Rosa's TurbineBenchmark

Percentage Accurate: 84.9% → 98.5%

Time: 14.9s

Alternatives: 11

Speedup: 1.3×

• 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: 84.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.3 \cdot 10^{+143}:\\ \;\;\;\;t\_0 + \left(-1.5 + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t\_0 + 3\right) + \left(\left(3 + v \cdot -2\right) \cdot \left(r \cdot \left(w \cdot 0.125\right)\right)\right) \cdot \frac{r \cdot w}{v + -1}\right) - 4.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -3.3e+143)
     (+ t_0 (+ -1.5 (* -0.25 (* w (* r (* r w))))))
     (-
      (+
       (+ t_0 3.0)
       (* (* (+ 3.0 (* v -2.0)) (* r (* w 0.125))) (/ (* r w) (+ v -1.0))))
      4.5))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -3.3e+143) {
		tmp = t_0 + (-1.5 + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = ((t_0 + 3.0) + (((3.0 + (v * -2.0)) * (r * (w * 0.125))) * ((r * w) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (v <= (-3.3d+143)) then
        tmp = t_0 + ((-1.5d0) + ((-0.25d0) * (w * (r * (r * w)))))
    else
        tmp = ((t_0 + 3.0d0) + (((3.0d0 + (v * (-2.0d0))) * (r * (w * 0.125d0))) * ((r * w) / (v + (-1.0d0))))) - 4.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -3.3e+143) {
		tmp = t_0 + (-1.5 + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = ((t_0 + 3.0) + (((3.0 + (v * -2.0)) * (r * (w * 0.125))) * ((r * w) / (v + -1.0)))) - 4.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= -3.3e+143:
		tmp = t_0 + (-1.5 + (-0.25 * (w * (r * (r * w)))))
	else:
		tmp = ((t_0 + 3.0) + (((3.0 + (v * -2.0)) * (r * (w * 0.125))) * ((r * w) / (v + -1.0)))) - 4.5
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -3.3e+143)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(-0.25 * Float64(w * Float64(r * Float64(r * w))))));
	else
		tmp = Float64(Float64(Float64(t_0 + 3.0) + Float64(Float64(Float64(3.0 + Float64(v * -2.0)) * Float64(r * Float64(w * 0.125))) * Float64(Float64(r * w) / Float64(v + -1.0)))) - 4.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -3.3e+143)
		tmp = t_0 + (-1.5 + (-0.25 * (w * (r * (r * w)))));
	else
		tmp = ((t_0 + 3.0) + (((3.0 + (v * -2.0)) * (r * (w * 0.125))) * ((r * w) / (v + -1.0)))) - 4.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.3e+143], N[(t$95$0 + N[(-1.5 + N[(-0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + 3.0), $MachinePrecision] + N[(N[(N[(3.0 + N[(v * -2.0), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -3.3e143

   1. Initial program 76.3%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) + \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{\frac{-3}{2}} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(r \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \color{blue}{r}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(\color{blue}{\left(r \cdot w\right)} \cdot w\right) \cdot r\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(\left(r \cdot w\right) \cdot \color{blue}{w}\right) \cdot r\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot r\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot r\right)}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \color{blue}{\left(\left(r \cdot w\right) \cdot r\right)}\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot \color{blue}{\left(r \cdot w\right)}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot w\right)}\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 95.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 95.6%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 + \left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\color{blue}{\frac{-1}{4}}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 95.6%

