Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.7%

Time: 13.4s

Alternatives: 13

Speedup: 1.7×

• 52.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}{v + -1} + -1.5\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+
  (/ 2.0 (* r r))
  (+ (* (+ 0.375 (* v -0.25)) (/ (* (* r w) (* r w)) (+ v -1.0))) -1.5)))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (((0.375 + (v * -0.25)) * (((r * w) * (r * w)) / (v + -1.0))) + -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)) + (((0.375d0 + (v * (-0.25d0))) * (((r * w) * (r * w)) / (v + (-1.0d0)))) + (-1.5d0))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (((0.375 + (v * -0.25)) * (((r * w) * (r * w)) / (v + -1.0))) + -1.5);
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (((0.375 + (v * -0.25)) * (((r * w) * (r * w)) / (v + -1.0))) + -1.5)

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(Float64(r * w) * Float64(r * w)) / Float64(v + -1.0))) + -1.5))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (((0.375 + (v * -0.25)) * (((r * w) * (r * w)) / (v + -1.0))) + -1.5);
end

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. associate--l- N/A

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

   7. associate--r+ N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate--l+ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

3. Simplified 95.3%

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

   1. associate-*r* N/A

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

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

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

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

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

   4. *-lowering-*.f64 99.8%

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

6. Applied egg-rr 99.8%

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

Alternative 2: 93.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 6.8 \cdot 10^{-29}:\\ \;\;\;\;t\_0 + \left(-1.5 - w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 6.8e-29)
     (+ t_0 (- -1.5 (* w (* w (* (* r r) 0.25)))))
     (+
      t_0
      (+ -1.5 (* (+ 0.375 (* v -0.25)) (/ (* r (* w (* r w))) (+ v -1.0))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.8e-29) {
		tmp = t_0 + (-1.5 - (w * (w * ((r * r) * 0.25))));
	} else {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (w * (r * w))) / (v + -1.0))));
	}
	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) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 6.8d-29) then
        tmp = t_0 + ((-1.5d0) - (w * (w * ((r * r) * 0.25d0))))
    else
        tmp = t_0 + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) * ((r * (w * (r * w))) / (v + (-1.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 6.8e-29) {
		tmp = t_0 + (-1.5 - (w * (w * ((r * r) * 0.25))));
	} else {
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (w * (r * w))) / (v + -1.0))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 6.8e-29:
		tmp = t_0 + (-1.5 - (w * (w * ((r * r) * 0.25))))
	else:
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (w * (r * w))) / (v + -1.0))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 6.8e-29)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(w * Float64(w * Float64(Float64(r * r) * 0.25)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) * Float64(Float64(r * Float64(w * Float64(r * w))) / Float64(v + -1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 6.8e-29)
		tmp = t_0 + (-1.5 - (w * (w * ((r * r) * 0.25))));
	else
		tmp = t_0 + (-1.5 + ((0.375 + (v * -0.25)) * ((r * (w * (r * w))) / (v + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 6.8e-29], N[(t$95$0 + N[(-1.5 - N[(w * N[(w * N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] * N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(v + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if r < 6.79999999999999945e-29

   1. Initial program 84.6%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 93.6%

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

   5. Taylor expanded in v around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. associate--l+ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. unpow2 N/A

