Rosa's TurbineBenchmark

Percentage Accurate: 84.8% → 97.4%

Time: 11.3s

Alternatives: 13

Speedup: 1.7×

• 53.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 0:\\ \;\;\;\;t_0 + \left(-1.5 + r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{elif}\;w \cdot w \leq 2 \cdot 10^{+233}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 0.0)
     (+ t_0 (+ -1.5 (* r (* w (* (* r w) -0.375)))))
     (if (<= (* w w) 2e+233)
       (+
        t_0
        (- -1.5 (* (/ (+ 0.375 (* v -0.25)) (- 1.0 v)) (* r (* r (* w w))))))
       (+ t_0 (- -1.5 (* 0.375 (* w (* (* r r) w)))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 0.0) {
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	} else if ((w * w) <= 2e+233) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * (w * ((r * r) * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((w * w) <= 0.0d0) then
        tmp = t_0 + ((-1.5d0) + (r * (w * ((r * w) * (-0.375d0)))))
    else if ((w * w) <= 2d+233) then
        tmp = t_0 + ((-1.5d0) - (((0.375d0 + (v * (-0.25d0))) / (1.0d0 - v)) * (r * (r * (w * w)))))
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (w * ((r * r) * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 0.0) {
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	} else if ((w * w) <= 2e+233) {
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * (w * ((r * r) * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 0.0:
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))))
	elif (w * w) <= 2e+233:
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))))
	else:
		tmp = t_0 + (-1.5 - (0.375 * (w * ((r * r) * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 0.0)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(r * Float64(w * Float64(Float64(r * w) * -0.375)))));
	elseif (Float64(w * w) <= 2e+233)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)) * Float64(r * Float64(r * Float64(w * w))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(w * Float64(Float64(r * r) * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 0.0)
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	elseif ((w * w) <= 2e+233)
		tmp = t_0 + (-1.5 - (((0.375 + (v * -0.25)) / (1.0 - v)) * (r * (r * (w * w)))));
	else
		tmp = t_0 + (-1.5 - (0.375 * (w * ((r * r) * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 0.0], N[(t$95$0 + N[(-1.5 + N[(r * N[(w * N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(w * w), $MachinePrecision], 2e+233], N[(t$95$0 + N[(-1.5 - N[(N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(r * N[(r * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(w * N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 w w) < 0.0

   1. Initial program 88.8%

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

      1. sub-neg 88.8%

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

      2. +-commutative 88.8%

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

      3. associate--l+ 88.8%

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

      4. associate-/l* 88.8%

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

      5. distribute-neg-frac 88.8%

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

      6. associate-/r/ 88.8%

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

      7. fma-def 88.8%

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

      8. sub-neg 88.8%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in v around 0 82.2%

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

      1. associate--l+ 82.2%

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

      2. associate-*r/ 82.2%

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

      3. metadata-eval 82.2%

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

      4. unpow2 82.2%

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

      5. *-commutative 82.2%

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

      6. *-commutative 82.2%

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

      7. unpow2 82.2%

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

      8. associate-*r* 88.9%

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

      9. unpow2 88.9%

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

      10. associate-*l* 88.9%

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

      11. fma-neg 88.9%

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

      12. associate-*r* 96.2%

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

      13. *-commutative 96.2%

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

      14. *-commutative 96.2%

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

      15. metadata-eval 96.2%

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

   6. Simplified 96.2%

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

      1. fma-udef 96.2%

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

      2. associate-*l* 96.3%

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

   8. Applied egg-rr 96.3%

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

   if 0.0 < (*.f64 w w) < 1.99999999999999995e233

   1. Initial program 94.2%

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

      1. associate--l- 94.2%

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

      2. +-commutative 94.2%

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

      3. associate--l+ 94.2%

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

      4. +-commutative 94.2%

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

      5. associate--r+ 94.2%

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

      6. metadata-eval 94.2%

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

      7. associate-*l/ 99.7%

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

      8. *-commutative 99.7%

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

      9. *-commutative 99.7%

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

      10. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   if 1.99999999999999995e233 < (*.f64 w w)

   1. Initial program 79.6%

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

      1. associate--l- 79.6%

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

      2. +-commutative 79.6%

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

      3. associate--l+ 79.6%

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

      4. +-commutative 79.6%

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

      5. associate--r+ 79.6%

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

      6. metadata-eval 79.6%

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

      7. associate-*l/ 79.6%

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

      8. *-commutative 79.6%

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

      9. *-commutative 79.6%

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

      10. *-commutative 79.6%

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

   3. Simplified 79.6%

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

   4. Taylor expanded in r around 0 79.6%

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

      1. unpow2 79.6%

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

      2. associate-*l* 92.0%

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

   6. Simplified 92.0%

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

      1. clear-num 92.0%

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

      2. inv-pow 92.0%

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

      3. +-commutative 92.0%

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

      4. fma-def 92.0%

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

   8. Applied egg-rr 92.0%

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

      1. unpow-1 92.0%

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

      2. fma-udef 92.0%

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

      3. *-commutative 92.0%

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

      4. fma-def 92.0%

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

   10. Simplified 92.0%

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

   11. Taylor expanded in v around 0 79.6%

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

       1. unpow2 79.6%

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

       2. associate-*l* 97.3%

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

       3. unpow2 97.3%

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

   13. Simplified 97.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 0:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{elif}\;w \cdot w \leq 2 \cdot 10^{+233}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 2: 96.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. associate--l- 88.7%

