Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 98.3%

Time: 12.2s

Alternatives: 12

Speedup: 1.5×

• 53.3% 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: 84.8% 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: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := \frac{w}{\frac{1 - v}{r\_m}}\\ \mathbf{if}\;r\_m \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - t\_0 \cdot \left(w \cdot \left(r\_m \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(\left(r\_m \cdot w\right) \cdot t\_0\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) + 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (/ w (/ (- 1.0 v) r_m))))
   (if (<= r_m 4e+17)
     (-
      (-
       (+ 3.0 (/ 2.0 (* r_m r_m)))
       (* t_0 (* w (* r_m (+ (* v -0.25) 0.375)))))
      4.5)
     (- 3.0 (+ (* (* (* r_m w) t_0) (* 0.125 (+ 3.0 (* -2.0 v)))) 4.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = w / ((1.0 - v) / r_m);
	double tmp;
	if (r_m <= 4e+17) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - (t_0 * (w * (r_m * ((v * -0.25) + 0.375))))) - 4.5;
	} else {
		tmp = 3.0 - ((((r_m * w) * t_0) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = w / ((1.0d0 - v) / r_m)
    if (r_m <= 4d+17) then
        tmp = ((3.0d0 + (2.0d0 / (r_m * r_m))) - (t_0 * (w * (r_m * ((v * (-0.25d0)) + 0.375d0))))) - 4.5d0
    else
        tmp = 3.0d0 - ((((r_m * w) * t_0) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))) + 4.5d0)
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = w / ((1.0 - v) / r_m);
	double tmp;
	if (r_m <= 4e+17) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - (t_0 * (w * (r_m * ((v * -0.25) + 0.375))))) - 4.5;
	} else {
		tmp = 3.0 - ((((r_m * w) * t_0) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = w / ((1.0 - v) / r_m)
	tmp = 0
	if r_m <= 4e+17:
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - (t_0 * (w * (r_m * ((v * -0.25) + 0.375))))) - 4.5
	else:
		tmp = 3.0 - ((((r_m * w) * t_0) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5)
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(w / Float64(Float64(1.0 - v) / r_m))
	tmp = 0.0
	if (r_m <= 4e+17)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(t_0 * Float64(w * Float64(r_m * Float64(Float64(v * -0.25) + 0.375))))) - 4.5);
	else
		tmp = Float64(3.0 - Float64(Float64(Float64(Float64(r_m * w) * t_0) * Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))) + 4.5));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = w / ((1.0 - v) / r_m);
	tmp = 0.0;
	if (r_m <= 4e+17)
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - (t_0 * (w * (r_m * ((v * -0.25) + 0.375))))) - 4.5;
	else
		tmp = 3.0 - ((((r_m * w) * t_0) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(w / N[(N[(1.0 - v), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 4e+17], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[(w * N[(r$95$m * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(N[(N[(N[(r$95$m * w), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 4e17

   1. Initial program 84.9%

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

      1. associate-/l* 87.7%

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

      2. cancel-sign-sub-inv 87.7%

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

      3. metadata-eval 87.7%

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

      4. +-commutative 87.7%

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

      5. *-commutative 87.7%

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

      6. fma-undefine 87.7%

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

      7. *-commutative 87.7%

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

      8. *-commutative 87.7%

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

      9. associate-/l* 87.7%

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

      10. *-commutative 87.7%

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

      11. associate-*r/ 87.7%

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

      12. associate-*r* 86.5%

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

      13. associate-*l* 95.4%

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

      14. associate-*r* 96.9%

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

   4. Applied egg-rr 96.9%

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

   if 4e17 < r

   1. Initial program 89.9%

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

      1. associate--l- 89.9%

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

      2. associate-*l* 78.3%

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

      3. sqr-neg 78.3%

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

      4. associate-*l* 89.9%

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

      5. associate-/l* 94.6%

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

      6. fma-define 94.6%

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

   3. Simplified 94.6%

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

      1. associate-/l* 94.6%

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

      2. *-commutative 94.6%

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

      3. associate-*r/ 94.5%

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

      4. associate-*l* 97.2%

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

      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. clear-num 99.7%

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

      7. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{w}{\frac{1 - v}{r}} \cdot \left(w \cdot \left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(\left(r \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r}}\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) + 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