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

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

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

      14. --lowering--.f64 N/A

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

      15. associate-*r* N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

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

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

   7. Simplified 93.5%

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

   if 6.79999999999999945e-29 < r

   1. Initial program 91.7%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.8%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 6.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 2.8 \cdot 10^{-116}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;r \leq 3.8 \cdot 10^{+151}:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= r 2.8e-116)
     t_0
     (if (<= r 3.8e+151)
       (+ t_0 (+ -1.5 (* (* r r) (* -0.375 (* w w)))))
       (* (* r w) (* r (* w -0.375)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.8e-116) {
		tmp = t_0;
	} else if (r <= 3.8e+151) {
		tmp = t_0 + (-1.5 + ((r * r) * (-0.375 * (w * w))));
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if (r <= 2.8d-116) then
        tmp = t_0
    else if (r <= 3.8d+151) then
        tmp = t_0 + ((-1.5d0) + ((r * r) * ((-0.375d0) * (w * w))))
    else
        tmp = (r * w) * (r * (w * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (r <= 2.8e-116) {
		tmp = t_0;
	} else if (r <= 3.8e+151) {
		tmp = t_0 + (-1.5 + ((r * r) * (-0.375 * (w * w))));
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if r <= 2.8e-116:
		tmp = t_0
	elif r <= 3.8e+151:
		tmp = t_0 + (-1.5 + ((r * r) * (-0.375 * (w * w))))
	else:
		tmp = (r * w) * (r * (w * -0.375))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (r <= 2.8e-116)
		tmp = t_0;
	elseif (r <= 3.8e+151)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * r) * Float64(-0.375 * Float64(w * w)))));
	else
		tmp = Float64(Float64(r * w) * Float64(r * Float64(w * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (r <= 2.8e-116)
		tmp = t_0;
	elseif (r <= 3.8e+151)
		tmp = t_0 + (-1.5 + ((r * r) * (-0.375 * (w * w))));
	else
		tmp = (r * w) * (r * (w * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, 2.8e-116], t$95$0, If[LessEqual[r, 3.8e+151], N[(t$95$0 + N[(-1.5 + N[(N[(r * r), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < 2.7999999999999999e-116

   1. Initial program 84.7%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 94.2%

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

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. 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 62.5%

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

   7. Simplified 62.5%

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

   if 2.7999999999999999e-116 < r < 3.8e151

   1. Initial program 91.9%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 96.6%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 88.4%

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

   7. Simplified 88.4%

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

   if 3.8e151 < r

   1. Initial program 85.7%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 65.7%

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

   7. Simplified 65.7%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right) \]

      10. *-lowering-*.f64 65.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right) \]

   10. Simplified 65.7%

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

       1. associate-*r* N/A

          \[\leadsto \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{-3}{8}} \]

       2. swap-sqr N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8} \]

       3. associate-*r* N/A

          \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right)} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)} \]

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

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

       6. associate-*l* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \left(r \cdot w\right)\right) \]

       9. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right) \]

   12. Applied egg-rr 79.7%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 2.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{elif}\;r \leq 3.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(r \cdot r\right) \cdot 3}{r \cdot r} - 4.5\\ \mathbf{elif}\;r \leq 3.8 \cdot 10^{+151}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.7e-5)
   (- (/ (+ 2.0 (* (* r r) 3.0)) (* r r)) 4.5)
   (if (<= r 3.8e+151)
     (* (* r r) (+ (* -0.375 (* w w)) (/ -1.5 (* r r))))
     (* (* r w) (* r (* w -0.375))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.7e-5) {
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	} else if (r <= 3.8e+151) {
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	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 <= 5.7d-5) then
        tmp = ((2.0d0 + ((r * r) * 3.0d0)) / (r * r)) - 4.5d0
    else if (r <= 3.8d+151) then
        tmp = (r * r) * (((-0.375d0) * (w * w)) + ((-1.5d0) / (r * r)))
    else
        tmp = (r * w) * (r * (w * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.7e-5) {
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	} else if (r <= 3.8e+151) {
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.7e-5:
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5
	elif r <= 3.8e+151:
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)))
	else:
		tmp = (r * w) * (r * (w * -0.375))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.7e-5)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(r * r) * 3.0)) / Float64(r * r)) - 4.5);
	elseif (r <= 3.8e+151)
		tmp = Float64(Float64(r * r) * Float64(Float64(-0.375 * Float64(w * w)) + Float64(-1.5 / Float64(r * r))));
	else
		tmp = Float64(Float64(r * w) * Float64(r * Float64(w * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.7e-5)
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	elseif (r <= 3.8e+151)
		tmp = (r * r) * ((-0.375 * (w * w)) + (-1.5 / (r * r)));
	else
		tmp = (r * w) * (r * (w * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.7e-5], N[(N[(N[(2.0 + N[(N[(r * r), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 3.8e+151], N[(N[(r * r), $MachinePrecision] * N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.7000000000000003e-5

   1. Initial program 85.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 68.7%

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

   5. Simplified 68.7%

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

   if 5.7000000000000003e-5 < r < 3.8e151

   1. Initial program 94.7%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in r around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. *-commutative N/A

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

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

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

      8. unpow2 N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. /-lowering-/.f64 N/A

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 89.0%

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

   10. Simplified 89.0%

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

   if 3.8e151 < r

   1. Initial program 85.7%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 65.7%

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

   7. Simplified 65.7%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right) \]

      10. *-lowering-*.f64 65.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right) \]

   10. Simplified 65.7%

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

       1. associate-*r* N/A

          \[\leadsto \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{-3}{8}} \]

