Rosa's TurbineBenchmark

Percentage Accurate: 85.0% → 99.2%

Time: 12.5s

Alternatives: 6

Speedup: 1.5×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot w}\\ t_1 := \left(r \cdot w\right) \cdot 0.5\\ t_2 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -54000000000000:\\ \;\;\;\;t\_2 + \left(-1.5 + \frac{-1}{t\_0 \cdot t\_0}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;t\_2 + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{\frac{1}{r}}{w} \cdot \frac{-1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(-1.5 - t\_1 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r w))) (t_1 (* (* r w) 0.5)) (t_2 (/ 2.0 (* r r))))
   (if (<= v -54000000000000.0)
     (+ t_2 (+ -1.5 (/ -1.0 (* t_0 t_0))))
     (if (<= v 2.5e-33)
       (+
        t_2
        (+
         -1.5
         (/ (+ 0.375 (* v -0.25)) (* (/ (/ 1.0 r) w) (/ -1.0 (* r w))))))
       (+ t_2 (- -1.5 (* t_1 t_1)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * w);
	double t_1 = (r * w) * 0.5;
	double t_2 = 2.0 / (r * r);
	double tmp;
	if (v <= -54000000000000.0) {
		tmp = t_2 + (-1.5 + (-1.0 / (t_0 * t_0)));
	} else if (v <= 2.5e-33) {
		tmp = t_2 + (-1.5 + ((0.375 + (v * -0.25)) / (((1.0 / r) / w) * (-1.0 / (r * w)))));
	} else {
		tmp = t_2 + (-1.5 - (t_1 * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 / (r * w)
    t_1 = (r * w) * 0.5d0
    t_2 = 2.0d0 / (r * r)
    if (v <= (-54000000000000.0d0)) then
        tmp = t_2 + ((-1.5d0) + ((-1.0d0) / (t_0 * t_0)))
    else if (v <= 2.5d-33) then
        tmp = t_2 + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) / (((1.0d0 / r) / w) * ((-1.0d0) / (r * w)))))
    else
        tmp = t_2 + ((-1.5d0) - (t_1 * t_1))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * w);
	double t_1 = (r * w) * 0.5;
	double t_2 = 2.0 / (r * r);
	double tmp;
	if (v <= -54000000000000.0) {
		tmp = t_2 + (-1.5 + (-1.0 / (t_0 * t_0)));
	} else if (v <= 2.5e-33) {
		tmp = t_2 + (-1.5 + ((0.375 + (v * -0.25)) / (((1.0 / r) / w) * (-1.0 / (r * w)))));
	} else {
		tmp = t_2 + (-1.5 - (t_1 * t_1));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * w)
	t_1 = (r * w) * 0.5
	t_2 = 2.0 / (r * r)
	tmp = 0
	if v <= -54000000000000.0:
		tmp = t_2 + (-1.5 + (-1.0 / (t_0 * t_0)))
	elif v <= 2.5e-33:
		tmp = t_2 + (-1.5 + ((0.375 + (v * -0.25)) / (((1.0 / r) / w) * (-1.0 / (r * w)))))
	else:
		tmp = t_2 + (-1.5 - (t_1 * t_1))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * w))
	t_1 = Float64(Float64(r * w) * 0.5)
	t_2 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -54000000000000.0)
		tmp = Float64(t_2 + Float64(-1.5 + Float64(-1.0 / Float64(t_0 * t_0))));
	elseif (v <= 2.5e-33)
		tmp = Float64(t_2 + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(Float64(Float64(1.0 / r) / w) * Float64(-1.0 / Float64(r * w))))));
	else
		tmp = Float64(t_2 + Float64(-1.5 - Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * w);
	t_1 = (r * w) * 0.5;
	t_2 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -54000000000000.0)
		tmp = t_2 + (-1.5 + (-1.0 / (t_0 * t_0)));
	elseif (v <= 2.5e-33)
		tmp = t_2 + (-1.5 + ((0.375 + (v * -0.25)) / (((1.0 / r) / w) * (-1.0 / (r * w)))));
	else
		tmp = t_2 + (-1.5 - (t_1 * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -54000000000000.0], N[(t$95$2 + N[(-1.5 + N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.5e-33], N[(t$95$2 + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(1.0 / r), $MachinePrecision] / w), $MachinePrecision] * N[(-1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(-1.5 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.4e13