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

   if -3.3e143 < v

   1. Initial program 83.2%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(w, \left(\left(w \cdot r\right) \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\left(w \cdot r\right), r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\left(r \cdot w\right), r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      6. *-lowering-*.f64 93.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 93.6%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot \left(r \cdot \left(r \cdot w\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot r\right) \cdot \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right) \cdot r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot w\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\left(3 - 2 \cdot v\right) \cdot \frac{1}{8}\right) \cdot w\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(3 - 2 \cdot v\right) \cdot \left(\frac{1}{8} \cdot w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(3 - 2 \cdot v\right), \left(\frac{1}{8} \cdot w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(3 + \left(\mathsf{neg}\left(2 \cdot v\right)\right)\right), \left(\frac{1}{8} \cdot w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \left(\mathsf{neg}\left(2 \cdot v\right)\right)\right), \left(\frac{1}{8} \cdot w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \left(\mathsf{neg}\left(v \cdot 2\right)\right)\right), \left(\frac{1}{8} \cdot w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      12. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \left(v \cdot \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(\frac{1}{8} \cdot w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \mathsf{*.f64}\left(v, \left(\mathsf{neg}\left(2\right)\right)\right)\right), \left(\frac{1}{8} \cdot w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \mathsf{*.f64}\left(v, -2\right)\right), \left(\frac{1}{8} \cdot w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \mathsf{*.f64}\left(v, -2\right)\right), \mathsf{*.f64}\left(\frac{1}{8}, w\right)\right), r\right), \left(r \cdot w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      16. *-lowering-*.f64 95.3%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \mathsf{*.f64}\left(v, -2\right)\right), \mathsf{*.f64}\left(\frac{1}{8}, w\right)\right), r\right), \mathsf{*.f64}\left(r, w\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   6. Applied egg-rr 95.3%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\left(3 + v \cdot -2\right) \cdot \left(\frac{1}{8} \cdot w\right)\right) \cdot r\right) \cdot \frac{r \cdot w}{1 - v}\right)\right), \frac{9}{2}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\frac{r \cdot w}{1 - v} \cdot \left(\left(\left(3 + v \cdot -2\right) \cdot \left(\frac{1}{8} \cdot w\right)\right) \cdot r\right)\right)\right), \frac{9}{2}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\frac{r \cdot w}{1 - v}\right), \left(\left(\left(3 + v \cdot -2\right) \cdot \left(\frac{1}{8} \cdot w\right)\right) \cdot r\right)\right)\right), \frac{9}{2}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(r \cdot w\right), \left(1 - v\right)\right), \left(\left(\left(3 + v \cdot -2\right) \cdot \left(\frac{1}{8} \cdot w\right)\right) \cdot r\right)\right)\right), \frac{9}{2}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \left(1 - v\right)\right), \left(\left(\left(3 + v \cdot -2\right) \cdot \left(\frac{1}{8} \cdot w\right)\right) \cdot r\right)\right)\right), \frac{9}{2}\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \left(\left(\left(3 + v \cdot -2\right) \cdot \left(\frac{1}{8} \cdot w\right)\right) \cdot r\right)\right)\right), \frac{9}{2}\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \left(\left(3 + v \cdot -2\right) \cdot \left(\left(\frac{1}{8} \cdot w\right) \cdot r\right)\right)\right)\right), \frac{9}{2}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{*.f64}\left(\left(3 + v \cdot -2\right), \left(\left(\frac{1}{8} \cdot w\right) \cdot r\right)\right)\right)\right), \frac{9}{2}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \left(v \cdot -2\right)\right), \left(\left(\frac{1}{8} \cdot w\right) \cdot r\right)\right)\right)\right), \frac{9}{2}\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \mathsf{*.f64}\left(v, -2\right)\right), \left(\left(\frac{1}{8} \cdot w\right) \cdot r\right)\right)\right)\right), \frac{9}{2}\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \mathsf{*.f64}\left(v, -2\right)\right), \left(r \cdot \left(\frac{1}{8} \cdot w\right)\right)\right)\right)\right), \frac{9}{2}\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \mathsf{*.f64}\left(v, -2\right)\right), \mathsf{*.f64}\left(r, \left(\frac{1}{8} \cdot w\right)\right)\right)\right)\right), \frac{9}{2}\right) \]

      13. *-lowering-*.f64 98.9%

          \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(r, w\right), \mathsf{\_.f64}\left(1, v\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(3, \mathsf{*.f64}\left(v, -2\right)\right), \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\frac{1}{8}, w\right)\right)\right)\right)\right), \frac{9}{2}\right) \]