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

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (-
  (+ 3.0 (/ 2.0 (* r_m r_m)))
  (+
   (* (* (* r_m w) (/ w (/ (- 1.0 v) r_m))) (* 0.125 (+ 3.0 (* -2.0 v))))
   4.5)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(r$95$m * w), $MachinePrecision] * N[(w / N[(N[(1.0 - v), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

   1. associate--l- 86.0%

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

   2. associate-*l* 82.4%

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

   3. sqr-neg 82.4%

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

   4. associate-*l* 86.0%

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

   5. associate-/l* 89.2%

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

   6. fma-define 89.2%

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

3. Simplified 89.2%

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

   1. associate-/l* 89.2%

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

   2. *-commutative 89.2%

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

   3. associate-*r/ 89.2%

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

   4. associate-*l* 97.1%

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

   5. associate-*r* 99.5%

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

   6. clear-num 99.4%

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

   7. un-div-inv 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 2.6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(r\_m \cdot w\right) \cdot \left(w \cdot \left(r\_m \cdot 0.375\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(\left(r\_m \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r\_m}}\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) + 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 2.6e+17)
   (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* r_m w) (* w (* r_m 0.375)))) 4.5)
   (-
    3.0
    (+
     (* (* (* r_m w) (/ w (/ (- 1.0 v) r_m))) (* 0.125 (+ 3.0 (* -2.0 v))))
     4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2.6e+17) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
	} else {
		tmp = 3.0 - ((((r_m * w) * (w / ((1.0 - v) / r_m))) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 2.6d+17) then
        tmp = ((3.0d0 + (2.0d0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375d0)))) - 4.5d0
    else
        tmp = 3.0d0 - ((((r_m * w) * (w / ((1.0d0 - v) / r_m))) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))) + 4.5d0)
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2.6e+17) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
	} else {
		tmp = 3.0 - ((((r_m * w) * (w / ((1.0 - v) / r_m))) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 2.6e+17:
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5
	else:
		tmp = 3.0 - ((((r_m * w) * (w / ((1.0 - v) / r_m))) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5)
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 2.6e+17)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(r_m * w) * Float64(w * Float64(r_m * 0.375)))) - 4.5);
	else
		tmp = Float64(3.0 - Float64(Float64(Float64(Float64(r_m * w) * Float64(w / Float64(Float64(1.0 - v) / r_m))) * Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))) + 4.5));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 2.6e+17)
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
	else
		tmp = 3.0 - ((((r_m * w) * (w / ((1.0 - v) / r_m))) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 2.6e+17], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r$95$m * w), $MachinePrecision] * N[(w * N[(r$95$m * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(N[(N[(N[(r$95$m * w), $MachinePrecision] * N[(w / N[(N[(1.0 - v), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.6e17