   1. Initial program 78.6%

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

   3. Simplified 89.6%

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

      1. associate-*l* 89.6%

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

      2. fma-undefine 89.6%

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

      3. *-commutative 89.6%

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

      4. +-commutative 89.6%

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

      5. associate-*r/ 89.6%

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

      6. *-commutative 89.6%

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

      7. associate-/l* 89.6%

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

      8. associate-*l* 89.6%

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

      9. associate-*r/ 78.6%

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

      10. clear-num 78.6%

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

      11. *-commutative 78.6%

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

      12. associate-*r* 72.0%

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

      13. pow2 72.0%

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

      14. pow2 72.0%

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

      15. pow-prod-down 83.6%

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

      16. *-commutative 83.6%

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

      17. distribute-rgt-in 83.6%

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

   6. Applied egg-rr 83.6%

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

   7. Taylor expanded in v around inf 78.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{r}^{2} \cdot {w}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. *-commutative 78.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{w}^{2} \cdot {r}^{2}}}}\right) \]

      2. unpow2 78.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot {r}^{2}}}\right) \]

      3. unpow2 78.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\left(w \cdot w\right) \cdot \color{blue}{\left(r \cdot r\right)}}}\right) \]

      4. swap-sqr 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

      5. unpow2 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{\left(w \cdot r\right)}^{2}}}}\right) \]

      6. *-commutative 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(r \cdot w\right)}}^{2}}}\right) \]

   9. Simplified 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(w \cdot r\right)}}^{2}}}\right) \]

       2. pow2 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       3. add-sqr-sqrt 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}}\right) \]

       4. sqrt-div 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{\sqrt{4}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       5. metadata-eval 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{\color{blue}{2}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       6. pow2 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       7. *-commutative 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       8. sqrt-pow1 77.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       9. metadata-eval 77.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{{\left(r \cdot w\right)}^{\color{blue}{1}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       10. pow1 77.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{r \cdot w}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       11. *-commutative 77.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{w \cdot r}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       12. sqrt-div 77.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{w \cdot r} \cdot \color{blue}{\frac{\sqrt{4}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}}\right) \]

       13. metadata-eval 77.9%

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

       14. pow2 77.9%

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

       15. *-commutative 77.9%

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

       16. sqrt-pow1 99.8%

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

       17. metadata-eval 99.8%

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

       18. pow1 99.8%

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

       19. *-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   if -5.4e13 < v < 2.50000000000000014e-33

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

      1. fma-undefine 86.4%

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

      2. *-commutative 86.4%

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

      3. +-commutative 86.4%

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

      4. associate-*r/ 86.4%

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

      5. *-commutative 86.4%

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

      6. associate-/l* 86.4%

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

      7. clear-num 86.4%

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

      8. un-div-inv 86.4%

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

      9. distribute-rgt-in 86.4%

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

      10. metadata-eval 86.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      11. *-commutative 86.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      12. associate-*l* 86.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      13. metadata-eval 86.4%

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

      14. associate-*r* 82.8%

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

      15. pow2 82.8%

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

      16. pow2 82.8%

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

      17. pow-prod-down 99.8%

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

      18. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow2 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-/r* 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-/l/ 99.8%