       2. swap-sqr N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8} \]

       3. associate-*r* N/A

          \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right)} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)} \]

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

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

       6. associate-*l* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \left(r \cdot w\right)\right) \]

       9. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right) \]

   12. Applied egg-rr 79.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(r \cdot r\right) \cdot 3}{r \cdot r} - 4.5\\ \mathbf{elif}\;r \leq 3.8 \cdot 10^{+151}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(r \cdot r\right) \cdot 3}{r \cdot r} - 4.5\\ \mathbf{elif}\;r \leq 3.8 \cdot 10^{+151}:\\ \;\;\;\;r \cdot \left(r \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.7e-5)
   (- (/ (+ 2.0 (* (* r r) 3.0)) (* r r)) 4.5)
   (if (<= r 3.8e+151)
     (* r (* r (+ (* -0.375 (* w w)) (/ -1.5 (* r r)))))
     (* (* r w) (* r (* w -0.375))))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.7e-5) {
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	} else if (r <= 3.8e+151) {
		tmp = r * (r * ((-0.375 * (w * w)) + (-1.5 / (r * r))));
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	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 <= 5.7d-5) then
        tmp = ((2.0d0 + ((r * r) * 3.0d0)) / (r * r)) - 4.5d0
    else if (r <= 3.8d+151) then
        tmp = r * (r * (((-0.375d0) * (w * w)) + ((-1.5d0) / (r * r))))
    else
        tmp = (r * w) * (r * (w * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.7e-5) {
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	} else if (r <= 3.8e+151) {
		tmp = r * (r * ((-0.375 * (w * w)) + (-1.5 / (r * r))));
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.7e-5:
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5
	elif r <= 3.8e+151:
		tmp = r * (r * ((-0.375 * (w * w)) + (-1.5 / (r * r))))
	else:
		tmp = (r * w) * (r * (w * -0.375))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.7e-5)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(r * r) * 3.0)) / Float64(r * r)) - 4.5);
	elseif (r <= 3.8e+151)
		tmp = Float64(r * Float64(r * Float64(Float64(-0.375 * Float64(w * w)) + Float64(-1.5 / Float64(r * r)))));
	else
		tmp = Float64(Float64(r * w) * Float64(r * Float64(w * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.7e-5)
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	elseif (r <= 3.8e+151)
		tmp = r * (r * ((-0.375 * (w * w)) + (-1.5 / (r * r))));
	else
		tmp = (r * w) * (r * (w * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.7e-5], N[(N[(N[(2.0 + N[(N[(r * r), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], If[LessEqual[r, 3.8e+151], N[(r * N[(r * N[(N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision] + N[(-1.5 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if r < 5.7000000000000003e-5

   1. Initial program 85.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 68.7%

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

   5. Simplified 68.7%

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

   if 5.7000000000000003e-5 < r < 3.8e151

   1. Initial program 94.7%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.8%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 89.1%

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

   7. Simplified 89.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. associate-*l* N/A

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

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

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

      7. *-lowering-*.f64 89.1%

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

   9. Applied egg-rr 89.1%

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

   10. Taylor expanded in r around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

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

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

       5. sub-neg N/A

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

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

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

       7. *-commutative N/A

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

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

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

       9. unpow2 N/A

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

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

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. distribute-neg-frac N/A

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

       14. metadata-eval N/A

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

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

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

       16. unpow2 N/A

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

       17. *-lowering-*.f64 88.8%

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

   12. Simplified 88.8%

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

   if 3.8e151 < r

   1. Initial program 85.7%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 65.7%

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

   7. Simplified 65.7%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right) \]

      10. *-lowering-*.f64 65.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right) \]

   10. Simplified 65.7%

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

       1. associate-*r* N/A

          \[\leadsto \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{-3}{8}} \]

       2. swap-sqr N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8} \]

       3. associate-*r* N/A

          \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right)} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)} \]

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

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

       6. associate-*l* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \left(r \cdot w\right)\right) \]

       9. *-lowering-*.f64 79.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right) \]