   1. Initial program 84.9%

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

      1. associate-/l* 87.7%

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

      2. cancel-sign-sub-inv 87.7%

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

      3. metadata-eval 87.7%

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

      4. +-commutative 87.7%

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

      5. *-commutative 87.7%

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

      6. fma-undefine 87.7%

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

      7. *-commutative 87.7%

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

      8. *-commutative 87.7%

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

      9. associate-/l* 87.7%

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

      10. *-commutative 87.7%

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

      11. associate-*r/ 87.7%

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

      12. associate-*r* 86.5%

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

      13. associate-*l* 95.4%

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

      14. associate-*r* 96.9%

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in v around 0 83.2%

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

   6. Taylor expanded in v around 0 95.1%

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

      1. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   if 2.6e17 < r

   1. Initial program 89.9%

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

      1. associate--l- 89.9%

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

      2. associate-*l* 78.3%

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

      3. sqr-neg 78.3%

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

      4. associate-*l* 89.9%

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

      5. associate-/l* 94.6%

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

      6. fma-define 94.6%

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

   3. Simplified 94.6%

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

      1. associate-/l* 94.6%

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

      2. *-commutative 94.6%

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

      3. associate-*r/ 94.5%

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

      4. associate-*l* 97.2%

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

      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. clear-num 99.7%

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

      7. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in r around inf 99.9%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(\left(r \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r}}\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) + 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 2.6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(r\_m \cdot w\right) \cdot \left(w \cdot \left(r\_m \cdot 0.375\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(r\_m \cdot \left(w \cdot \left(r\_m \cdot \frac{w}{1 - v}\right)\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) + 4.5\right)\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 2.6e+17)
   (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* r_m w) (* w (* r_m 0.375)))) 4.5)
   (-
    3.0
    (+
     (* (* r_m (* w (* r_m (/ w (- 1.0 v))))) (* 0.125 (+ 3.0 (* -2.0 v))))
     4.5))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2.6e+17) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
	} else {
		tmp = 3.0 - (((r_m * (w * (r_m * (w / (1.0 - v))))) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 2.6d+17) then
        tmp = ((3.0d0 + (2.0d0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375d0)))) - 4.5d0
    else
        tmp = 3.0d0 - (((r_m * (w * (r_m * (w / (1.0d0 - v))))) * (0.125d0 * (3.0d0 + ((-2.0d0) * v)))) + 4.5d0)
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 2.6e+17) {
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
	} else {
		tmp = 3.0 - (((r_m * (w * (r_m * (w / (1.0 - v))))) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 2.6e+17:
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5
	else:
		tmp = 3.0 - (((r_m * (w * (r_m * (w / (1.0 - v))))) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5)
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 2.6e+17)
		tmp = Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(r_m * w) * Float64(w * Float64(r_m * 0.375)))) - 4.5);
	else
		tmp = Float64(3.0 - Float64(Float64(Float64(r_m * Float64(w * Float64(r_m * Float64(w / Float64(1.0 - v))))) * Float64(0.125 * Float64(3.0 + Float64(-2.0 * v)))) + 4.5));
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 2.6e+17)
		tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
	else
		tmp = 3.0 - (((r_m * (w * (r_m * (w / (1.0 - v))))) * (0.125 * (3.0 + (-2.0 * v)))) + 4.5);
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 2.6e+17], N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r$95$m * w), $MachinePrecision] * N[(w * N[(r$95$m * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(3.0 - N[(N[(N[(r$95$m * N[(w * N[(r$95$m * N[(w / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.125 * N[(3.0 + N[(-2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 4.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 2.6e17

   1. Initial program 84.9%

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

      1. associate-/l* 87.7%

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

      2. cancel-sign-sub-inv 87.7%

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

      3. metadata-eval 87.7%

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

      4. +-commutative 87.7%

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

      5. *-commutative 87.7%

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

      6. fma-undefine 87.7%

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

      7. *-commutative 87.7%

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

      8. *-commutative 87.7%

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

      9. associate-/l* 87.7%

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

      10. *-commutative 87.7%

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

      11. associate-*r/ 87.7%

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

      12. associate-*r* 86.5%

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

      13. associate-*l* 95.4%

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

      14. associate-*r* 96.9%

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

   4. Applied egg-rr 96.9%

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

   5. Taylor expanded in v around 0 83.2%

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

   6. Taylor expanded in v around 0 95.1%

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

      1. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   if 2.6e17 < r

   1. Initial program 89.9%

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

      1. associate--l- 89.9%

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

      2. associate-*l* 78.3%

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

      3. sqr-neg 78.3%

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

      4. associate-*l* 89.9%

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

      5. associate-/l* 94.6%

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

      6. fma-define 94.6%

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

   3. Simplified 94.6%

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

      1. associate-/r* 94.6%

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

      2. div-inv 94.6%

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

   6. Applied egg-rr 94.6%

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

      1. associate-*r/ 94.6%

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

      2. *-rgt-identity 94.6%

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

   8. Simplified 94.6%

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

      1. associate-/l* 94.6%

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

      2. *-commutative 94.6%

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

      3. associate-*r/ 94.5%

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

      4. associate-*l* 97.2%

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

      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. clear-num 99.7%

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

      7. un-div-inv 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. associate-*l* 97.2%

