.5;
	elseif (r <= 24.0)
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * t_1))));
	else
		tmp = 3.0 + (((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (w * t_1))) - 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 6.4e-90], N[(t$95$0 + -1.5), $MachinePrecision], If[LessEqual[r, 24.0], N[(t$95$0 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 + N[(N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(w * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if r < 6.40000000000000014e-90

   1. Initial program 83.3%

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

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 73.8%

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

   if 6.40000000000000014e-90 < r < 24

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 99.8%

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

      1. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   if 24 < r

   1. Initial program 92.0%

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

      1. associate--l- 92.0%

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

      2. associate-*l* 86.5%

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

      3. sqr-neg 86.5%

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

      4. associate-*l* 92.0%

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

      5. associate-/l* 98.3%

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

      6. fma-define 98.3%

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

   3. Simplified 98.3%

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

      1. associate-/l* 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r/ 95.6%

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

      4. associate-*l* 96.9%

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

      5. associate-*r* 99.7%

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

      6. add-sqr-sqrt 99.8%

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

      7. sqrt-prod 90.5%

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

      8. add-sqr-sqrt 47.4%

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

      9. sqrt-prod 56.7%

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

      10. sqrt-prod 56.8%

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

      11. associate-*r* 63.6%

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

      12. *-commutative 63.6%

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

      13. sqrt-prod 63.6%

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

      14. *-commutative 63.6%

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

      15. sqrt-prod 63.6%

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

      16. sqrt-prod 52.7%

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

      17. add-sqr-sqrt 99.8%

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

      18. associate-*r* 99.8%

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

      19. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in r around inf 99.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.4 \cdot 10^{-90}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 24:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;3 + \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 1.1 \cdot 10^{-89}:\\ \;\;\;\;t\_0 + -1.5\\ \mathbf{elif}\;r \leq 23:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 1.1e-89)
     (+ t_0 -1.5)
     (if (<= r 23.0)
       (+ t_0 (+ -1.5 (* (* v -0.25) (* r (* (* w w) (/ r (+ v -1.0)))))))
       (-
        (+ 3.0 (* r (* (+ 0.375 (* v -0.25)) (/ (* w (* r w)) (+ v -1.0)))))
        4.5)))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.1e-89) {
		tmp = t_0 + -1.5;
	} else if (r <= 23.0) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 1.1d-89) then
        tmp = t_0 + (-1.5d0)
    else if (r <= 23.0d0) then
        tmp = t_0 + ((-1.5d0) + ((v * (-0.25d0)) * (r * ((w * w) * (r / (v + (-1.0d0)))))))
    else
        tmp = (3.0d0 + (r * ((0.375d0 + (v * (-0.25d0))) * ((w * (r * w)) / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 1.1e-89) {
		tmp = t_0 + -1.5;
	} else if (r <= 23.0) {
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	} else {
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 1.1e-89:
		tmp = t_0 + -1.5
	elif r <= 23.0:
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))))
	else:
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 1.1e-89)
		tmp = Float64(t_0 + -1.5);
	elseif (r <= 23.0)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(v * -0.25) * Float64(r * Float64(Float64(w * w) * Float64(r / Float64(v + -1.0)))))));
	else
		tmp = Float64(Float64(3.0 + Float64(r * Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(w * Float64(r * w)) / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 1.1e-89)
		tmp = t_0 + -1.5;
	elseif (r <= 23.0)
		tmp = t_0 + (-1.5 + ((v * -0.25) * (r * ((w * w) * (r / (v + -1.0))))));
	else
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 1.1e-89], N[(t$95$0 + -1.5), $MachinePrecision], If[LessEqual[r, 23.0], N[(t$95$0 + N[(-1.5 + N[(N[(v * -0.25), $MachinePrecision] * N[(r * N[(N[(w * w), $MachinePrecision] * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 + N[(r * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.10000000000000006e-89