   12. Applied egg-rr 79.7%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2 + \left(r \cdot r\right) \cdot 3}{r \cdot r} - 4.5\\ \mathbf{elif}\;r \leq 3.8 \cdot 10^{+151}:\\ \;\;\;\;r \cdot \left(r \cdot \left(-0.375 \cdot \left(w \cdot w\right) + \frac{-1.5}{r \cdot r}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 2.7:\\ \;\;\;\;t\_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \left(-1.5 - w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 2.7)
     (+ t_0 (+ -1.5 (* (* r w) (* (* r w) -0.375))))
     (+ t_0 (- -1.5 (* w (* w (* (* r r) 0.25))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 2.7) {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * -0.375)));
	} else {
		tmp = t_0 + (-1.5 - (w * (w * ((r * r) * 0.25))));
	}
	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 <= 2.7d0) then
        tmp = t_0 + ((-1.5d0) + ((r * w) * ((r * w) * (-0.375d0))))
    else
        tmp = t_0 + ((-1.5d0) - (w * (w * ((r * r) * 0.25d0))))
    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 <= 2.7) {
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * -0.375)));
	} else {
		tmp = t_0 + (-1.5 - (w * (w * ((r * r) * 0.25))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= 2.7:
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * -0.375)))
	else:
		tmp = t_0 + (-1.5 - (w * (w * ((r * r) * 0.25))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 2.7)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * -0.375))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(w * Float64(w * Float64(Float64(r * r) * 0.25)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= 2.7)
		tmp = t_0 + (-1.5 + ((r * w) * ((r * w) * -0.375)));
	else
		tmp = t_0 + (-1.5 - (w * (w * ((r * r) * 0.25))));
	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, 2.7], N[(t$95$0 + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(w * N[(w * N[(N[(r * r), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 2.7000000000000002

   1. Initial program 86.2%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 95.1%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 82.3%

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

   7. Simplified 82.3%

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

      1. associate-*r* N/A

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

      2. swap-sqr N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. *-lowering-*.f64 97.2%

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

   9. Applied egg-rr 97.2%

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

   if 2.7000000000000002 < v

   1. Initial program 87.3%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 95.9%

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

   5. Taylor expanded in v around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. metadata-eval N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. associate--l+ N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. unpow2 N/A

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

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

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

      14. --lowering--.f64 N/A

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

      15. associate-*r* N/A

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

      16. unpow2 N/A

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

      17. associate-*r* N/A

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

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

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

   7. Simplified 97.2%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq 2.7:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5e-5)
   (- (/ (+ 2.0 (* (* r r) 3.0)) (* r r)) 4.5)
   (* (* r w) (* r (* w -0.375)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e-5) {
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	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-5) then
        tmp = ((2.0d0 + ((r * r) * 3.0d0)) / (r * r)) - 4.5d0
    else
        tmp = (r * w) * (r * (w * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5e-5) {
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5e-5:
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5
	else:
		tmp = (r * w) * (r * (w * -0.375))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5e-5)
		tmp = Float64(Float64(Float64(2.0 + Float64(Float64(r * r) * 3.0)) / Float64(r * r)) - 4.5);
	else
		tmp = Float64(Float64(r * w) * Float64(r * Float64(w * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5e-5)
		tmp = ((2.0 + ((r * r) * 3.0)) / (r * r)) - 4.5;
	else
		tmp = (r * w) * (r * (w * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5e-5], N[(N[(N[(2.0 + N[(N[(r * r), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] / N[(r * r), $MachinePrecision]), $MachinePrecision] - 4.5), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.00000000000000024e-5

   1. Initial program 85.0%

      \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   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. /-lowering-/.f64 N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 68.7%

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

   5. Simplified 68.7%

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

   if 5.00000000000000024e-5 < r

   1. Initial program 91.0%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right) \]

      10. *-lowering-*.f64 53.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right) \]

   10. Simplified 53.9%

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

       1. associate-*r* N/A

          \[\leadsto \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{-3}{8}} \]

       2. swap-sqr N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8} \]

       3. associate-*r* N/A

          \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right)} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)} \]

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

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

       6. associate-*l* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \left(r \cdot w\right)\right) \]

       9. *-lowering-*.f64 59.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right) \]

   12. Applied egg-rr 59.9%

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

4. Final simplification 66.5%

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

Alternative 8: 93.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right) \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (+ (/ 2.0 (* r r)) (+ -1.5 (* (* r w) (* (* r w) -0.375)))))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r * w) * -0.375)));
}

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) + ((r * w) * ((r * w) * (-0.375d0))))
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r * w) * -0.375)));
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r * w) * -0.375)))