   8. Applied egg-rr 98.9%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.3 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \left(\left(3 + v \cdot -2\right) \cdot \left(r \cdot \left(w \cdot 0.125\right)\right)\right) \cdot \frac{r \cdot w}{v + -1}\right) - 4.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(r \cdot \left(r \cdot w\right)\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\left(t\_1 + -1.5\right) + r \cdot \left(\left(-0.25 + \frac{0.125}{v}\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 2.55 \cdot 10^{-46}:\\ \;\;\;\;\left(\left(t\_1 + 3\right) + \frac{0.375 \cdot t\_0}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(-1.5 + -0.25 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* r (* r w)))) (t_1 (/ 2.0 (* r r))))
   (if (<= v -3.9e-13)
     (+ (+ t_1 -1.5) (* r (* (+ -0.25 (/ 0.125 v)) (* w (* r w)))))
     (if (<= v 2.55e-46)
       (- (+ (+ t_1 3.0) (/ (* 0.375 t_0) (+ v -1.0))) 4.5)
       (+ t_1 (+ -1.5 (* -0.25 t_0)))))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r * (r * w));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -3.9e-13) {
		tmp = (t_1 + -1.5) + (r * ((-0.25 + (0.125 / v)) * (w * (r * w))));
	} else if (v <= 2.55e-46) {
		tmp = ((t_1 + 3.0) + ((0.375 * t_0) / (v + -1.0))) - 4.5;
	} else {
		tmp = t_1 + (-1.5 + (-0.25 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = w * (r * (r * w))
    t_1 = 2.0d0 / (r * r)
    if (v <= (-3.9d-13)) then
        tmp = (t_1 + (-1.5d0)) + (r * (((-0.25d0) + (0.125d0 / v)) * (w * (r * w))))
    else if (v <= 2.55d-46) then
        tmp = ((t_1 + 3.0d0) + ((0.375d0 * t_0) / (v + (-1.0d0)))) - 4.5d0
    else
        tmp = t_1 + ((-1.5d0) + ((-0.25d0) * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * (r * (r * w));
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -3.9e-13) {
		tmp = (t_1 + -1.5) + (r * ((-0.25 + (0.125 / v)) * (w * (r * w))));
	} else if (v <= 2.55e-46) {
		tmp = ((t_1 + 3.0) + ((0.375 * t_0) / (v + -1.0))) - 4.5;
	} else {
		tmp = t_1 + (-1.5 + (-0.25 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * (r * (r * w))
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= -3.9e-13:
		tmp = (t_1 + -1.5) + (r * ((-0.25 + (0.125 / v)) * (w * (r * w))))
	elif v <= 2.55e-46:
		tmp = ((t_1 + 3.0) + ((0.375 * t_0) / (v + -1.0))) - 4.5
	else:
		tmp = t_1 + (-1.5 + (-0.25 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r * Float64(r * w)))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -3.9e-13)
		tmp = Float64(Float64(t_1 + -1.5) + Float64(r * Float64(Float64(-0.25 + Float64(0.125 / v)) * Float64(w * Float64(r * w)))));
	elseif (v <= 2.55e-46)
		tmp = Float64(Float64(Float64(t_1 + 3.0) + Float64(Float64(0.375 * t_0) / Float64(v + -1.0))) - 4.5);
	else
		tmp = Float64(t_1 + Float64(-1.5 + Float64(-0.25 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * (r * (r * w));
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -3.9e-13)
		tmp = (t_1 + -1.5) + (r * ((-0.25 + (0.125 / v)) * (w * (r * w))));
	elseif (v <= 2.55e-46)
		tmp = ((t_1 + 3.0) + ((0.375 * t_0) / (v + -1.0))) - 4.5;
	else
		tmp = t_1 + (-1.5 + (-0.25 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -3.9e-13], N[(N[(t$95$1 + -1.5), $MachinePrecision] + N[(r * N[(N[(-0.25 + N[(0.125 / v), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.55e-46], N[(N[(N[(t$95$1 + 3.0), $MachinePrecision] + N[(N[(0.375 * t$95$0), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(t$95$1 + N[(-1.5 + N[(-0.25 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.90000000000000004e-13