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

       2. associate-/r/ 96.5%

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

   12. Simplified 96.5%

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

   13. Taylor expanded in r around inf 96.5%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.6 \cdot 10^{+17}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;3 - \left(\left(r \cdot \left(w \cdot \left(r \cdot \frac{w}{1 - v}\right)\right)\right) \cdot \left(0.125 \cdot \left(3 + -2 \cdot v\right)\right) + 4.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} t_0 := r\_m \cdot \left(-0.25 \cdot \left(v \cdot w\right)\right)\\ \mathbf{if}\;r\_m \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r\_m}}{r\_m}\right) - 4.5\\ \mathbf{elif}\;r\_m \leq 4 \cdot 10^{+200}:\\ \;\;\;\;\left(3 + t\_0 \cdot \left(r\_m \cdot \frac{w}{v}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + t\_0 \cdot \frac{w}{\frac{v}{r\_m}}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (let* ((t_0 (* r_m (* -0.25 (* v w)))))
   (if (<= r_m 7e-18)
     (- (+ 3.0 (/ (/ 2.0 r_m) r_m)) 4.5)
     (if (<= r_m 4e+200)
       (- (+ 3.0 (* t_0 (* r_m (/ w v)))) 4.5)
       (- (+ 3.0 (* t_0 (/ w (/ v r_m)))) 4.5)))))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double t_0 = r_m * (-0.25 * (v * w));
	double tmp;
	if (r_m <= 7e-18) {
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	} else if (r_m <= 4e+200) {
		tmp = (3.0 + (t_0 * (r_m * (w / v)))) - 4.5;
	} else {
		tmp = (3.0 + (t_0 * (w / (v / r_m)))) - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r_m * ((-0.25d0) * (v * w))
    if (r_m <= 7d-18) then
        tmp = (3.0d0 + ((2.0d0 / r_m) / r_m)) - 4.5d0
    else if (r_m <= 4d+200) then
        tmp = (3.0d0 + (t_0 * (r_m * (w / v)))) - 4.5d0
    else
        tmp = (3.0d0 + (t_0 * (w / (v / r_m)))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double t_0 = r_m * (-0.25 * (v * w));
	double tmp;
	if (r_m <= 7e-18) {
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	} else if (r_m <= 4e+200) {
		tmp = (3.0 + (t_0 * (r_m * (w / v)))) - 4.5;
	} else {
		tmp = (3.0 + (t_0 * (w / (v / r_m)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	t_0 = r_m * (-0.25 * (v * w))
	tmp = 0
	if r_m <= 7e-18:
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5
	elif r_m <= 4e+200:
		tmp = (3.0 + (t_0 * (r_m * (w / v)))) - 4.5
	else:
		tmp = (3.0 + (t_0 * (w / (v / r_m)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	t_0 = Float64(r_m * Float64(-0.25 * Float64(v * w)))
	tmp = 0.0
	if (r_m <= 7e-18)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r_m) / r_m)) - 4.5);
	elseif (r_m <= 4e+200)
		tmp = Float64(Float64(3.0 + Float64(t_0 * Float64(r_m * Float64(w / v)))) - 4.5);
	else
		tmp = Float64(Float64(3.0 + Float64(t_0 * Float64(w / Float64(v / r_m)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	t_0 = r_m * (-0.25 * (v * w));
	tmp = 0.0;
	if (r_m <= 7e-18)
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	elseif (r_m <= 4e+200)
		tmp = (3.0 + (t_0 * (r_m * (w / v)))) - 4.5;
	else
		tmp = (3.0 + (t_0 * (w / (v / r_m)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := Block[{t$95$0 = N[(r$95$m * N[(-0.25 * N[(v * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r$95$m, 7e-18], N[(N[(3.0 + N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r$95$m, 4e+200], N[(N[(3.0 + N[(t$95$0 * N[(r$95$m * N[(w / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 + N[(t$95$0 * N[(w / N[(v / r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 6.9999999999999997e-18

   1. Initial program 85.1%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in r around 0 68.6%

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

      1. associate-/r* 87.9%

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

      2. div-inv 87.8%

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

   7. Applied egg-rr 68.5%

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

      1. associate-*r/ 87.9%

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

      2. *-rgt-identity 87.9%

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

   9. Simplified 68.6%

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

   if 6.9999999999999997e-18 < r < 3.9999999999999999e200

   1. Initial program 91.9%

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

      1. associate-/l* 97.3%

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

      2. cancel-sign-sub-inv 97.3%

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

      3. metadata-eval 97.3%

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

      4. +-commutative 97.3%

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

      5. *-commutative 97.3%

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

      6. fma-undefine 97.3%

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

      7. *-commutative 97.3%

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

      8. *-commutative 97.3%

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

      9. associate-/l* 97.3%

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

      10. *-commutative 97.3%

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

      11. associate-*r/ 97.3%

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

      12. associate-*r* 91.1%

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

      13. associate-*l* 91.1%

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

      14. associate-*r* 91.1%

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

   4. Applied egg-rr 91.1%

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

   5. Taylor expanded in v around inf 81.9%

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

      1. *-commutative 81.9%

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

      2. associate-*l* 81.9%

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

      3. *-commutative 81.9%

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

      4. associate-*r* 81.9%

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

      5. *-commutative 81.9%

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

      6. *-commutative 81.9%

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

      7. associate-*l* 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in r around inf 81.4%