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

   2. +-commutative 88.7%

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

   3. associate--l+ 88.7%

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

   4. +-commutative 88.7%

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

   5. associate--r+ 88.7%

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

   6. metadata-eval 88.7%

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

   7. associate-*l/ 91.2%

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

   8. *-commutative 91.2%

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

   9. *-commutative 91.2%

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

   10. *-commutative 91.2%

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

3. Simplified 91.2%

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

4. Taylor expanded in r around 0 91.2%

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

   1. unpow2 91.2%

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

   2. associate-*l* 97.5%

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

6. Simplified 97.5%

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

   1. clear-num 97.5%

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

   2. inv-pow 97.5%

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

   3. +-commutative 97.5%

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

   4. fma-def 97.5%

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

8. Applied egg-rr 97.5%

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

   1. unpow-1 97.5%

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

   2. fma-udef 97.5%

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

   3. *-commutative 97.5%

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

   4. fma-def 97.5%

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

10. Simplified 97.5%

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

11. Final simplification 97.5%

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

Alternative 3: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := r \cdot t_0\\ \mathbf{if}\;v \leq -2.8 \cdot 10^{+15}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - r \cdot \left(t_0 \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 10^{-28}:\\ \;\;\;\;t_1 + \left(-1.5 - t_2 \cdot \left(0.375 + v \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - t_2 \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* r w))) (t_1 (/ 2.0 (* r r))) (t_2 (* r t_0)))
   (if (<= v -2.8e+15)
     (- (+ -1.5 (/ (/ 2.0 r) r)) (* r (* t_0 0.25)))
     (if (<= v 1e-28)
       (+ t_1 (- -1.5 (* t_2 (+ 0.375 (* v 0.125)))))
       (+ t_1 (- -1.5 (* t_2 0.25)))))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double t_2 = r * t_0;
	double tmp;
	if (v <= -2.8e+15) {
		tmp = (-1.5 + ((2.0 / r) / r)) - (r * (t_0 * 0.25));
	} else if (v <= 1e-28) {
		tmp = t_1 + (-1.5 - (t_2 * (0.375 + (v * 0.125))));
	} else {
		tmp = t_1 + (-1.5 - (t_2 * 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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = w * (r * w)
    t_1 = 2.0d0 / (r * r)
    t_2 = r * t_0
    if (v <= (-2.8d+15)) then
        tmp = ((-1.5d0) + ((2.0d0 / r) / r)) - (r * (t_0 * 0.25d0))
    else if (v <= 1d-28) then
        tmp = t_1 + ((-1.5d0) - (t_2 * (0.375d0 + (v * 0.125d0))))
    else
        tmp = t_1 + ((-1.5d0) - (t_2 * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double t_2 = r * t_0;
	double tmp;
	if (v <= -2.8e+15) {
		tmp = (-1.5 + ((2.0 / r) / r)) - (r * (t_0 * 0.25));
	} else if (v <= 1e-28) {
		tmp = t_1 + (-1.5 - (t_2 * (0.375 + (v * 0.125))));
	} else {
		tmp = t_1 + (-1.5 - (t_2 * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * (r * w)
	t_1 = 2.0 / (r * r)
	t_2 = r * t_0
	tmp = 0
	if v <= -2.8e+15:
		tmp = (-1.5 + ((2.0 / r) / r)) - (r * (t_0 * 0.25))
	elif v <= 1e-28:
		tmp = t_1 + (-1.5 - (t_2 * (0.375 + (v * 0.125))))
	else:
		tmp = t_1 + (-1.5 - (t_2 * 0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	t_2 = Float64(r * t_0)
	tmp = 0.0
	if (v <= -2.8e+15)
		tmp = Float64(Float64(-1.5 + Float64(Float64(2.0 / r) / r)) - Float64(r * Float64(t_0 * 0.25)));
	elseif (v <= 1e-28)
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_2 * Float64(0.375 + Float64(v * 0.125)))));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_2 * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * (r * w);
	t_1 = 2.0 / (r * r);
	t_2 = r * t_0;
	tmp = 0.0;
	if (v <= -2.8e+15)
		tmp = (-1.5 + ((2.0 / r) / r)) - (r * (t_0 * 0.25));
	elseif (v <= 1e-28)
		tmp = t_1 + (-1.5 - (t_2 * (0.375 + (v * 0.125))));
	else
		tmp = t_1 + (-1.5 - (t_2 * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(r * t$95$0), $MachinePrecision]}, If[LessEqual[v, -2.8e+15], N[(N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - N[(r * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 1e-28], N[(t$95$1 + N[(-1.5 - N[(t$95$2 * N[(0.375 + N[(v * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(t$95$2 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -2.8e15

   1. Initial program 80.7%

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

      1. associate--l- 80.7%

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

      2. +-commutative 80.7%

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

      3. associate--l+ 80.7%

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

      4. +-commutative 80.7%

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

      5. associate--r+ 80.7%

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

      6. metadata-eval 80.7%

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

      7. associate-*l/ 86.9%

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

      8. *-commutative 86.9%

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

      9. *-commutative 86.9%

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

      10. *-commutative 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in r around 0 86.9%

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

      1. unpow2 86.9%

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

      2. associate-*l* 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in v around inf 96.8%

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

      1. associate-+r- 96.8%

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

      2. associate-/r* 96.9%

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

      3. associate-*l* 96.9%

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

      4. *-commutative 96.9%

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

   9. Applied egg-rr 96.9%

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

   if -2.8e15 < v < 9.99999999999999971e-29

   1. Initial program 91.9%

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

      1. associate--l- 91.9%

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

      2. +-commutative 91.9%

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

      3. associate--l+ 92.0%

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

      4. +-commutative 92.0%

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

      5. associate--r+ 92.0%

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

      6. metadata-eval 92.0%

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

      7. associate-*l/ 92.0%

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

      8. *-commutative 92.0%

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

      9. *-commutative 92.0%

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

      10. *-commutative 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in r around 0 92.0%

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

      1. unpow2 92.0%

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

      2. associate-*l* 97.3%

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

   6. Simplified 97.3%

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

   7. Taylor expanded in v around 0 97.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(w \cdot r\right)\right)\right) \cdot \color{blue}{\left(0.125 \cdot v + 0.375\right)}\right) \]

   if 9.99999999999999971e-29 < v

   1. Initial program 89.7%

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

      1. associate--l- 89.7%

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

      2. +-commutative 89.7%

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

      3. associate--l+ 89.8%

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

      4. +-commutative 89.8%

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

      5. associate--r+ 89.8%

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

      6. metadata-eval 89.8%

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

      7. associate-*l/ 93.4%

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

      8. *-commutative 93.4%

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

      9. *-commutative 93.4%

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

      10. *-commutative 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in r around 0 93.4%