   1. Initial program 78.6%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 87.4%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(r \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \cdot \left(\frac{\frac{1}{8}}{\color{blue}{v}} - \frac{1}{4}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(r \cdot \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right) \cdot \color{blue}{r}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right), \color{blue}{r}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \left(r \cdot w\right)\right)\right), r\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), r\right)\right) \]

      13. *-lowering-*.f64 94.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]

   7. Applied egg-rr 94.1%

      \[\leadsto \left(-1.5 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r} \]

   if -3.90000000000000004e-13 < v < 2.5499999999999999e-46

   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* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(\left(w \cdot \left(w \cdot r\right)\right) \cdot r\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \left(w \cdot \left(\left(w \cdot r\right) \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(w, \left(\left(w \cdot r\right) \cdot r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\left(w \cdot r\right), r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\left(r \cdot w\right), r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

      6. *-lowering-*.f64 97.0%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, \mathsf{\_.f64}\left(3, \mathsf{*.f64}\left(2, v\right)\right)\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in v around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{\_.f64}\left(\mathsf{+.f64}\left(3, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{3}{8}}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, w\right), r\right)\right)\right), \mathsf{\_.f64}\left(1, v\right)\right)\right), \frac{9}{2}\right) \]
   6. Step-by-step derivation

   7. Simplified 97.0%

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

   if 2.5499999999999999e-46 < v

   1. Initial program 81.3%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) + \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{\frac{-3}{2}} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(r \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \color{blue}{r}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(\color{blue}{\left(r \cdot w\right)} \cdot w\right) \cdot r\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(\left(r \cdot w\right) \cdot \color{blue}{w}\right) \cdot r\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot r\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot r\right)}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \color{blue}{\left(\left(r \cdot w\right) \cdot r\right)}\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot \color{blue}{\left(r \cdot w\right)}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot w\right)}\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 94.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 94.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 + \left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\color{blue}{\frac{-1}{4}}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 98.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.9 \cdot 10^{-13}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + r \cdot \left(\left(-0.25 + \frac{0.125}{v}\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{elif}\;v \leq 2.55 \cdot 10^{-46}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \frac{0.375 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)}{v + -1}\right) - 4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 1.95 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 10^{+85}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1.95e-122)
   (/ (/ 2.0 r) r)
   (if (<= r 1e+85)
     (+ (+ (/ 2.0 (* r r)) -1.5) (* r (* r (* -0.25 (* w w)))))
     (+ -1.5 (* r (* -0.25 (* w (* r w))))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.95e-122) {
		tmp = (2.0 / r) / r;
	} else if (r <= 1e+85) {
		tmp = ((2.0 / (r * r)) + -1.5) + (r * (r * (-0.25 * (w * w))));
	} else {
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1.95d-122) then
        tmp = (2.0d0 / r) / r
    else if (r <= 1d+85) then
        tmp = ((2.0d0 / (r * r)) + (-1.5d0)) + (r * (r * ((-0.25d0) * (w * w))))
    else
        tmp = (-1.5d0) + (r * ((-0.25d0) * (w * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1.95e-122) {
		tmp = (2.0 / r) / r;
	} else if (r <= 1e+85) {
		tmp = ((2.0 / (r * r)) + -1.5) + (r * (r * (-0.25 * (w * w))));
	} else {
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1.95e-122:
		tmp = (2.0 / r) / r
	elif r <= 1e+85:
		tmp = ((2.0 / (r * r)) + -1.5) + (r * (r * (-0.25 * (w * w))))
	else:
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1.95e-122)
		tmp = Float64(Float64(2.0 / r) / r);
	elseif (r <= 1e+85)
		tmp = Float64(Float64(Float64(2.0 / Float64(r * r)) + -1.5) + Float64(r * Float64(r * Float64(-0.25 * Float64(w * w)))));
	else
		tmp = Float64(-1.5 + Float64(r * Float64(-0.25 * Float64(w * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1.95e-122)
		tmp = (2.0 / r) / r;
	elseif (r <= 1e+85)
		tmp = ((2.0 / (r * r)) + -1.5) + (r * (r * (-0.25 * (w * w))));
	else
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1.95e-122], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], If[LessEqual[r, 1e+85], N[(N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision] + N[(r * N[(r * N[(-0.25 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r * N[(-0.25 * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 1.94999999999999995e-122