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

      4. div-inv 99.8%

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

      5. *-commutative 99.8%

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

      6. times-frac 99.8%

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

      7. associate-/l/ 99.9%

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

      8. *-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in v around 0 99.9%

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

   if 2.50000000000000014e-33 < v

   1. Initial program 80.0%

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

   3. Simplified 90.2%

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

      1. associate-*l* 90.2%

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

      2. fma-undefine 90.2%

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

      3. *-commutative 90.2%

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

      4. +-commutative 90.2%

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

      5. associate-*r/ 91.5%

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

      6. *-commutative 91.5%

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

      7. associate-/l* 92.6%

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

      8. associate-*l* 92.6%

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

      9. associate-*r/ 80.0%

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

      10. clear-num 79.9%

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

      11. *-commutative 79.9%

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

      12. associate-*r* 73.9%

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

      13. pow2 73.9%

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

      14. pow2 73.9%

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

      15. pow-prod-down 86.0%

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

      16. *-commutative 86.0%

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

      17. distribute-rgt-in 86.0%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in v around inf 84.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{r}^{2} \cdot {w}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. *-commutative 84.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{w}^{2} \cdot {r}^{2}}}}\right) \]

      2. unpow2 84.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot {r}^{2}}}\right) \]

      3. unpow2 84.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\left(w \cdot w\right) \cdot \color{blue}{\left(r \cdot r\right)}}}\right) \]

      4. swap-sqr 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

      5. unpow2 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{\left(w \cdot r\right)}^{2}}}}\right) \]

      6. *-commutative 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(r \cdot w\right)}}^{2}}}\right) \]

   9. Simplified 99.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. associate-/r/ 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{1}{4} \cdot {\left(r \cdot w\right)}^{2}}\right) \]

       2. *-commutative 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{4} \cdot {\color{blue}{\left(w \cdot r\right)}}^{2}\right) \]

       3. pow2 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{4} \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       4. add-sqr-sqrt 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)} \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}\right) \]

       5. sqrt-prod 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\sqrt{\frac{1}{4}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)} \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       6. metadata-eval 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\sqrt{\color{blue}{0.25}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       7. metadata-eval 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{0.5} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       8. pow2 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       9. *-commutative 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       10. sqrt-pow1 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       11. metadata-eval 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot {\left(r \cdot w\right)}^{\color{blue}{1}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       12. pow1 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       13. *-commutative 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       14. sqrt-prod 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{4}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)}\right) \]

       15. metadata-eval 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(\sqrt{\color{blue}{0.25}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)\right) \]

       16. metadata-eval 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(\color{blue}{0.5} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)\right) \]

       17. pow2 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}\right)\right) \]

       18. *-commutative 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}\right)\right) \]

       19. sqrt-pow1 99.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}\right)\right) \]

       20. metadata-eval 99.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot {\left(r \cdot w\right)}^{\color{blue}{1}}\right)\right) \]

       21. pow1 99.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(r \cdot w\right)}\right)\right) \]

       22. *-commutative 99.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \]