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

      1. unpow2 93.4%

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

      2. associate-*l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in v around inf 97.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -2.8 \cdot 10^{+15}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 10^{-28}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \left(0.375 + v \cdot 0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternative 4: 96.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 + (v * (-0.25d0))) / (1.0d0 - v))))
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 + (v * -0.25)) / (1.0 - v))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. associate--l- 88.7%

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

   2. +-commutative 88.7%

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

   3. associate--l+ 88.7%

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

   4. +-commutative 88.7%

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

   5. associate--r+ 88.7%

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

   6. metadata-eval 88.7%

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

   7. associate-*l/ 91.2%

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

   8. *-commutative 91.2%

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

   9. *-commutative 91.2%

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

   10. *-commutative 91.2%

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

3. Simplified 91.2%

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

4. Taylor expanded in r around 0 91.2%

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

   1. unpow2 91.2%

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

   2. associate-*l* 97.5%

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

6. Simplified 97.5%

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

7. Final simplification 97.5%

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

Alternative 5: 95.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -8.2 \cdot 10^{+30}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - r \cdot \left(t_0 \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-31}:\\ \;\;\;\;t_1 + \left(-1.5 + r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot t_0\right) \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* w (* r w))) (t_1 (/ 2.0 (* r r))))
   (if (<= v -8.2e+30)
     (- (+ -1.5 (/ (/ 2.0 r) r)) (* r (* t_0 0.25)))
     (if (<= v 2e-31)
       (+ t_1 (+ -1.5 (* r (* w (* (* r w) -0.375)))))
       (+ t_1 (- -1.5 (* (* r t_0) 0.25)))))))

C

double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -8.2e+30) {
		tmp = (-1.5 + ((2.0 / r) / r)) - (r * (t_0 * 0.25));
	} else if (v <= 2e-31) {
		tmp = t_1 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	} else {
		tmp = t_1 + (-1.5 - ((r * t_0) * 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) :: t_1
    real(8) :: tmp
    t_0 = w * (r * w)
    t_1 = 2.0d0 / (r * r)
    if (v <= (-8.2d+30)) then
        tmp = ((-1.5d0) + ((2.0d0 / r) / r)) - (r * (t_0 * 0.25d0))
    else if (v <= 2d-31) then
        tmp = t_1 + ((-1.5d0) + (r * (w * ((r * w) * (-0.375d0)))))
    else
        tmp = t_1 + ((-1.5d0) - ((r * t_0) * 0.25d0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = w * (r * w);
	double t_1 = 2.0 / (r * r);
	double tmp;
	if (v <= -8.2e+30) {
		tmp = (-1.5 + ((2.0 / r) / r)) - (r * (t_0 * 0.25));
	} else if (v <= 2e-31) {
		tmp = t_1 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	} else {
		tmp = t_1 + (-1.5 - ((r * t_0) * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = w * (r * w)
	t_1 = 2.0 / (r * r)
	tmp = 0
	if v <= -8.2e+30:
		tmp = (-1.5 + ((2.0 / r) / r)) - (r * (t_0 * 0.25))
	elif v <= 2e-31:
		tmp = t_1 + (-1.5 + (r * (w * ((r * w) * -0.375))))
	else:
		tmp = t_1 + (-1.5 - ((r * t_0) * 0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(w * Float64(r * w))
	t_1 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -8.2e+30)
		tmp = Float64(Float64(-1.5 + Float64(Float64(2.0 / r) / r)) - Float64(r * Float64(t_0 * 0.25)));
	elseif (v <= 2e-31)
		tmp = Float64(t_1 + Float64(-1.5 + Float64(r * Float64(w * Float64(Float64(r * w) * -0.375)))));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(Float64(r * t_0) * 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = w * (r * w);
	t_1 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -8.2e+30)
		tmp = (-1.5 + ((2.0 / r) / r)) - (r * (t_0 * 0.25));
	elseif (v <= 2e-31)
		tmp = t_1 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	else
		tmp = t_1 + (-1.5 - ((r * t_0) * 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -8.2e+30], N[(N[(-1.5 + N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] - N[(r * N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2e-31], N[(t$95$1 + N[(-1.5 + N[(r * N[(w * N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(N[(r * t$95$0), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -8.20000000000000011e30

   1. Initial program 79.3%

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

      1. associate--l- 79.3%

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

      2. +-commutative 79.3%

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

      3. associate--l+ 79.3%

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

      4. +-commutative 79.3%

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

      5. associate--r+ 79.3%

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

      6. metadata-eval 79.3%

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

      7. associate-*l/ 85.9%

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

      8. *-commutative 85.9%

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

      9. *-commutative 85.9%

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

      10. *-commutative 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in r around 0 85.9%

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

      1. unpow2 85.9%

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

      2. associate-*l* 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in v around inf 96.6%