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(Float64(r * w) * Float64(Float64(r * w) * -0.375))))
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + (-1.5 + ((r * w) * ((r * w) * -0.375)));
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.5%

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

   1. associate--l- N/A

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

   7. associate--r+ N/A

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

   8. sub-neg N/A

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

   9. +-commutative N/A

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

   10. associate--l+ N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

3. Simplified 95.3%

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

5. Taylor expanded in v around 0

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

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

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

   4. unpow2 N/A

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

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

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

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

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 82.7%

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

7. Simplified 82.7%

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

   1. associate-*r* N/A

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

   2. swap-sqr N/A

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

   3. associate-*l* N/A

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

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

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

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

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

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

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

   7. *-lowering-*.f64 95.5%

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

9. Applied egg-rr 95.5%

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

10. Final simplification 95.5%

    \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right) \]
11. Add Preprocessing

Alternative 9: 66.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.3e-5) (+ (/ 2.0 (* r r)) -1.5) (* (* r w) (* r (* w -0.375)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.3e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	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 <= 5.3d-5) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (r * w) * (r * (w * (-0.375d0)))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.3e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * w) * (r * (w * -0.375));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.3e-5:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (r * w) * (r * (w * -0.375))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.3e-5)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(r * w) * Float64(r * Float64(w * -0.375)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.3e-5)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (r * w) * (r * (w * -0.375));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.3e-5], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r * w), $MachinePrecision] * N[(r * N[(w * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.3000000000000001e-5

   1. Initial program 85.0%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 93.8%

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

   5. Taylor expanded in r around 0

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

   7. Simplified 70.0%

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

   if 5.3000000000000001e-5 < r

   1. Initial program 91.0%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right) \]

      10. *-lowering-*.f64 53.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right) \]

   10. Simplified 53.9%

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

       1. associate-*r* N/A

          \[\leadsto \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \color{blue}{\frac{-3}{8}} \]

       2. swap-sqr N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{-3}{8} \]

       3. associate-*r* N/A

          \[\leadsto \left(r \cdot w\right) \cdot \color{blue}{\left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right)} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(r \cdot w\right) \cdot \frac{-3}{8}\right) \cdot \color{blue}{\left(r \cdot w\right)} \]

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

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

       6. associate-*l* N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \left(r \cdot w\right)\right) \]

       9. *-lowering-*.f64 59.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, \mathsf{*.f64}\left(w, \frac{-3}{8}\right)\right), \mathsf{*.f64}\left(r, \color{blue}{w}\right)\right) \]

   12. Applied egg-rr 59.9%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.375\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (<= r 5.2e-5) (+ (/ 2.0 (* r r)) -1.5) (* (* r r) (* -0.375 (* w w)))))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.2e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * r) * (-0.375 * (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 <= 5.2d-5) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (r * r) * ((-0.375d0) * (w * w))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.2e-5) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = (r * r) * (-0.375 * (w * w));
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.2e-5:
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = (r * r) * (-0.375 * (w * w))
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.2e-5)
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(Float64(r * r) * Float64(-0.375 * Float64(w * w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if (r <= 5.2e-5)
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = (r * r) * (-0.375 * (w * w));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.2e-5], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(N[(r * r), $MachinePrecision] * N[(-0.375 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < 5.19999999999999968e-5

   1. Initial program 85.0%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 93.8%

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

   5. Taylor expanded in r around 0

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

   7. Simplified 70.0%

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

   if 5.19999999999999968e-5 < r

   1. Initial program 91.0%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in v around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 79.4%

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

   7. Simplified 79.4%

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

   8. Taylor expanded in r around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. unpow2 N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\left(w \cdot w\right), \frac{-3}{8}\right)\right) \]

      10. *-lowering-*.f64 53.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(r, r\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \frac{-3}{8}\right)\right) \]

   10. Simplified 53.9%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 5.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(-0.375 \cdot \left(w \cdot w\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (if (<= r 5.7e-5) (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	double tmp;
	if (r <= 5.7e-5) {
		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 <= 5.7d-5) 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 <= 5.7e-5) {
		tmp = 2.0 / (r * r);
	} else {
		tmp = -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if r <= 5.7e-5:
		tmp = 2.0 / (r * r)
	else:
		tmp = -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if (r <= 5.7e-5)
		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 <= 5.7e-5)
		tmp = 2.0 / (r * r);
	else
		tmp = -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[LessEqual[r, 5.7e-5], N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision], -1.5]