   1. Initial program 81.0%

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

   3. Taylor expanded in r around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 55.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 55.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 55.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   7. Applied egg-rr 55.2%

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

   if 1.94999999999999995e-122 < r < 1e85

   1. Initial program 89.4%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 80.8%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(r \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \cdot \left(\frac{\frac{1}{8}}{\color{blue}{v}} - \frac{1}{4}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(r \cdot \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right) \cdot \color{blue}{r}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right), \color{blue}{r}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \left(r \cdot w\right)\right)\right), r\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), r\right)\right) \]

      13. *-lowering-*.f64 80.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]

   7. Applied egg-rr 80.8%

      \[\leadsto \left(-1.5 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(r \cdot {w}^{2}\right)\right)}, r\right)\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(r \cdot {w}^{2}\right) \cdot \frac{-1}{4}\right), r\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(r \cdot \left({w}^{2} \cdot \frac{-1}{4}\right)\right), r\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \left({w}^{2} \cdot \frac{-1}{4}\right)\right), r\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\left({w}^{2}\right), \frac{-1}{4}\right)\right), r\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-1}{4}\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 92.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-1}{4}\right)\right), r\right)\right) \]

   10. Simplified 92.1%

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

   if 1e85 < r

   1. Initial program 79.9%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 79.6%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(r \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \cdot \left(\frac{\frac{1}{8}}{\color{blue}{v}} - \frac{1}{4}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(r \cdot \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right) \cdot \color{blue}{r}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right), \color{blue}{r}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \left(r \cdot w\right)\right)\right), r\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), r\right)\right) \]

      13. *-lowering-*.f64 89.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]

   7. Applied egg-rr 89.6%

      \[\leadsto \left(-1.5 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r} \]

   8. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\frac{-3}{2}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 89.6%

       \[\leadsto \color{blue}{-1.5} + \left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r \]

   11. Taylor expanded in v around inf

       \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{-1}{4}}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 94.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 1.95 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 10^{+85}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) + r \cdot \left(r \cdot \left(-0.25 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 10^{+148}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 1e+148)
   (+ (/ 2.0 (* r r)) (+ -1.5 (* -0.25 (* w (* r (* r w))))))
   (+ -1.5 (* r (* -0.25 (* w (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 1e+148) {
		tmp = (2.0 / (r * r)) + (-1.5 + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 1d+148) then
        tmp = (2.0d0 / (r * r)) + ((-1.5d0) + ((-0.25d0) * (w * (r * (r * w)))))
    else
        tmp = (-1.5d0) + (r * ((-0.25d0) * (w * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 1e+148) {
		tmp = (2.0 / (r * r)) + (-1.5 + (-0.25 * (w * (r * (r * w)))));
	} else {
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 1e+148:
		tmp = (2.0 / (r * r)) + (-1.5 + (-0.25 * (w * (r * (r * w)))))
	else:
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 1e+148)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(-0.25 * Float64(w * Float64(r * Float64(r * w))))));
	else
		tmp = Float64(-1.5 + Float64(r * Float64(-0.25 * Float64(w * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 1e+148)
		tmp = (2.0 / (r * r)) + (-1.5 + (-0.25 * (w * (r * (r * w)))));
	else
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 1e+148], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(-0.25 * N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.5 + N[(r * N[(-0.25 * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 1e148

   1. Initial program 82.6%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 79.7%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + \frac{-3}{2}\right) + \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)} \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \]

      2. associate-+l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{\frac{-3}{2}} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\frac{-3}{2} + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \color{blue}{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(r \cdot \left(\left(r \cdot w\right) \cdot \color{blue}{w}\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \color{blue}{r}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot r\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\color{blue}{\left(\left(r \cdot w\right) \cdot w\right)} \cdot r\right)\right)\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(\color{blue}{\left(r \cdot w\right)} \cdot w\right) \cdot r\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(\left(r \cdot w\right) \cdot \color{blue}{w}\right) \cdot r\right)\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot r\right)\right)\right)\right) \]