   1. Initial program 83.3%

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

   3. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 73.8%

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

   if 1.10000000000000006e-89 < r < 23

   1. Initial program 99.8%

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

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in v around inf 99.8%

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

      1. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   if 23 < r

   1. Initial program 92.0%

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

   3. Taylor expanded in r around inf 92.0%

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

      1. associate-/l* 98.3%

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

      2. *-commutative 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-/l* 95.7%

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

      5. *-commutative 95.7%

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

      6. associate-*r/ 95.7%

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

      7. *-commutative 95.7%

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

      8. associate-*r* 95.7%

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

   5. Applied egg-rr 95.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(0.375 + -0.25 \cdot v\right) \cdot \frac{{w}^{2} \cdot r}{1 - v}\right) \cdot r}\right) - 4.5 \]
   6. Step-by-step derivation

      1. pow2 95.7%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. *-un-lft-identity 97.0%

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

      5. times-frac 96.1%

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

   7. Applied egg-rr 96.1%

      \[\leadsto \left(3 - \left(\left(0.375 + -0.25 \cdot v\right) \cdot \color{blue}{\left(\frac{w \cdot r}{1} \cdot \frac{w}{1 - v}\right)}\right) \cdot r\right) - 4.5 \]
   8. Step-by-step derivation

      1. /-rgt-identity 96.1%

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

      2. associate-*r/ 97.0%

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

      3. *-commutative 97.0%

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

   9. Simplified 97.0%

      \[\leadsto \left(3 - \left(\left(0.375 + -0.25 \cdot v\right) \cdot \color{blue}{\frac{\left(r \cdot w\right) \cdot w}{1 - v}}\right) \cdot r\right) - 4.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 23:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(v \cdot -0.25\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{v + -1}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1120000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(w \cdot \left(r \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1120000.0)
   (-
    (+
     (+ 3.0 (/ 2.0 (* r r)))
     (* (* w (* r (+ 0.375 (* v -0.25)))) (* w (/ r (+ v -1.0)))))
    4.5)
   (-
    3.0
    (+
     4.5
     (* (* 0.125 (+ 3.0 (* -2.0 v))) (/ (* r (* w (* r w))) (- 1.0 v)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1120000.0) {
		tmp = ((3.0 + (2.0 / (r * r))) + ((w * (r * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * (w * (r * w))) / (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 <= 1120000.0d0) then
        tmp = ((3.0d0 + (2.0d0 / (r * r))) + ((w * (r * (0.375d0 + (v * (-0.25d0))))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * (w * (r * w))) / (1.0d0 - v))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1120000.0) {
		tmp = ((3.0 + (2.0 / (r * r))) + ((w * (r * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * (w * (r * w))) / (1.0 - v))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1120000.0:
		tmp = ((3.0 + (2.0 / (r * r))) + ((w * (r * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * (w * (r * w))) / (1.0 - v))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1120000.0)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(w * Float64(r * Float64(0.375 + Float64(v * -0.25)))) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * Float64(w * Float64(r * w))) / Float64(1.0 - v)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1120000.0)
		tmp = ((3.0 + (2.0 / (r * r))) + ((w * (r * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * (w * (r * w))) / (1.0 - v))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1120000.0], N[(N[(N[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(w * N[(r * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.12e6