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

   9. Taylor expanded in v around inf 83.4%

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

       1. mul-1-neg 83.4%

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

       2. associate-/l* 83.4%

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

       3. distribute-rgt-neg-in 83.4%

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

   11. Simplified 83.4%

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

   if 3.9999999999999999e200 < r

   1. Initial program 85.4%

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

      1. associate-/l* 88.8%

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

      2. cancel-sign-sub-inv 88.8%

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

      3. metadata-eval 88.8%

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

      4. +-commutative 88.8%

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

      5. *-commutative 88.8%

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

      6. fma-undefine 88.8%

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

      7. *-commutative 88.8%

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

      8. *-commutative 88.8%

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

      9. associate-/l* 88.7%

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

      10. *-commutative 88.7%

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

      11. associate-*r/ 88.7%

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

      12. associate-*r* 83.4%

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

      13. associate-*l* 85.7%

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

      14. associate-*r* 85.9%

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

   4. Applied egg-rr 85.9%

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

   5. Taylor expanded in v around inf 60.6%

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

      1. *-commutative 60.6%

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

      2. associate-*l* 60.6%

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

      3. *-commutative 60.6%

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

      4. associate-*r* 60.6%

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

      5. *-commutative 60.6%

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

      6. *-commutative 60.6%

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

      7. associate-*l* 60.6%

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

   7. Simplified 60.6%

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

   8. Taylor expanded in r around inf 60.6%

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

   9. Taylor expanded in v around inf 82.9%

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

       1. associate-*r/ 82.9%

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

       2. neg-mul-1 82.9%

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

   11. Simplified 82.9%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7 \cdot 10^{-18}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{elif}\;r \leq 4 \cdot 10^{+200}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(-0.25 \cdot \left(v \cdot w\right)\right)\right) \cdot \left(r \cdot \frac{w}{v}\right)\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(-0.25 \cdot \left(v \cdot w\right)\right)\right) \cdot \frac{w}{\frac{v}{r}}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 7.6 \cdot 10^{-18}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r\_m}}{r\_m}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r\_m \cdot \left(-0.25 \cdot \left(v \cdot w\right)\right)\right) \cdot \left(r\_m \cdot \frac{w}{v}\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 7.6e-18)
   (- (+ 3.0 (/ (/ 2.0 r_m) r_m)) 4.5)
   (- (+ 3.0 (* (* r_m (* -0.25 (* v w))) (* r_m (/ w v)))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 7.6e-18) {
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	} else {
		tmp = (3.0 + ((r_m * (-0.25 * (v * w))) * (r_m * (w / v)))) - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 7.6d-18) then
        tmp = (3.0d0 + ((2.0d0 / r_m) / r_m)) - 4.5d0
    else
        tmp = (3.0d0 + ((r_m * ((-0.25d0) * (v * w))) * (r_m * (w / v)))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 7.6e-18) {
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	} else {
		tmp = (3.0 + ((r_m * (-0.25 * (v * w))) * (r_m * (w / v)))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 7.6e-18:
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5
	else:
		tmp = (3.0 + ((r_m * (-0.25 * (v * w))) * (r_m * (w / v)))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 7.6e-18)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r_m) / r_m)) - 4.5);
	else
		tmp = Float64(Float64(3.0 + Float64(Float64(r_m * Float64(-0.25 * Float64(v * w))) * Float64(r_m * Float64(w / v)))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 7.6e-18)
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	else
		tmp = (3.0 + ((r_m * (-0.25 * (v * w))) * (r_m * (w / v)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 7.6e-18], N[(N[(3.0 + N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 + N[(N[(r$95$m * N[(-0.25 * N[(v * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(r$95$m * N[(w / v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 7.5999999999999996e-18