      16. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot r\right)}\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \color{blue}{\left(\left(r \cdot w\right) \cdot r\right)}\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot \color{blue}{\left(r \cdot w\right)}\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \color{blue}{\left(r \cdot w\right)}\right)\right)\right)\right)\right) \]

      20. *-lowering-*.f64 85.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right)\right)\right)\right)\right) \]

   7. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 + \left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)} \]

   8. Taylor expanded in v around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\color{blue}{\frac{-1}{4}}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(r, w\right)\right)\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 93.4%

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

   if 1e148 < r

   1. Initial program 78.5%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 75.6%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(r \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \cdot \left(\frac{\frac{1}{8}}{\color{blue}{v}} - \frac{1}{4}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(r \cdot \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right) \cdot \color{blue}{r}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right), \color{blue}{r}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \left(r \cdot w\right)\right)\right), r\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), r\right)\right) \]

      13. *-lowering-*.f64 89.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]

   7. Applied egg-rr 89.0%

      \[\leadsto \left(-1.5 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r} \]

   8. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\frac{-3}{2}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 89.0%

       \[\leadsto \color{blue}{-1.5} + \left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r \]

   11. Taylor expanded in v around inf

       \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{-1}{4}}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 95.4%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 10^{+148}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + -0.25 \cdot \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e-27) (/ (/ 2.0 r) r) (+ -1.5 (* r (* -0.25 (* w (* r w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e-27) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 5d-27) then
        tmp = (2.0d0 / r) / r
    else
        tmp = (-1.5d0) + (r * ((-0.25d0) * (w * (r * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e-27) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5e-27:
		tmp = (2.0 / r) / r
	else:
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e-27)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(-1.5 + Float64(r * Float64(-0.25 * Float64(w * Float64(r * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5e-27)
		tmp = (2.0 / r) / r;
	else
		tmp = -1.5 + (r * (-0.25 * (w * (r * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e-27], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(-1.5 + N[(r * N[(-0.25 * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.0000000000000002e-27

   1. Initial program 81.7%

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

   3. Taylor expanded in r around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 57.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   7. Applied egg-rr 57.2%

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

   if 5.0000000000000002e-27 < r

   1. Initial program 83.0%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 79.4%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(r \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \cdot \left(\frac{\frac{1}{8}}{\color{blue}{v}} - \frac{1}{4}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(r \cdot \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right) \cdot \color{blue}{r}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right), \color{blue}{r}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \left(r \cdot w\right)\right)\right), r\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), r\right)\right) \]

      13. *-lowering-*.f64 87.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]

   7. Applied egg-rr 87.1%

      \[\leadsto \left(-1.5 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r} \]

   8. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\frac{-3}{2}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 87.1%

       \[\leadsto \color{blue}{-1.5} + \left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r \]

   11. Taylor expanded in v around inf

       \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\frac{-1}{4}}, \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 96.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 4.6e-27) (/ (/ 2.0 r) r) (+ -1.5 (* r (* -0.25 (* r (* w w)))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.6e-27) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 4.6d-27) then
        tmp = (2.0d0 / r) / r
    else
        tmp = (-1.5d0) + (r * ((-0.25d0) * (r * (w * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 4.6e-27) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 4.6e-27:
		tmp = (2.0 / r) / r
	else:
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 4.6e-27)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = Float64(-1.5 + Float64(r * Float64(-0.25 * Float64(r * Float64(w * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 4.6e-27)
		tmp = (2.0 / r) / r;
	else
		tmp = -1.5 + (r * (-0.25 * (r * (w * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 4.6e-27], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], N[(-1.5 + N[(r * N[(-0.25 * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 4.5999999999999999e-27

   1. Initial program 81.7%

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

   3. Taylor expanded in r around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 57.2%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 57.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   7. Applied egg-rr 57.2%