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

      1. associate-+r- 96.6%

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

      2. associate-/r* 96.7%

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

      3. associate-*l* 96.7%

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

      4. *-commutative 96.7%

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

   9. Applied egg-rr 96.7%

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

   if -8.20000000000000011e30 < v < 2e-31

   1. Initial program 92.2%

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

      1. sub-neg 92.2%

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

      2. +-commutative 92.2%

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

      3. associate--l+ 92.2%

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

      4. associate-/l* 92.2%

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

      5. distribute-neg-frac 92.2%

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

      6. associate-/r/ 92.2%

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

      7. fma-def 92.2%

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

      8. sub-neg 92.2%

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

   3. Simplified 86.6%

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

   4. Taylor expanded in v around 0 86.6%

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

      1. associate--l+ 86.6%

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

      2. associate-*r/ 86.6%

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

      3. metadata-eval 86.6%

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

      4. unpow2 86.6%

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

      5. *-commutative 86.6%

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

      6. *-commutative 86.6%

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

      7. unpow2 86.6%

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

      8. associate-*r* 92.2%

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

      9. unpow2 92.2%

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

      10. associate-*l* 92.2%

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

      11. fma-neg 92.2%

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

      12. associate-*r* 97.1%

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

      13. *-commutative 97.1%

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

      14. *-commutative 97.1%

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

      15. metadata-eval 97.1%

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

   6. Simplified 97.1%

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

      1. fma-udef 97.1%

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

      2. associate-*l* 97.2%

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

   8. Applied egg-rr 97.2%

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

   if 2e-31 < v

   1. Initial program 89.7%

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

      1. associate--l- 89.7%

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

      2. +-commutative 89.7%

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

      3. associate--l+ 89.8%

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

      4. +-commutative 89.8%

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

      5. associate--r+ 89.8%

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

      6. metadata-eval 89.8%

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

      7. associate-*l/ 93.4%

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

      8. *-commutative 93.4%

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

      9. *-commutative 93.4%

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

      10. *-commutative 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in r around 0 93.4%

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

      1. unpow2 93.4%

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

      2. associate-*l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in v around inf 97.6%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -8.2 \cdot 10^{+30}:\\ \;\;\;\;\left(-1.5 + \frac{\frac{2}{r}}{r}\right) - r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternative 6: 93.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{-110}:\\ \;\;\;\;t_0 + \left(-1.5 + r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 1e-110)
     (+ t_0 (+ -1.5 (* r (* w (* (* r w) -0.375)))))
     (+ t_0 (- -1.5 (* 0.375 (* w (* (* r r) w))))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e-110) {
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * (w * ((r * r) * w))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 / (r * r)
    if ((w * w) <= 1d-110) then
        tmp = t_0 + ((-1.5d0) + (r * (w * ((r * w) * (-0.375d0)))))
    else
        tmp = t_0 + ((-1.5d0) - (0.375d0 * (w * ((r * r) * w))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 1e-110) {
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * (w * ((r * r) * w))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if (w * w) <= 1e-110:
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))))
	else:
		tmp = t_0 + (-1.5 - (0.375 * (w * ((r * r) * w))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 1e-110)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(r * Float64(w * Float64(Float64(r * w) * -0.375)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(w * Float64(Float64(r * r) * w)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * r);
	tmp = 0.0;
	if ((w * w) <= 1e-110)
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	else
		tmp = t_0 + (-1.5 - (0.375 * (w * ((r * r) * w))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 1e-110], N[(t$95$0 + N[(-1.5 + N[(r * N[(w * N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(w * N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 1.0000000000000001e-110

   1. Initial program 91.2%

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

      1. sub-neg 91.2%

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

      2. +-commutative 91.2%

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

      3. associate--l+ 91.2%

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

      4. associate-/l* 93.5%

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

      5. distribute-neg-frac 93.5%

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

      6. associate-/r/ 93.5%

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

      7. fma-def 93.5%

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

      8. sub-neg 93.5%

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

   3. Simplified 82.6%

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

   4. Taylor expanded in v around 0 81.9%

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

      1. associate--l+ 81.9%

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

      2. associate-*r/ 81.9%

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

      3. metadata-eval 81.9%

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

      4. unpow2 81.9%

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

      5. *-commutative 81.9%

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

      6. *-commutative 81.9%

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

      7. unpow2 81.9%

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

      8. associate-*r* 90.8%

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

      9. unpow2 90.8%

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

      10. associate-*l* 90.8%

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

      11. fma-neg 90.8%

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

      12. associate-*r* 95.0%

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

      13. *-commutative 95.0%

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

      14. *-commutative 95.0%

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

      15. metadata-eval 95.0%

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

   6. Simplified 95.0%

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

      1. fma-udef 95.0%

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

      2. associate-*l* 95.0%

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

   8. Applied egg-rr 95.0%

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

   if 1.0000000000000001e-110 < (*.f64 w w)

   1. Initial program 86.5%

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

      1. associate--l- 86.5%

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

      2. +-commutative 86.5%

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

      3. associate--l+ 86.5%

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

      4. +-commutative 86.5%

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

      5. associate--r+ 86.5%

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

      6. metadata-eval 86.5%

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

      7. associate-*l/ 89.2%

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

      8. *-commutative 89.2%

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

      9. *-commutative 89.2%

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

      10. *-commutative 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in r around 0 89.2%

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

      1. unpow2 89.2%

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

      2. associate-*l* 95.7%

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

   6. Simplified 95.7%

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

      1. clear-num 95.7%

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

      2. inv-pow 95.7%

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

      3. +-commutative 95.7%

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

      4. fma-def 95.7%

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

   8. Applied egg-rr 95.7%

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

      1. unpow-1 95.7%

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

      2. fma-udef 95.7%

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

      3. *-commutative 95.7%

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

      4. fma-def 95.7%

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

   10. Simplified 95.7%

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

   11. Taylor expanded in v around 0 85.7%

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

       1. unpow2 85.7%

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

       2. associate-*l* 95.0%

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

       3. unpow2 95.0%

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

   13. Simplified 95.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 10^{-110}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - 0.375 \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)\right)\\ \end{array} \]

Alternative 7: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 5 \cdot 10^{-29}:\\ \;\;\;\;t_0 + \left(-1.5 + r \cdot \left(w \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v 5e-29)
     (+ t_0 (+ -1.5 (* r (* w (* (* r w) -0.375)))))
     (+ t_0 (- -1.5 (* (* r (* w (* r w))) 0.25))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= 5e-29) {
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	} else {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 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 <= 5d-29) then
        tmp = t_0 + ((-1.5d0) + (r * (w * ((r * w) * (-0.375d0)))))
    else
        tmp = t_0 + ((-1.5d0) - ((r * (w * (r * w))) * 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 <= 5e-29) {
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	} else {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * r)
	tmp = 0
	if v <= 5e-29:
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))))
	else:
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 0.25))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= 5e-29)
		tmp = Float64(t_0 + Float64(-1.5 + Float64(r * Float64(w * Float64(Float64(r * w) * -0.375)))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * 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 <= 5e-29)
		tmp = t_0 + (-1.5 + (r * (w * ((r * w) * -0.375))));
	else
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * 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, 5e-29], N[(t$95$0 + N[(-1.5 + N[(r * N[(w * N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < 4.99999999999999986e-29