   1. Initial program 84.5%

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

      1. associate-/l* 87.0%

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

      2. cancel-sign-sub-inv 87.0%

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

      3. metadata-eval 87.0%

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

      4. +-commutative 87.0%

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

      5. *-commutative 87.0%

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

      6. fma-undefine 87.0%

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

      7. *-commutative 87.0%

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

      8. *-commutative 87.0%

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

      9. associate-/l* 87.1%

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

      10. *-commutative 87.1%

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

      11. associate-*r/ 87.1%

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

      12. associate-*r* 83.8%

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

      13. associate-*l* 93.5%

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

      14. associate-*r* 94.6%

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

   4. Applied egg-rr 94.6%

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

   if 1.12e6 < r

   1. Initial program 92.0%

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

      1. associate--l- 92.0%

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

      2. associate-*l* 86.5%

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

      3. sqr-neg 86.5%

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

      4. associate-*l* 92.0%

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

      5. associate-/l* 98.3%

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

      6. fma-define 98.3%

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

   3. Simplified 98.3%

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

      1. add-sqr-sqrt 98.2%

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

      2. pow2 98.2%

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

      3. *-commutative 98.2%

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

      4. sqrt-prod 98.2%

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

      5. sqrt-prod 52.6%

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

      6. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-prod-down 98.3%

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

      2. pow2 98.3%

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

      3. pow2 98.3%

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

      4. add-sqr-sqrt 98.3%

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

      5. associate-*l* 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in r around inf 99.8%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1120000:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(w \cdot \left(r \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{1 - v}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 7.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 9 \cdot 10^{+261}:\\ \;\;\;\;\left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 7.2e-22)
   (+ (/ 2.0 (* r r)) -1.5)
   (if (<= r 9e+261)
     (- (+ 3.0 (* r (* (+ 0.375 (* v -0.25)) (/ (* w (* r w)) v)))) 4.5)
     (- 3.0 (+ 4.5 (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* r w))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.2e-22) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 9e+261) {
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / v)))) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 7.2d-22) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else if (r <= 9d+261) then
        tmp = (3.0d0 + (r * ((0.375d0 + (v * (-0.25d0))) * ((w * (r * w)) / v)))) - 4.5d0
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 7.2e-22) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else if (r <= 9e+261) {
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / v)))) - 4.5;
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 7.2e-22:
		tmp = (2.0 / (r * r)) + -1.5
	elif r <= 9e+261:
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / v)))) - 4.5
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 7.2e-22)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	elseif (r <= 9e+261)
		tmp = Float64(Float64(3.0 + Float64(r * Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(w * Float64(r * w)) / v)))) - 4.5);
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 7.2e-22)
		tmp = (2.0 / (r * r)) + -1.5;
	elseif (r <= 9e+261)
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / v)))) - 4.5;
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 7.2e-22], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], If[LessEqual[r, 9e+261], N[(N[(3.0 + N[(r * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 7.1999999999999996e-22

   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 87.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 74.4%

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

   if 7.1999999999999996e-22 < r < 8.9999999999999998e261

   1. Initial program 92.2%

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

   3. Taylor expanded in r around inf 92.2%

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

      1. associate-/l* 99.6%

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

      2. *-commutative 99.6%

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

      3. *-commutative 99.6%

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

      4. associate-/l* 96.5%

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

      5. *-commutative 96.5%

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

      6. associate-*r/ 96.5%

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

      7. *-commutative 96.5%

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

      8. associate-*r* 96.5%

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

   5. Applied egg-rr 96.5%

      \[\leadsto \left(3 - \color{blue}{\left(\left(0.375 + -0.25 \cdot v\right) \cdot \frac{{w}^{2} \cdot r}{1 - v}\right) \cdot r}\right) - 4.5 \]
   6. Step-by-step derivation

      1. pow2 96.5%

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

      2. associate-*r* 96.5%

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

      3. *-commutative 96.5%

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

      4. *-un-lft-identity 96.5%

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

      5. times-frac 95.4%

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

   7. Applied egg-rr 95.4%

      \[\leadsto \left(3 - \left(\left(0.375 + -0.25 \cdot v\right) \cdot \color{blue}{\left(\frac{w \cdot r}{1} \cdot \frac{w}{1 - v}\right)}\right) \cdot r\right) - 4.5 \]
   8. Step-by-step derivation

      1. /-rgt-identity 95.4%

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

      2. associate-*r/ 96.5%

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

      3. *-commutative 96.5%

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

   9. Simplified 96.5%

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

   10. Taylor expanded in v around inf 72.6%

       \[\leadsto \left(3 - \left(\left(0.375 + -0.25 \cdot v\right) \cdot \frac{\left(r \cdot w\right) \cdot w}{\color{blue}{-1 \cdot v}}\right) \cdot r\right) - 4.5 \]
   11. Step-by-step derivation