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

   if 4.5999999999999999e-27 < r

   1. Initial program 83.0%

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

   3. Taylor expanded in v around inf

      \[\leadsto \color{blue}{\left(\frac{-1}{8} \cdot \frac{-3 \cdot \left({r}^{2} \cdot {w}^{2}\right) - -2 \cdot \left({r}^{2} \cdot {w}^{2}\right)}{v} + 2 \cdot \frac{1}{{r}^{2}}\right) - \left(\frac{3}{2} + \frac{1}{4} \cdot \left({r}^{2} \cdot {w}^{2}\right)\right)} \]

   5. Simplified 79.4%

      \[\leadsto \color{blue}{\left(-1.5 + \frac{2}{r \cdot r}\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(\frac{0.125}{v} - 0.25\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(r \cdot \left(\left(r \cdot w\right) \cdot w\right)\right) \cdot \left(\frac{\frac{1}{8}}{\color{blue}{v}} - \frac{1}{4}\right)\right)\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(r \cdot \color{blue}{\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right) \cdot \color{blue}{r}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\left(r \cdot w\right) \cdot w\right) \cdot \left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right)\right), \color{blue}{r}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right) \cdot \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} - \frac{1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{8}}{v} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{v}\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(\left(r \cdot w\right) \cdot w\right)\right), r\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \left(w \cdot \left(r \cdot w\right)\right)\right), r\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \left(r \cdot w\right)\right)\right), r\right)\right) \]

      13. *-lowering-*.f64 87.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]

   7. Applied egg-rr 87.1%

      \[\leadsto \left(-1.5 + \frac{2}{r \cdot r}\right) + \color{blue}{\left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r} \]

   8. Taylor expanded in r around inf

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\frac{-3}{2}}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, v\right), \frac{-1}{4}\right), \mathsf{*.f64}\left(w, \mathsf{*.f64}\left(r, w\right)\right)\right), r\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 87.1%

       \[\leadsto \color{blue}{-1.5} + \left(\left(\frac{0.125}{v} + -0.25\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot r \]

   11. Taylor expanded in v around inf

       \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\color{blue}{\left(\frac{-1}{4} \cdot \left(r \cdot {w}^{2}\right)\right)}, r\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(r \cdot {w}^{2}\right)\right), r\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \left({w}^{2}\right)\right)\right), r\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \left(w \cdot w\right)\right)\right), r\right)\right) \]

       4. *-lowering-*.f64 88.3%

          \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, w\right)\right)\right), r\right)\right) \]

   13. Simplified 88.3%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5 + r \cdot \left(-0.25 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 9.2e-15) (/ (/ 2.0 r) r) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 9.2e-15) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 9.2d-15) then
        tmp = (2.0d0 / r) / r
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 9.2e-15) {
		tmp = (2.0 / r) / r;
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 9.2e-15:
		tmp = (2.0 / r) / r
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 9.2e-15)
		tmp = Float64(Float64(2.0 / r) / r);
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 9.2e-15)
		tmp = (2.0 / r) / r;
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 9.2e-15], N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 9.19999999999999961e-15

   1. Initial program 81.8%

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

   3. Taylor expanded in r around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 56.9%

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

      1. associate-/r* N/A

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

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{r}\right), \color{blue}{r}\right) \]

      3. /-lowering-/.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right) \]

   7. Applied egg-rr 56.9%

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

   if 9.19999999999999961e-15 < r

   1. Initial program 82.7%

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

   3. Taylor expanded in r around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{2 + 3 \cdot {r}^{2}}{{r}^{2}}\right)}, \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r \cdot r}\right), \frac{9}{2}\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}\right), \frac{9}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r}\right), r\right), \frac{9}{2}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + 3 \cdot {r}^{2}\right), r\right), r\right), \frac{9}{2}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(3 \cdot {r}^{2}\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({r}^{2} \cdot 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({r}^{2}\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(r \cdot r\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      9. *-lowering-*.f64 15.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   5. Simplified 15.9%

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

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 28.1%

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

Alternative 8: 50.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 9.2e-15) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 9.2e-15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: tmp
    if (r <= 9.2d-15) then
        tmp = 2.0d0 / (r * r)
    else
        tmp = -1.5d0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 9.2e-15) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 9.2e-15:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 9.2e-15)
		tmp = Float64(2.0 / Float64(r * r));
	else
		tmp = -1.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 9.2e-15)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 9.2e-15], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 9.19999999999999961e-15