   1. Initial program 88.2%

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

      1. sub-neg 88.2%

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

      2. +-commutative 88.2%

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

      3. associate--l+ 88.2%

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

      4. associate-/l* 90.2%

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

      5. distribute-neg-frac 90.2%

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

      6. associate-/r/ 90.3%

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

      7. fma-def 90.2%

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

      8. sub-neg 90.2%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in v around 0 84.6%

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

      1. associate--l+ 84.6%

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

      2. associate-*r/ 84.6%

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

      3. metadata-eval 84.6%

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

      4. unpow2 84.6%

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

      5. *-commutative 84.6%

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

      6. *-commutative 84.6%

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

      7. unpow2 84.6%

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

      8. associate-*r* 88.5%

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

      9. unpow2 88.5%

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

      10. associate-*l* 88.5%

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

      11. fma-neg 88.5%

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

      12. associate-*r* 94.8%

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

      13. *-commutative 94.8%

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

      14. *-commutative 94.8%

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

      15. metadata-eval 94.8%

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

   6. Simplified 94.8%

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

      1. fma-udef 94.8%

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

      2. associate-*l* 94.8%

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

   8. Applied egg-rr 94.8%

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

   if 4.99999999999999986e-29 < v

   1. Initial program 89.7%

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

      1. associate--l- 89.7%

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

      2. +-commutative 89.7%

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

      3. associate--l+ 89.8%

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

      4. +-commutative 89.8%

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

      5. associate--r+ 89.8%

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

      6. metadata-eval 89.8%

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

      7. associate-*l/ 93.4%

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

      8. *-commutative 93.4%

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

      9. *-commutative 93.4%

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

      10. *-commutative 93.4%

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

   3. Simplified 93.4%

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

   4. Taylor expanded in r around 0 93.4%

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

      1. unpow2 93.4%

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

      2. associate-*l* 98.5%

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

   6. Simplified 98.5%

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

   7. Taylor expanded in v around inf 97.6%

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

4. Final simplification 95.6%

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

Alternative 8: 75.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq -16600000 \lor \neg \left(r \leq 9.8 \cdot 10^{-33} \lor \neg \left(r \leq 2.9 \cdot 10^{-9}\right) \land r \leq 2.3 \cdot 10^{+97}\right):\\ \;\;\;\;-0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (if (or (<= r -16600000.0)
         (not (or (<= r 9.8e-33) (and (not (<= r 2.9e-9)) (<= r 2.3e+97)))))
   (* -0.375 (* (* r w) (* r w)))
   (+ (/ 2.0 (* r r)) -1.5)))

C

double code(double v, double w, double r) {
	double tmp;
	if ((r <= -16600000.0) || !((r <= 9.8e-33) || (!(r <= 2.9e-9) && (r <= 2.3e+97)))) {
		tmp = -0.375 * ((r * w) * (r * w));
	} else {
		tmp = (2.0 / (r * r)) + -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 <= (-16600000.0d0)) .or. (.not. (r <= 9.8d-33) .or. (.not. (r <= 2.9d-9)) .and. (r <= 2.3d+97))) then
        tmp = (-0.375d0) * ((r * w) * (r * w))
    else
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double tmp;
	if ((r <= -16600000.0) || !((r <= 9.8e-33) || (!(r <= 2.9e-9) && (r <= 2.3e+97)))) {
		tmp = -0.375 * ((r * w) * (r * w));
	} else {
		tmp = (2.0 / (r * r)) + -1.5;
	}
	return tmp;
}

Python

def code(v, w, r):
	tmp = 0
	if (r <= -16600000.0) or not ((r <= 9.8e-33) or (not (r <= 2.9e-9) and (r <= 2.3e+97))):
		tmp = -0.375 * ((r * w) * (r * w))
	else:
		tmp = (2.0 / (r * r)) + -1.5
	return tmp

Julia

function code(v, w, r)
	tmp = 0.0
	if ((r <= -16600000.0) || !((r <= 9.8e-33) || (!(r <= 2.9e-9) && (r <= 2.3e+97))))
		tmp = Float64(-0.375 * Float64(Float64(r * w) * Float64(r * w)));
	else
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	tmp = 0.0;
	if ((r <= -16600000.0) || ~(((r <= 9.8e-33) || (~((r <= 2.9e-9)) && (r <= 2.3e+97)))))
		tmp = -0.375 * ((r * w) * (r * w));
	else
		tmp = (2.0 / (r * r)) + -1.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := If[Or[LessEqual[r, -16600000.0], N[Not[Or[LessEqual[r, 9.8e-33], And[N[Not[LessEqual[r, 2.9e-9]], $MachinePrecision], LessEqual[r, 2.3e+97]]]], $MachinePrecision]], N[(-0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if r < -1.66e7 or 9.7999999999999996e-33 < r < 2.89999999999999991e-9 or 2.30000000000000006e97 < r

   1. Initial program 87.6%

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

      1. sub-neg 87.6%

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

      2. +-commutative 87.6%

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

      3. associate--l+ 87.7%

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

      4. associate-/l* 92.8%

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

      5. distribute-neg-frac 92.8%

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

      6. associate-/r/ 92.9%

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

      7. fma-def 92.8%

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

      8. sub-neg 92.8%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in r around inf 69.3%