   1. Initial program 85.1%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in r around 0 68.6%

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

      1. associate-/r* 87.9%

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

      2. div-inv 87.8%

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

   7. Applied egg-rr 68.5%

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

      1. associate-*r/ 87.9%

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

      2. *-rgt-identity 87.9%

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

   9. Simplified 68.6%

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

   if 7.5999999999999996e-18 < r

   1. Initial program 89.0%

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

      1. associate-/l* 93.4%

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

      2. cancel-sign-sub-inv 93.4%

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

      3. metadata-eval 93.4%

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

      4. +-commutative 93.4%

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

      5. *-commutative 93.4%

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

      6. fma-undefine 93.4%

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

      7. *-commutative 93.4%

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

      8. *-commutative 93.4%

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

      9. associate-/l* 93.4%

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

      10. *-commutative 93.4%

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

      11. associate-*r/ 93.4%

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

      12. associate-*r* 87.6%

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

      13. associate-*l* 88.7%

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

      14. associate-*r* 88.8%

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

   4. Applied egg-rr 88.8%

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

   5. Taylor expanded in v around inf 72.3%

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

      1. *-commutative 72.3%

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

      2. associate-*l* 72.3%

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

      3. *-commutative 72.3%

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

      4. associate-*r* 72.3%

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

      5. *-commutative 72.3%

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

      6. *-commutative 72.3%

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

      7. associate-*l* 72.3%

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

   7. Simplified 72.3%

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

   8. Taylor expanded in r around inf 72.0%

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

   9. Taylor expanded in v around inf 83.6%

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

       1. mul-1-neg 83.6%

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

       2. associate-/l* 80.4%

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

       3. distribute-rgt-neg-in 80.4%

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

   11. Simplified 80.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.6 \cdot 10^{-18}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 + \left(r \cdot \left(-0.25 \cdot \left(v \cdot w\right)\right)\right) \cdot \left(r \cdot \frac{w}{v}\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 7.6 \cdot 10^{-18}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r\_m}}{r\_m}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r\_m \cdot w\right) \cdot \left(w \cdot \left(r\_m \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 7.6e-18)
   (- (+ 3.0 (/ (/ 2.0 r_m) r_m)) 4.5)
   (- (- 3.0 (* (* r_m w) (* w (* r_m (+ (* v -0.25) 0.375))))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 7.6e-18) {
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	} else {
		tmp = (3.0 - ((r_m * w) * (w * (r_m * ((v * -0.25) + 0.375))))) - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 7.6d-18) then
        tmp = (3.0d0 + ((2.0d0 / r_m) / r_m)) - 4.5d0
    else
        tmp = (3.0d0 - ((r_m * w) * (w * (r_m * ((v * (-0.25d0)) + 0.375d0))))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 7.6e-18) {
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	} else {
		tmp = (3.0 - ((r_m * w) * (w * (r_m * ((v * -0.25) + 0.375))))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 7.6e-18:
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5
	else:
		tmp = (3.0 - ((r_m * w) * (w * (r_m * ((v * -0.25) + 0.375))))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 7.6e-18)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r_m) / r_m)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r_m * w) * Float64(w * Float64(r_m * Float64(Float64(v * -0.25) + 0.375))))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 7.6e-18)
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	else
		tmp = (3.0 - ((r_m * w) * (w * (r_m * ((v * -0.25) + 0.375))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 7.6e-18], N[(N[(3.0 + N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(r$95$m * w), $MachinePrecision] * N[(w * N[(r$95$m * N[(N[(v * -0.25), $MachinePrecision] + 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 7.5999999999999996e-18