       1. neg-mul-1 72.6%

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

   12. Simplified 72.6%

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

   if 8.9999999999999998e261 < r

   1. Initial program 92.0%

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

      1. associate--l- 92.0%

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

      2. associate-*l* 83.9%

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

      3. sqr-neg 83.9%

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

      4. associate-*l* 92.0%

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

      5. associate-/l* 92.0%

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

      6. fma-define 92.0%

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

   3. Simplified 92.0%

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

      1. associate-/l* 92.0%

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

      2. *-commutative 92.0%

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

      3. associate-*r/ 92.0%

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

      4. associate-*l* 99.9%

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

      5. associate-*r* 99.9%

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

      6. add-sqr-sqrt 99.7%

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

      7. sqrt-prod 84.2%

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

      8. add-sqr-sqrt 59.2%

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

      9. sqrt-prod 58.9%

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

      10. sqrt-prod 58.9%

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

      11. associate-*r* 67.0%

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

      12. *-commutative 67.0%

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

      13. sqrt-prod 67.0%

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

      14. *-commutative 67.0%

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

      15. sqrt-prod 67.1%

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

      16. sqrt-prod 74.7%

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

      17. add-sqr-sqrt 99.9%

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

      18. associate-*r* 99.7%

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

      19. add-sqr-sqrt 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in v around 0 91.5%

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

   8. Taylor expanded in r around inf 91.5%

      \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(w \cdot r\right) \cdot \left(r \cdot w\right)\right) + 4.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.2 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 9 \cdot 10^{+261}:\\ \;\;\;\;\left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 9.5e-22)
   (+ (/ 2.0 (* r r)) -1.5)
   (-
    (+ 3.0 (* r (* (+ 0.375 (* v -0.25)) (/ (* w (* r w)) (+ v -1.0)))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 9.5e-22) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 9.5d-22) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (3.0d0 + (r * ((0.375d0 + (v * (-0.25d0))) * ((w * (r * w)) / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 9.5e-22) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 9.5e-22:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 9.5e-22)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(3.0 + Float64(r * Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(w * Float64(r * w)) / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 9.5e-22)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (3.0 + (r * ((0.375 + (v * -0.25)) * ((w * (r * w)) / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 9.5e-22], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(3.0 + N[(r * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 9.4999999999999994e-22

   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 87.0%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 74.4%

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

   if 9.4999999999999994e-22 < r

   1. Initial program 92.1%

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

   3. Taylor expanded in r around inf 92.1%

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

      1. associate-/l* 98.3%

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

      2. *-commutative 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-/l* 95.7%

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

      5. *-commutative 95.7%

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

      6. associate-*r/ 95.7%

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

      7. *-commutative 95.7%

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

      8. associate-*r* 95.7%

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

   5. Applied egg-rr 95.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(0.375 + -0.25 \cdot v\right) \cdot \frac{{w}^{2} \cdot r}{1 - v}\right) \cdot r}\right) - 4.5 \]
   6. Step-by-step derivation

      1. pow2 95.7%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. *-un-lft-identity 97.0%

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

      5. times-frac 96.1%

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

   7. Applied egg-rr 96.1%

      \[\leadsto \left(3 - \left(\left(0.375 + -0.25 \cdot v\right) \cdot \color{blue}{\left(\frac{w \cdot r}{1} \cdot \frac{w}{1 - v}\right)}\right) \cdot r\right) - 4.5 \]
   8. Step-by-step derivation