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

      1. associate-/l* 71.9%

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

      2. unpow2 71.9%

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

      3. *-commutative 71.9%

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

      4. associate-/l* 72.4%

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

      5. unpow2 72.4%

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

      6. *-commutative 72.4%

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

      7. fma-neg 72.4%

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

      8. metadata-eval 72.4%

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

   6. Simplified 72.4%

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

   7. Taylor expanded in v around 0 69.9%

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

      1. associate-*r* 69.9%

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

      2. unpow2 69.9%

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

      3. unpow2 69.9%

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

   9. Simplified 69.9%

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

   10. Taylor expanded in w around 0 69.9%

       \[\leadsto \color{blue}{-0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 69.9%

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

       2. unpow2 69.9%

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

       3. unswap-sqr 80.5%

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

   12. Simplified 80.5%

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

   if -1.66e7 < r < 9.7999999999999996e-33 or 2.89999999999999991e-9 < r < 2.30000000000000006e97

   1. Initial program 89.4%

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

      1. sub-neg 89.4%

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

      2. +-commutative 89.4%

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

      3. associate--l+ 89.4%

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

      4. associate-/l* 90.0%

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

      5. distribute-neg-frac 90.0%

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

      6. associate-/r/ 90.0%

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

      7. fma-def 90.0%

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

      8. sub-neg 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in r around 0 84.8%

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

      1. sub-neg 84.8%

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

      2. associate-*r/ 84.8%

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

      3. metadata-eval 84.8%

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

      4. unpow2 84.8%

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

      5. metadata-eval 84.8%

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

   6. Simplified 84.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -16600000 \lor \neg \left(r \leq 9.8 \cdot 10^{-33} \lor \neg \left(r \leq 2.9 \cdot 10^{-9}\right) \land r \leq 2.3 \cdot 10^{+97}\right):\\ \;\;\;\;-0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]

Alternative 9: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;r \leq -54000000:\\ \;\;\;\;-0.25 \cdot t_0\\ \mathbf{elif}\;r \leq 10^{-32} \lor \neg \left(r \leq 3.6 \cdot 10^{-9}\right) \land r \leq 1.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))))
   (if (<= r -54000000.0)
     (* -0.25 t_0)
     (if (or (<= r 1e-32) (and (not (<= r 3.6e-9)) (<= r 1.1e+97)))
       (+ (/ 2.0 (* r r)) -1.5)
       (* -0.375 t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if (r <= -54000000.0) {
		tmp = -0.25 * t_0;
	} else if ((r <= 1e-32) || (!(r <= 3.6e-9) && (r <= 1.1e+97))) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -0.375 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    if (r <= (-54000000.0d0)) then
        tmp = (-0.25d0) * t_0
    else if ((r <= 1d-32) .or. (.not. (r <= 3.6d-9)) .and. (r <= 1.1d+97)) then
        tmp = (2.0d0 / (r * r)) + (-1.5d0)
    else
        tmp = (-0.375d0) * t_0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double tmp;
	if (r <= -54000000.0) {
		tmp = -0.25 * t_0;
	} else if ((r <= 1e-32) || (!(r <= 3.6e-9) && (r <= 1.1e+97))) {
		tmp = (2.0 / (r * r)) + -1.5;
	} else {
		tmp = -0.375 * t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	tmp = 0
	if r <= -54000000.0:
		tmp = -0.25 * t_0
	elif (r <= 1e-32) or (not (r <= 3.6e-9) and (r <= 1.1e+97)):
		tmp = (2.0 / (r * r)) + -1.5
	else:
		tmp = -0.375 * t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if (r <= -54000000.0)
		tmp = Float64(-0.25 * t_0);
	elseif ((r <= 1e-32) || (!(r <= 3.6e-9) && (r <= 1.1e+97)))
		tmp = Float64(Float64(2.0 / Float64(r * r)) + -1.5);
	else
		tmp = Float64(-0.375 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	tmp = 0.0;
	if (r <= -54000000.0)
		tmp = -0.25 * t_0;
	elseif ((r <= 1e-32) || (~((r <= 3.6e-9)) && (r <= 1.1e+97)))
		tmp = (2.0 / (r * r)) + -1.5;
	else
		tmp = -0.375 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[r, -54000000.0], N[(-0.25 * t$95$0), $MachinePrecision], If[Or[LessEqual[r, 1e-32], And[N[Not[LessEqual[r, 3.6e-9]], $MachinePrecision], LessEqual[r, 1.1e+97]]], N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision], N[(-0.375 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if r < -5.4e7

   1. Initial program 81.7%

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

      1. associate--l- 81.7%

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

      2. +-commutative 81.7%

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

      3. associate--l+ 81.7%

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

      4. +-commutative 81.7%

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

      5. associate--r+ 81.8%

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

      6. metadata-eval 81.8%

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

      7. associate-*l/ 91.5%

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

      8. *-commutative 91.5%

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

      9. *-commutative 91.5%

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

      10. *-commutative 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in r around 0 91.5%

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

      1. unpow2 91.5%

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

      2. associate-*l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in v around inf 89.7%

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

   8. Taylor expanded in r around inf 67.9%

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

      1. unpow2 67.9%

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

      2. unpow2 67.9%

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

      3. unswap-sqr 77.9%

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

   10. Simplified 77.9%

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

   if -5.4e7 < r < 1.00000000000000006e-32 or 3.6e-9 < r < 1.1e97

   1. Initial program 89.4%

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

      1. sub-neg 89.4%

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

      2. +-commutative 89.4%

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

      3. associate--l+ 89.4%

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

      4. associate-/l* 90.0%

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

      5. distribute-neg-frac 90.0%

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

      6. associate-/r/ 90.0%

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

      7. fma-def 90.0%

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

      8. sub-neg 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in r around 0 84.8%