   1. Initial program 85.1%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in r around 0 68.6%

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

      1. associate-/r* 87.9%

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

      2. div-inv 87.8%

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

   7. Applied egg-rr 68.5%

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

      1. associate-*r/ 87.9%

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

      2. *-rgt-identity 87.9%

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

   9. Simplified 68.6%

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

   if 7.5999999999999996e-18 < r

   1. Initial program 89.0%

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

      1. associate-/l* 93.4%

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

      2. cancel-sign-sub-inv 93.4%

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

      3. metadata-eval 93.4%

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

      4. +-commutative 93.4%

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

      5. *-commutative 93.4%

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

      6. fma-undefine 93.4%

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

      7. *-commutative 93.4%

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

      8. *-commutative 93.4%

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

      9. associate-/l* 93.4%

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

      10. *-commutative 93.4%

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

      11. associate-*r/ 93.4%

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

      12. associate-*r* 87.6%

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

      13. associate-*l* 88.7%

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

      14. associate-*r* 88.8%

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

   4. Applied egg-rr 88.8%

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

   5. Taylor expanded in v around 0 65.9%

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

   6. Taylor expanded in r around inf 65.7%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 7.6 \cdot 10^{-18}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot \left(v \cdot -0.25 + 0.375\right)\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \begin{array}{l} \mathbf{if}\;r\_m \leq 1.9 \cdot 10^{+76}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r\_m}}{r\_m}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r\_m \cdot w\right) \cdot \left(r\_m \cdot \left(-0.25 \cdot \left(v \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (if (<= r_m 1.9e+76)
   (- (+ 3.0 (/ (/ 2.0 r_m) r_m)) 4.5)
   (- (- 3.0 (* (* r_m w) (* r_m (* -0.25 (* v w))))) 4.5)))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.9e+76) {
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	} else {
		tmp = (3.0 - ((r_m * w) * (r_m * (-0.25 * (v * w))))) - 4.5;
	}
	return tmp;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    real(8) :: tmp
    if (r_m <= 1.9d+76) then
        tmp = (3.0d0 + ((2.0d0 / r_m) / r_m)) - 4.5d0
    else
        tmp = (3.0d0 - ((r_m * w) * (r_m * ((-0.25d0) * (v * w))))) - 4.5d0
    end if
    code = tmp
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	double tmp;
	if (r_m <= 1.9e+76) {
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	} else {
		tmp = (3.0 - ((r_m * w) * (r_m * (-0.25 * (v * w))))) - 4.5;
	}
	return tmp;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	tmp = 0
	if r_m <= 1.9e+76:
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5
	else:
		tmp = (3.0 - ((r_m * w) * (r_m * (-0.25 * (v * w))))) - 4.5
	return tmp

Julia

r_m = abs(r)
function code(v, w, r_m)
	tmp = 0.0
	if (r_m <= 1.9e+76)
		tmp = Float64(Float64(3.0 + Float64(Float64(2.0 / r_m) / r_m)) - 4.5);
	else
		tmp = Float64(Float64(3.0 - Float64(Float64(r_m * w) * Float64(r_m * Float64(-0.25 * Float64(v * w))))) - 4.5);
	end
	return tmp
end

MATLAB

r_m = abs(r);
function tmp_2 = code(v, w, r_m)
	tmp = 0.0;
	if (r_m <= 1.9e+76)
		tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
	else
		tmp = (3.0 - ((r_m * w) * (r_m * (-0.25 * (v * w))))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := If[LessEqual[r$95$m, 1.9e+76], N[(N[(3.0 + N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(3.0 - N[(N[(r$95$m * w), $MachinePrecision] * N[(r$95$m * N[(-0.25 * N[(v * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1.90000000000000012e76

   1. Initial program 85.5%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in r around 0 67.9%

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

      1. associate-/r* 88.2%

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

      2. div-inv 88.1%

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

   7. Applied egg-rr 67.9%

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

      1. associate-*r/ 88.2%

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

      2. *-rgt-identity 88.2%

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

   9. Simplified 68.0%

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

   if 1.90000000000000012e76 < r

   1. Initial program 88.0%

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

      1. associate-/l* 93.6%

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

      2. cancel-sign-sub-inv 93.6%

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

      3. metadata-eval 93.6%

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

      4. +-commutative 93.6%

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

      5. *-commutative 93.6%

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

      6. fma-undefine 93.6%

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

      7. *-commutative 93.6%

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

      8. *-commutative 93.6%

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

      9. associate-/l* 93.6%

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

      10. *-commutative 93.6%

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

      11. associate-*r/ 93.5%

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

      12. associate-*r* 86.6%

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

      13. associate-*l* 88.0%

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

      14. associate-*r* 88.1%

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

   4. Applied egg-rr 88.1%

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

   5. Taylor expanded in v around inf 71.9%

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

      1. *-commutative 71.9%

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

      2. associate-*l* 71.9%

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

      3. *-commutative 71.9%

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

      4. associate-*r* 71.9%

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

      5. *-commutative 71.9%

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

      6. *-commutative 71.9%

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

      7. associate-*l* 71.9%

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

   7. Simplified 71.9%

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

   8. Taylor expanded in r around inf 71.9%

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

   9. Taylor expanded in v around 0 39.4%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.9 \cdot 10^{+76}:\\ \;\;\;\;\left(3 + \frac{\frac{2}{r}}{r}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\left(3 - \left(r \cdot w\right) \cdot \left(r \cdot \left(-0.25 \cdot \left(v \cdot w\right)\right)\right)\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \left(\left(3 + \frac{2}{r\_m \cdot r\_m}\right) - \left(r\_m \cdot w\right) \cdot \left(w \cdot \left(r\_m \cdot 0.375\right)\right)\right) - 4.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m)
 :precision binary64
 (- (- (+ 3.0 (/ 2.0 (* r_m r_m))) (* (* r_m w) (* w (* r_m 0.375)))) 4.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = ((3.0d0 + (2.0d0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375d0)))) - 4.5d0
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - Float64(Float64(r_m * w) * Float64(w * Float64(r_m * 0.375)))) - 4.5)
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = ((3.0 + (2.0 / (r_m * r_m))) - ((r_m * w) * (w * (r_m * 0.375)))) - 4.5;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(r$95$m * w), $MachinePrecision] * N[(w * N[(r$95$m * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