      1. /-rgt-identity 96.1%

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

      2. associate-*r/ 97.0%

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

      3. *-commutative 97.0%

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

   9. Simplified 97.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{w \cdot \left(r \cdot w\right)}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 2.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 2.5e-39)
   (+ (/ 2.0 (* r r)) -1.5)
   (-
    (+ 3.0 (* (* w (* r (+ 0.375 (* v -0.25)))) (* w (/ r (+ v -1.0)))))
    4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.5e-39) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (3.0 + ((w * (r * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 2.5d-39) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (3.0d0 + ((w * (r * (0.375d0 + (v * (-0.25d0))))) * (w * (r / (v + (-1.0d0)))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 2.5e-39) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (3.0 + ((w * (r * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 2.5e-39:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (3.0 + ((w * (r * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 2.5e-39)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(w * Float64(r * Float64(0.375 + Float64(v * -0.25)))) * Float64(w * Float64(r / Float64(v + -1.0))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 2.5e-39)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (3.0 + ((w * (r * (0.375 + (v * -0.25)))) * (w * (r / (v + -1.0))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 2.5e-39], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(3.0 + N[(N[(w * N[(r * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.4999999999999999e-39

   1. Initial program 84.2%

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

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 74.1%

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

   if 2.4999999999999999e-39 < r

   1. Initial program 92.3%

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

   3. Taylor expanded in r around inf 89.7%

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

      1. associate-/l* 98.4%

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

      2. cancel-sign-sub-inv 98.4%

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

      3. metadata-eval 98.4%

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

      4. +-commutative 98.4%

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

      5. *-commutative 98.4%

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

      6. fma-undefine 98.4%

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

      7. *-commutative 98.4%

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

      8. *-commutative 98.4%

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

      9. associate-/l* 95.8%

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

      10. *-commutative 95.8%

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

      11. associate-*r/ 95.8%

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

      12. associate-*r* 88.2%

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

      13. associate-*l* 89.5%

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

      14. associate-*r* 88.4%

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

   5. Applied egg-rr 85.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(w \cdot \left(r \cdot \left(0.375 + v \cdot -0.25\right)\right)\right) \cdot \left(w \cdot \frac{r}{v + -1}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 15.8:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 15.8)
   (+ (/ 2.0 (* r r)) -1.5)
   (- 3.0 (+ 4.5 (* (* 0.125 (+ 3.0 (* -2.0 v))) (* (* r w) (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 15.8) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 15.8d0) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = 3.0d0 - (4.5d0 + ((0.125d0 * (3.0d0 + ((-2.0d0) * v))) * ((r * w) * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 15.8) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 15.8:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 15.8)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(3.0 - Float64(4.5 + Float64(Float64(0.125 * Float64(3.0 + Float64(-2.0 * v))) * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 15.8)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = 3.0 - (4.5 + ((0.125 * (3.0 + (-2.0 * v))) * ((r * w) * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 15.8], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(3.0 - N[(4.5 + N[(N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 15.800000000000001

   1. Initial program 84.5%

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

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 74.0%

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

   if 15.800000000000001 < r

   1. Initial program 92.0%

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

      1. associate--l- 92.0%

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

      2. associate-*l* 86.5%

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

      3. sqr-neg 86.5%

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

      4. associate-*l* 92.0%

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

      5. associate-/l* 98.3%

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

      6. fma-define 98.3%

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

   3. Simplified 98.3%

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

      1. associate-/l* 95.6%

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

      2. *-commutative 95.6%

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

      3. associate-*r/ 95.6%

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

      4. associate-*l* 96.9%

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

      5. associate-*r* 99.7%

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

      6. add-sqr-sqrt 99.8%

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

      7. sqrt-prod 90.5%

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

      8. add-sqr-sqrt 47.4%

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

      9. sqrt-prod 56.7%

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

      10. sqrt-prod 56.8%

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

      11. associate-*r* 63.6%

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

      12. *-commutative 63.6%

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

      13. sqrt-prod 63.6%

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

      14. *-commutative 63.6%

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

      15. sqrt-prod 63.6%

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

      16. sqrt-prod 52.7%

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

      17. add-sqr-sqrt 99.8%

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

      18. associate-*r* 99.8%

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

      19. add-sqr-sqrt 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in v around 0 71.3%