   1. Initial program 81.8%

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

   3. Taylor expanded in r around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right) \]

      3. *-lowering-*.f64 56.9%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right) \]

   5. Simplified 56.9%

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

   if 9.19999999999999961e-15 < r

   1. Initial program 82.7%

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

   3. Taylor expanded in r around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{2 + 3 \cdot {r}^{2}}{{r}^{2}}\right)}, \frac{9}{2}\right) \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r \cdot r}\right), \frac{9}{2}\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}\right), \frac{9}{2}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r}\right), r\right), \frac{9}{2}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + 3 \cdot {r}^{2}\right), r\right), r\right), \frac{9}{2}\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(3 \cdot {r}^{2}\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({r}^{2} \cdot 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({r}^{2}\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(r \cdot r\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

      9. *-lowering-*.f64 15.9%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   5. Simplified 15.9%

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

   6. Taylor expanded in r around inf

      \[\leadsto \color{blue}{\frac{-3}{2}} \]
   7. Step-by-step derivation

   8. Simplified 28.1%

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

Alternative 9: 57.0% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in w around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2}\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto 2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{-3}{2} + \color{blue}{2 \cdot \frac{1}{{r}^{2}}} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \color{blue}{\left(2 \cdot \frac{1}{{r}^{2}}\right)}\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\frac{2 \cdot 1}{\color{blue}{{r}^{2}}}\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \left(\frac{2}{{\color{blue}{r}}^{2}}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \color{blue}{\left({r}^{2}\right)}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \left(r \cdot \color{blue}{r}\right)\right)\right) \]

   9. *-lowering-*.f64 55.6%

      \[\leadsto \mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, \color{blue}{r}\right)\right)\right) \]

5. Simplified 55.6%

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

6. Final simplification 55.6%

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

Alternative 10: 14.1% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in r around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{2 + 3 \cdot {r}^{2}}{{r}^{2}}\right)}, \frac{9}{2}\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r \cdot r}\right), \frac{9}{2}\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}\right), \frac{9}{2}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r}\right), r\right), \frac{9}{2}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + 3 \cdot {r}^{2}\right), r\right), r\right), \frac{9}{2}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(3 \cdot {r}^{2}\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({r}^{2} \cdot 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({r}^{2}\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(r \cdot r\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   9. *-lowering-*.f64 51.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

5. Simplified 51.0%

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

6. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{-3}{2}} \]
7. Step-by-step derivation

8. Simplified 12.7%

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

Alternative 11: 4.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

double code(double v, double w, double r) {
	return -4.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -4.5d0
end function

Java

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

Python

def code(v, w, r):
	return -4.5

Julia

function code(v, w, r)
	return -4.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -4.5;
end

Wolfram

code[v_, w_, r_] := -4.5

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in r around 0

   \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\frac{2 + 3 \cdot {r}^{2}}{{r}^{2}}\right)}, \frac{9}{2}\right) \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r \cdot r}\right), \frac{9}{2}\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{\_.f64}\left(\left(\frac{\frac{2 + 3 \cdot {r}^{2}}{r}}{r}\right), \frac{9}{2}\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 + 3 \cdot {r}^{2}}{r}\right), r\right), \frac{9}{2}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 + 3 \cdot {r}^{2}\right), r\right), r\right), \frac{9}{2}\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left(3 \cdot {r}^{2}\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \left({r}^{2} \cdot 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({r}^{2}\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(r \cdot r\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

   9. *-lowering-*.f64 51.0%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), 3\right)\right), r\right), r\right), \frac{9}{2}\right) \]

5. Simplified 51.0%

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

6. Taylor expanded in r around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(\frac{2}{r}\right)}, r\right), \frac{9}{2}\right) \]
7. Step-by-step derivation

   1. /-lowering-/.f64 46.6%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, r\right), r\right), \frac{9}{2}\right) \]

8. Simplified 46.6%

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

9. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{-9}{2}} \]
10. Step-by-step derivation

11. Simplified 4.1%

    \[\leadsto \color{blue}{-4.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))