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

      1. sub-neg 84.8%

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

      2. associate-*r/ 84.8%

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

      3. metadata-eval 84.8%

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

      4. unpow2 84.8%

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

      5. metadata-eval 84.8%

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

   6. Simplified 84.8%

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

   if 1.00000000000000006e-32 < r < 3.6e-9 or 1.1e97 < r

   1. Initial program 94.4%

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

      1. sub-neg 94.4%

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

      2. +-commutative 94.4%

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

      3. associate--l+ 94.4%

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

      4. associate-/l* 94.3%

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

      5. distribute-neg-frac 94.3%

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

      6. associate-/r/ 94.4%

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

      7. fma-def 94.3%

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

      8. sub-neg 94.3%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in r around inf 75.3%

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

      1. associate-/l* 75.3%

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

      2. unpow2 75.3%

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

      3. *-commutative 75.3%

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

      4. associate-/l* 75.7%

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

      5. unpow2 75.7%

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

      6. *-commutative 75.7%

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

      7. fma-neg 75.7%

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

      8. metadata-eval 75.7%

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

   6. Simplified 75.7%

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

   7. Taylor expanded in v around 0 75.5%

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

      1. associate-*r* 75.5%

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

      2. unpow2 75.5%

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

      3. unpow2 75.5%

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

   9. Simplified 75.5%

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

   10. Taylor expanded in w around 0 75.5%

       \[\leadsto \color{blue}{-0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 75.5%

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

       2. unpow2 75.5%

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

       3. unswap-sqr 86.7%

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

   12. Simplified 86.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -54000000:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{elif}\;r \leq 10^{-32} \lor \neg \left(r \leq 3.6 \cdot 10^{-9}\right) \land r \leq 1.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 10: 75.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;r \leq -24000000:\\ \;\;\;\;-0.25 \cdot t_0\\ \mathbf{elif}\;r \leq 9 \cdot 10^{-33}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \mathbf{elif}\;r \leq 3.2 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (+ (/ 2.0 (* r r)) -1.5)))
   (if (<= r -24000000.0)
     (* -0.25 t_0)
     (if (<= r 9e-33)
       t_1
       (if (<= r 2.8e-9)
         (* (* r r) (* (* w w) -0.375))
         (if (<= r 3.2e+102) t_1 (* -0.375 t_0)))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (r <= -24000000.0) {
		tmp = -0.25 * t_0;
	} else if (r <= 9e-33) {
		tmp = t_1;
	} else if (r <= 2.8e-9) {
		tmp = (r * r) * ((w * w) * -0.375);
	} else if (r <= 3.2e+102) {
		tmp = t_1;
	} else {
		tmp = -0.375 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = (2.0d0 / (r * r)) + (-1.5d0)
    if (r <= (-24000000.0d0)) then
        tmp = (-0.25d0) * t_0
    else if (r <= 9d-33) then
        tmp = t_1
    else if (r <= 2.8d-9) then
        tmp = (r * r) * ((w * w) * (-0.375d0))
    else if (r <= 3.2d+102) then
        tmp = t_1
    else
        tmp = (-0.375d0) * t_0
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / (r * r)) + -1.5;
	double tmp;
	if (r <= -24000000.0) {
		tmp = -0.25 * t_0;
	} else if (r <= 9e-33) {
		tmp = t_1;
	} else if (r <= 2.8e-9) {
		tmp = (r * r) * ((w * w) * -0.375);
	} else if (r <= 3.2e+102) {
		tmp = t_1;
	} else {
		tmp = -0.375 * t_0;
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = (2.0 / (r * r)) + -1.5
	tmp = 0
	if r <= -24000000.0:
		tmp = -0.25 * t_0
	elif r <= 9e-33:
		tmp = t_1
	elif r <= 2.8e-9:
		tmp = (r * r) * ((w * w) * -0.375)
	elif r <= 3.2e+102:
		tmp = t_1
	else:
		tmp = -0.375 * t_0
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(Float64(2.0 / Float64(r * r)) + -1.5)
	tmp = 0.0
	if (r <= -24000000.0)
		tmp = Float64(-0.25 * t_0);
	elseif (r <= 9e-33)
		tmp = t_1;
	elseif (r <= 2.8e-9)
		tmp = Float64(Float64(r * r) * Float64(Float64(w * w) * -0.375));
	elseif (r <= 3.2e+102)
		tmp = t_1;
	else
		tmp = Float64(-0.375 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = (2.0 / (r * r)) + -1.5;
	tmp = 0.0;
	if (r <= -24000000.0)
		tmp = -0.25 * t_0;
	elseif (r <= 9e-33)
		tmp = t_1;
	elseif (r <= 2.8e-9)
		tmp = (r * r) * ((w * w) * -0.375);
	elseif (r <= 3.2e+102)
		tmp = t_1;
	else
		tmp = -0.375 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]}, If[LessEqual[r, -24000000.0], N[(-0.25 * t$95$0), $MachinePrecision], If[LessEqual[r, 9e-33], t$95$1, If[LessEqual[r, 2.8e-9], N[(N[(r * r), $MachinePrecision] * N[(N[(w * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision], If[LessEqual[r, 3.2e+102], t$95$1, N[(-0.375 * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if r < -2.4e7

   1. Initial program 81.7%

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

      1. associate--l- 81.7%

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

      2. +-commutative 81.7%

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

      3. associate--l+ 81.7%

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

      4. +-commutative 81.7%

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

      5. associate--r+ 81.8%

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

      6. metadata-eval 81.8%

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

      7. associate-*l/ 91.5%

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

      8. *-commutative 91.5%

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

      9. *-commutative 91.5%

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

      10. *-commutative 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in r around 0 91.5%

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

      1. unpow2 91.5%

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

      2. associate-*l* 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in v around inf 89.7%

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

   8. Taylor expanded in r around inf 67.9%

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

      1. unpow2 67.9%

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

      2. unpow2 67.9%

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

      3. unswap-sqr 77.9%

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

   10. Simplified 77.9%

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

   if -2.4e7 < r < 8.99999999999999982e-33 or 2.79999999999999984e-9 < r < 3.1999999999999999e102