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

   8. Taylor expanded in r around inf 71.3%

      \[\leadsto \color{blue}{3} - \left(\left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(w \cdot r\right) \cdot \left(r \cdot w\right)\right) + 4.5\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 15.8:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(4.5 + \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 0.065:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 0.065)
   (+ (/ 2.0 (* r r)) -1.5)
   (- (- 3.0 (* r (* (+ 0.375 (* v -0.25)) (* w (* r w))))) 4.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.065) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (3.0 - (r * ((0.375 + (v * -0.25)) * (w * (r * w))))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 0.065d0) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (3.0d0 - (r * ((0.375d0 + (v * (-0.25d0))) * (w * (r * w))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 0.065) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (3.0 - (r * ((0.375 + (v * -0.25)) * (w * (r * w))))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 0.065:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (3.0 - (r * ((0.375 + (v * -0.25)) * (w * (r * w))))) - 4.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 0.065)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(3.0 - Float64(r * Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(w * Float64(r * w))))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 0.065)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (3.0 - (r * ((0.375 + (v * -0.25)) * (w * (r * w))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 0.065], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(3.0 - N[(r * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 0.065000000000000002

   1. Initial program 84.5%

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

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

   7. Taylor expanded in w around 0 74.0%

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

   if 0.065000000000000002 < r

   1. Initial program 92.0%

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

   3. Taylor expanded in r around inf 92.0%

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

      1. associate-/l* 98.3%

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

      2. *-commutative 98.3%

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

      3. *-commutative 98.3%

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

      4. associate-/l* 95.7%

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

      5. *-commutative 95.7%

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

      6. associate-*r/ 95.7%

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

      7. *-commutative 95.7%

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

      8. associate-*r* 95.7%

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

   5. Applied egg-rr 95.7%

      \[\leadsto \left(3 - \color{blue}{\left(\left(0.375 + -0.25 \cdot v\right) \cdot \frac{{w}^{2} \cdot r}{1 - v}\right) \cdot r}\right) - 4.5 \]
   6. Step-by-step derivation

      1. pow2 95.7%

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

      2. associate-*r* 97.0%

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

      3. *-commutative 97.0%

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

      4. *-un-lft-identity 97.0%

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

      5. times-frac 96.1%

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

   7. Applied egg-rr 96.1%

      \[\leadsto \left(3 - \left(\left(0.375 + -0.25 \cdot v\right) \cdot \color{blue}{\left(\frac{w \cdot r}{1} \cdot \frac{w}{1 - v}\right)}\right) \cdot r\right) - 4.5 \]
   8. Step-by-step derivation

      1. /-rgt-identity 96.1%

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

      2. associate-*r/ 97.0%

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

      3. *-commutative 97.0%

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

   9. Simplified 97.0%

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

   10. Taylor expanded in v around 0 72.8%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 0.065:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - r \cdot \left(\left(0.375 + v \cdot -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.2% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

3. Simplified 89.5%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot \frac{r}{1 - v}\right)\right)\right)} \]
4. Add Preprocessing

6. Applied egg-rr 99.7%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{{\left(\sqrt[3]{\frac{{\left(w \cdot r\right)}^{2}}{1 - v} \cdot \left(0.375 + -0.25 \cdot v\right)}\right)}^{3}}\right) \]

7. Taylor expanded in w around 0 60.6%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]
8. Add Preprocessing

Alternative 13: 13.9% 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 86.6%

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

3. Simplified 83.1%

   \[\leadsto \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(0.375 + 0.125 \cdot \left(v \cdot -2\right), \left(r \cdot r\right) \cdot \frac{w \cdot w}{1 - v}, 4.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0 60.6%

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

6. Taylor expanded in r around inf 16.2%

   \[\leadsto \color{blue}{-1.5} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(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))