Derivation

1. Split input into 2 regimes

2. if r < 5.7000000000000003e-5

   1. Initial program 85.0%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 93.8%

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

   5. Taylor expanded in r around 0

      \[\leadsto \color{blue}{\frac{2}{{r}^{2}}} \]
   6. 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 62.8%

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

   7. Simplified 62.8%

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

   if 5.7000000000000003e-5 < r

   1. Initial program 91.0%

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

      1. associate--l- N/A

         \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

      3. associate--l+ N/A

         \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

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

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

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

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

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

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

      7. associate--r+ N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. associate--l+ N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around inf

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

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

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

      2. unpow2 N/A

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

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

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

      4. associate--l+ N/A

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

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

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

      6. associate-/r* N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

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

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

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

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

      13. unpow2 N/A

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

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

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

      15. unpow2 N/A

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

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

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

      17. sub-neg N/A

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

   7. Simplified 59.1%

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

   8. Taylor expanded in w around 0

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

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

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. /-lowering-/.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 8.2%

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

   10. Simplified 8.2%

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

   11. Taylor expanded in r around inf

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

   13. Simplified 33.1%

       \[\leadsto \color{blue}{-1.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 57.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{r \cdot r} + -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 (+ (/ 2.0 (* r r)) -1.5))

C

double code(double v, double w, double r) {
	return (2.0 / (r * r)) + -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = (2.0d0 / (r * r)) + (-1.5d0)
end function

Java

public static double code(double v, double w, double r) {
	return (2.0 / (r * r)) + -1.5;
}

Python

def code(v, w, r):
	return (2.0 / (r * r)) + -1.5

Julia

function code(v, w, r)
	return Float64(Float64(2.0 / Float64(r * r)) + -1.5)
end

MATLAB

function tmp = code(v, w, r)
	tmp = (2.0 / (r * r)) + -1.5;
end

Wolfram

code[v_, w_, r_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Step-by-step derivation

   1. associate--l- N/A

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

   10. associate--l+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

3. Simplified 95.3%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in r around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \color{blue}{\frac{-3}{2}}\right) \]
6. Step-by-step derivation

7. Simplified 60.9%

   \[\leadsto \frac{2}{r \cdot r} + \color{blue}{-1.5} \]
8. Add Preprocessing

Alternative 13: 13.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (v w r) :precision binary64 -1.5)

C

double code(double v, double w, double r) {
	return -1.5;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = -1.5d0
end function

Java

public static double code(double v, double w, double r) {
	return -1.5;
}

Python

def code(v, w, r):
	return -1.5

Julia

function code(v, w, r)
	return -1.5
end

MATLAB

function tmp = code(v, w, r)
	tmp = -1.5;
end

Wolfram

code[v_, w_, r_] := -1.5

Derivation

1. Initial program 86.5%

   \[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
2. Step-by-step derivation

   1. associate--l- N/A

      \[\leadsto \left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto \left(\frac{2}{r \cdot r} + 3\right) - \left(\color{blue}{\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}} + \frac{9}{2}\right) \]

   3. associate--l+ N/A

      \[\leadsto \frac{2}{r \cdot r} + \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)} \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{2}{r \cdot r}\right), \color{blue}{\left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left(\color{blue}{3} - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(3 - \left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + \frac{9}{2}\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 - \frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - \color{blue}{\frac{9}{2}}\right)\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(3 + \left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)\right) - \frac{9}{2}\right)\right) \]

   9. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + 3\right) - \frac{9}{2}\right)\right) \]

   10. associate--l+ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \color{blue}{\left(3 - \frac{9}{2}\right)}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \frac{-3}{2}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\frac{-9}{2} + \color{blue}{3}\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(\left(\mathsf{neg}\left(\frac{\left(\frac{1}{8} \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right) + \left(\left(\mathsf{neg}\left(\frac{9}{2}\right)\right) + 3\right)\right)\right) \]

3. Simplified 95.3%

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(\left(0.375 + v \cdot -0.25\right) \cdot \frac{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}{v + -1} + -1.5\right)} \]
4. Add Preprocessing

5. Taylor expanded in w around inf

   \[\leadsto \color{blue}{{w}^{2} \cdot \left(\left(\frac{2}{{r}^{2} \cdot {w}^{2}} + \frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1}\right) - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({w}^{2}\right), \color{blue}{\left(\left(\frac{2}{{r}^{2} \cdot {w}^{2}} + \frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1}\right) - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(w \cdot w\right), \left(\color{blue}{\left(\frac{2}{{r}^{2} \cdot {w}^{2}} + \frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1}\right)} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\color{blue}{\left(\frac{2}{{r}^{2} \cdot {w}^{2}} + \frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1}\right)} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right) \]