   1. Initial program 89.4%

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

      1. sub-neg 89.4%

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

      2. +-commutative 89.4%

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

      3. associate--l+ 89.4%

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

      4. associate-/l* 90.0%

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

      5. distribute-neg-frac 90.0%

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

      6. associate-/r/ 90.0%

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

      7. fma-def 90.0%

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

      8. sub-neg 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in r around 0 84.8%

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

      1. sub-neg 84.8%

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

      2. associate-*r/ 84.8%

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

      3. metadata-eval 84.8%

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

      4. unpow2 84.8%

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

      5. metadata-eval 84.8%

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

   6. Simplified 84.8%

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

   if 8.99999999999999982e-33 < r < 2.79999999999999984e-9

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. associate--l+ 100.0%

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

      4. associate-/l* 99.7%

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

      5. distribute-neg-frac 99.7%

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

      6. associate-/r/ 100.0%

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

      7. fma-def 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in r around inf 89.8%

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

      1. associate-/l* 89.8%

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

      2. unpow2 89.8%

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

      3. *-commutative 89.8%

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

      4. associate-/l* 89.8%

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

      5. unpow2 89.8%

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

      6. *-commutative 89.8%

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

      7. fma-neg 89.8%

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

      8. metadata-eval 89.8%

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

   6. Simplified 89.8%

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

   7. Taylor expanded in v around 0 90.1%

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

      1. associate-*r* 90.1%

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

      2. unpow2 90.1%

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

      3. unpow2 90.1%

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

   9. Simplified 90.1%

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

   if 3.1999999999999999e102 < r

   1. Initial program 93.8%

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

      1. sub-neg 93.8%

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

      2. +-commutative 93.8%

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

      3. associate--l+ 93.8%

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

      4. associate-/l* 93.7%

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

      5. distribute-neg-frac 93.7%

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

      6. associate-/r/ 93.8%

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

      7. fma-def 93.7%

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

      8. sub-neg 93.7%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in r around inf 73.6%

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

      1. associate-/l* 73.6%

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

      2. unpow2 73.6%

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

      3. *-commutative 73.6%

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

      4. associate-/l* 74.1%

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

      5. unpow2 74.1%

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

      6. *-commutative 74.1%

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

      7. fma-neg 74.1%

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

      8. metadata-eval 74.1%

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

   6. Simplified 74.1%

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

   7. Taylor expanded in v around 0 73.8%

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

      1. associate-*r* 73.8%

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

      2. unpow2 73.8%

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

      3. unpow2 73.8%

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

   9. Simplified 73.8%

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

   10. Taylor expanded in w around 0 73.8%

       \[\leadsto \color{blue}{-0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 73.8%

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

       2. unpow2 73.8%

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

       3. unswap-sqr 86.4%

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

   12. Simplified 86.4%

       \[\leadsto \color{blue}{-0.375 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -24000000:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \mathbf{elif}\;r \leq 9 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 2.8 \cdot 10^{-9}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \mathbf{elif}\;r \leq 3.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;-0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\\ \end{array} \]

Alternative 11: 90.7% accurate, 1.7× speedup

Math

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

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(r * Float64(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[(r * N[(w * N[(N[(r * w), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.7%

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

   1. sub-neg 88.7%

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

   2. +-commutative 88.7%

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

   3. associate--l+ 88.7%

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

   4. associate-/l* 91.2%

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

   5. distribute-neg-frac 91.2%

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

   6. associate-/r/ 91.2%

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

   7. fma-def 91.2%

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

   8. sub-neg 91.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in v around 0 84.0%

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

   1. associate--l+ 84.0%

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

   2. associate-*r/ 84.0%

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

   3. metadata-eval 84.0%

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

   4. unpow2 84.0%

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

   5. *-commutative 84.0%

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

   6. *-commutative 84.0%

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

   7. unpow2 84.0%

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

   8. associate-*r* 88.1%

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

   9. unpow2 88.1%

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

   10. associate-*l* 88.1%

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

   11. fma-neg 88.1%

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

   12. associate-*r* 93.4%

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

   13. *-commutative 93.4%

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

   14. *-commutative 93.4%

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

   15. metadata-eval 93.4%

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

6. Simplified 93.4%

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

   1. fma-udef 93.4%

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

   2. associate-*l* 93.4%

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

8. Applied egg-rr 93.4%

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

9. Final simplification 93.4%

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

Alternative 12: 56.7% 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 88.7%

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

   1. sub-neg 88.7%

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

   2. +-commutative 88.7%

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

   3. associate--l+ 88.7%

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

   4. associate-/l* 91.2%

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

   5. distribute-neg-frac 91.2%

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

   6. associate-/r/ 91.2%

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

   7. fma-def 91.2%

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

   8. sub-neg 91.2%

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

3. Simplified 86.2%

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

4. Taylor expanded in r around 0 56.0%

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

   1. sub-neg 56.0%

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

   2. associate-*r/ 56.0%

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

   3. metadata-eval 56.0%

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

   4. unpow2 56.0%

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

   5. metadata-eval 56.0%

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

6. Simplified 56.0%

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

7. Final simplification 56.0%

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

Alternative 13: 43.8% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

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)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.7%

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

   1. associate--l- 88.7%

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

   2. +-commutative 88.7%

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

   3. associate--l+ 88.7%

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

   4. +-commutative 88.7%

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

   5. associate--r+ 88.7%

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

   6. metadata-eval 88.7%

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

   7. associate-*l/ 91.2%

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

   8. *-commutative 91.2%

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

   9. *-commutative 91.2%

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

   10. *-commutative 91.2%

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

3. Simplified 91.2%

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

4. Taylor expanded in r around 0 91.2%

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

   1. unpow2 91.2%

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

   2. associate-*l* 97.5%

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

6. Simplified 97.5%

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

7. Taylor expanded in v around inf 91.7%

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

8. Taylor expanded in r around 0 43.8%

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

   1. unpow2 43.8%

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

10. Simplified 43.8%

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

11. Final simplification 43.8%

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

Reproduce

herbie shell --seed 2023171 
(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))