   1. associate-/l* 89.2%

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

   2. cancel-sign-sub-inv 89.2%

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

   3. metadata-eval 89.2%

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

   4. +-commutative 89.2%

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

   5. *-commutative 89.2%

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

   6. fma-undefine 89.2%

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

   7. *-commutative 89.2%

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

   8. *-commutative 89.2%

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

   9. associate-/l* 89.2%

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

   10. *-commutative 89.2%

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

   11. associate-*r/ 89.2%

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

   12. associate-*r* 86.6%

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

   13. associate-*l* 93.8%

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

   14. associate-*r* 95.0%

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

4. Applied egg-rr 95.0%

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

5. Taylor expanded in v around 0 78.9%

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

6. Taylor expanded in v around 0 93.4%

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

   1. *-commutative 93.4%

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

8. Simplified 93.4%

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

9. Final simplification 93.4%

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right) - 4.5 \]
10. Add Preprocessing

Alternative 10: 57.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \left(3 + \frac{\frac{2}{r\_m}}{r\_m}\right) - 4.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (- (+ 3.0 (/ (/ 2.0 r_m) r_m)) 4.5))

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
}

Fortran

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (3.0 + ((2.0 / r_m) / r_m)) - 4.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(3.0 + Float64(Float64(2.0 / r_m) / r_m)) - 4.5)
end

MATLAB

r_m = abs(r);
function tmp = code(v, w, r_m)
	tmp = (3.0 + ((2.0 / r_m) / r_m)) - 4.5;
end

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(3.0 + N[(N[(2.0 / r$95$m), $MachinePrecision] / r$95$m), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

3. Simplified 83.8%

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

5. Taylor expanded in r around 0 57.9%

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

   1. associate-/r* 89.2%

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

   2. div-inv 89.1%

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

7. Applied egg-rr 57.8%

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

   1. associate-*r/ 89.2%

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

   2. *-rgt-identity 89.2%

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

9. Simplified 57.9%

   \[\leadsto \left(3 + \color{blue}{\frac{\frac{2}{r}}{r}}\right) - 4.5 \]
10. Add Preprocessing

Alternative 11: 57.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ \left(3 + \frac{2}{r\_m \cdot r\_m}\right) - 4.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 (- (+ 3.0 (/ 2.0 (* r_m r_m))) 4.5))

C

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

Fortran

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

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return (3.0 + (2.0 / (r_m * r_m))) - 4.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return (3.0 + (2.0 / (r_m * r_m))) - 4.5

Julia

r_m = abs(r)
function code(v, w, r_m)
	return Float64(Float64(3.0 + Float64(2.0 / Float64(r_m * r_m))) - 4.5)
end

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := N[(N[(3.0 + N[(2.0 / N[(r$95$m * r$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]

Derivation

1. Initial program 86.0%

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

3. Simplified 83.8%

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

5. Taylor expanded in r around 0 57.9%

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

Alternative 12: 13.5% accurate, 29.0× speedup

Math

\[\begin{array}{l} r_m = \left|r\right| \\ -1.5 \end{array} \]

FPCore

r_m = (fabs.f64 r)
(FPCore (v w r_m) :precision binary64 -1.5)

C

r_m = fabs(r);
double code(double v, double w, double r_m) {
	return -1.5;
}

Fortran

r_m = abs(r)
real(8) function code(v, w, r_m)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r_m
    code = -1.5d0
end function

Java

r_m = Math.abs(r);
public static double code(double v, double w, double r_m) {
	return -1.5;
}

Python

r_m = math.fabs(r)
def code(v, w, r_m):
	return -1.5

Julia

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

MATLAB

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

Wolfram

r_m = N[Abs[r], $MachinePrecision]
code[v_, w_, r$95$m_] := -1.5

Derivation

1. Initial program 86.0%

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

3. Simplified 83.8%

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

5. Taylor expanded in r around 0 57.9%

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

6. Taylor expanded in r around inf 9.9%

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

Reproduce

herbie shell --seed 2024132 
(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))