   4. associate--l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{2}{{r}^{2} \cdot {w}^{2}} + \color{blue}{\left(\frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)}\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\left(\frac{2}{{r}^{2} \cdot {w}^{2}}\right), \color{blue}{\left(\frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)}\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\left(\frac{\frac{2}{{r}^{2}}}{{w}^{2}}\right), \left(\color{blue}{\frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1}} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\left(\frac{\frac{2 \cdot 1}{{r}^{2}}}{{w}^{2}}\right), \left(\frac{\color{blue}{{r}^{2}} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\left(\frac{2 \cdot \frac{1}{{r}^{2}}}{{w}^{2}}\right), \left(\frac{\color{blue}{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}}\right), \left({w}^{2}\right)\right), \left(\color{blue}{\frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1}} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot 1}{{r}^{2}}\right), \left({w}^{2}\right)\right), \left(\frac{\color{blue}{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\frac{2}{{r}^{2}}\right), \left({w}^{2}\right)\right), \left(\frac{\color{blue}{{r}^{2}} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left({r}^{2}\right)\right), \left({w}^{2}\right)\right), \left(\frac{\color{blue}{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \left(r \cdot r\right)\right), \left({w}^{2}\right)\right), \left(\frac{{r}^{2} \cdot \color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left({w}^{2}\right)\right), \left(\frac{{r}^{2} \cdot \color{blue}{\left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}}{v - 1} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \left(w \cdot w\right)\right), \left(\frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{\color{blue}{v - 1}} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{\color{blue}{v - 1}} - \frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)\right) \]

   17. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right), \mathsf{*.f64}\left(w, w\right)\right), \left(\frac{{r}^{2} \cdot \left(\frac{3}{8} + \frac{-1}{4} \cdot v\right)}{v - 1} + \color{blue}{\left(\mathsf{neg}\left(\frac{3}{2} \cdot \frac{1}{{w}^{2}}\right)\right)}\right)\right)\right) \]

7. Simplified 57.9%

   \[\leadsto \color{blue}{\left(w \cdot w\right) \cdot \left(\frac{\frac{2}{r \cdot r}}{w \cdot w} + \left(\frac{\left(r \cdot r\right) \cdot \left(0.375 + -0.25 \cdot v\right)}{v + -1} + \frac{-1.5}{w \cdot w}\right)\right)} \]

8. Taylor expanded in w around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \color{blue}{\left(\frac{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}}{{w}^{2}}\right)}\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}}{w \cdot \color{blue}{w}}\right)\right) \]

   2. associate-/r* N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \left(\frac{\frac{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}}{w}}{\color{blue}{w}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\left(\frac{2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}}{w}\right), \color{blue}{w}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}} - \frac{3}{2}\right), w\right), w\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}} + \left(\mathsf{neg}\left(\frac{3}{2}\right)\right)\right), w\right), w\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \frac{1}{{r}^{2}} + \frac{-3}{2}\right), w\right), w\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\frac{-3}{2} + 2 \cdot \frac{1}{{r}^{2}}\right), w\right), w\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \left(2 \cdot \frac{1}{{r}^{2}}\right)\right), w\right), w\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \left(\frac{2 \cdot 1}{{r}^{2}}\right)\right), w\right), w\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \left(\frac{2}{{r}^{2}}\right)\right), w\right), w\right)\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \left({r}^{2}\right)\right)\right), w\right), w\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \left(r \cdot r\right)\right)\right), w\right), w\right)\right) \]

   13. *-lowering-*.f64 33.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(w, w\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{-3}{2}, \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(r, r\right)\right)\right), w\right), w\right)\right) \]

10. Simplified 33.7%

    \[\leadsto \left(w \cdot w\right) \cdot \color{blue}{\frac{\frac{-1.5 + \frac{2}{r \cdot r}}{w}}{w}} \]

11. Taylor expanded in r around inf

    \[\leadsto \color{blue}{\frac{-3}{2}} \]
12. Step-by-step derivation

13. Simplified 14.5%

    \[\leadsto \color{blue}{-1.5} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(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))