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \left(w \cdot r\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -54000000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{-1}{\frac{2}{r \cdot w} \cdot \frac{2}{r \cdot w}}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{\frac{1}{r}}{w} \cdot \frac{-1}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot 0.5\right) \cdot \left(\left(r \cdot w\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot w}\\ t_1 := \left(r \cdot w\right) \cdot 0.5\\ t_2 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -54000000000000:\\ \;\;\;\;t\_2 + \left(-1.5 + \frac{-1}{t\_0 \cdot t\_0}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;t\_2 + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{\frac{-1}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(-1.5 - t\_1 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r w))) (t_1 (* (* r w) 0.5)) (t_2 (/ 2.0 (* r r))))
   (if (<= v -54000000000000.0)
     (+ t_2 (+ -1.5 (/ -1.0 (* t_0 t_0))))
     (if (<= v 2.5e-33)
       (+ t_2 (+ -1.5 (/ (+ 0.375 (* v -0.25)) (/ (/ -1.0 (* r w)) (* r w)))))
       (+ t_2 (- -1.5 (* t_1 t_1)))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * w);
	double t_1 = (r * w) * 0.5;
	double t_2 = 2.0 / (r * r);
	double tmp;
	if (v <= -54000000000000.0) {
		tmp = t_2 + (-1.5 + (-1.0 / (t_0 * t_0)));
	} else if (v <= 2.5e-33) {
		tmp = t_2 + (-1.5 + ((0.375 + (v * -0.25)) / ((-1.0 / (r * w)) / (r * w))));
	} else {
		tmp = t_2 + (-1.5 - (t_1 * t_1));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 / (r * w)
    t_1 = (r * w) * 0.5d0
    t_2 = 2.0d0 / (r * r)
    if (v <= (-54000000000000.0d0)) then
        tmp = t_2 + ((-1.5d0) + ((-1.0d0) / (t_0 * t_0)))
    else if (v <= 2.5d-33) then
        tmp = t_2 + ((-1.5d0) + ((0.375d0 + (v * (-0.25d0))) / (((-1.0d0) / (r * w)) / (r * w))))
    else
        tmp = t_2 + ((-1.5d0) - (t_1 * t_1))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * w);
	double t_1 = (r * w) * 0.5;
	double t_2 = 2.0 / (r * r);
	double tmp;
	if (v <= -54000000000000.0) {
		tmp = t_2 + (-1.5 + (-1.0 / (t_0 * t_0)));
	} else if (v <= 2.5e-33) {
		tmp = t_2 + (-1.5 + ((0.375 + (v * -0.25)) / ((-1.0 / (r * w)) / (r * w))));
	} else {
		tmp = t_2 + (-1.5 - (t_1 * t_1));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * w)
	t_1 = (r * w) * 0.5
	t_2 = 2.0 / (r * r)
	tmp = 0
	if v <= -54000000000000.0:
		tmp = t_2 + (-1.5 + (-1.0 / (t_0 * t_0)))
	elif v <= 2.5e-33:
		tmp = t_2 + (-1.5 + ((0.375 + (v * -0.25)) / ((-1.0 / (r * w)) / (r * w))))
	else:
		tmp = t_2 + (-1.5 - (t_1 * t_1))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * w))
	t_1 = Float64(Float64(r * w) * 0.5)
	t_2 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (v <= -54000000000000.0)
		tmp = Float64(t_2 + Float64(-1.5 + Float64(-1.0 / Float64(t_0 * t_0))));
	elseif (v <= 2.5e-33)
		tmp = Float64(t_2 + Float64(-1.5 + Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(Float64(-1.0 / Float64(r * w)) / Float64(r * w)))));
	else
		tmp = Float64(t_2 + Float64(-1.5 - Float64(t_1 * t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = 2.0 / (r * w);
	t_1 = (r * w) * 0.5;
	t_2 = 2.0 / (r * r);
	tmp = 0.0;
	if (v <= -54000000000000.0)
		tmp = t_2 + (-1.5 + (-1.0 / (t_0 * t_0)));
	elseif (v <= 2.5e-33)
		tmp = t_2 + (-1.5 + ((0.375 + (v * -0.25)) / ((-1.0 / (r * w)) / (r * w))));
	else
		tmp = t_2 + (-1.5 - (t_1 * t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -54000000000000.0], N[(t$95$2 + N[(-1.5 + N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.5e-33], N[(t$95$2 + N[(-1.5 + N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 / N[(r * w), $MachinePrecision]), $MachinePrecision] / N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(-1.5 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -5.4e13

   1. Initial program 78.6%

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

   3. Simplified 89.6%

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

      1. associate-*l* 89.6%

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

      2. fma-undefine 89.6%

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

      3. *-commutative 89.6%

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

      4. +-commutative 89.6%

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

      5. associate-*r/ 89.6%

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

      6. *-commutative 89.6%

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

      7. associate-/l* 89.6%

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

      8. associate-*l* 89.6%

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

      9. associate-*r/ 78.6%

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

      10. clear-num 78.6%

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

      11. *-commutative 78.6%

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

      12. associate-*r* 72.0%

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

      13. pow2 72.0%

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

      14. pow2 72.0%

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

      15. pow-prod-down 83.6%

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

      16. *-commutative 83.6%

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

      17. distribute-rgt-in 83.6%

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

   6. Applied egg-rr 83.6%

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

   7. Taylor expanded in v around inf 78.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{r}^{2} \cdot {w}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. *-commutative 78.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{w}^{2} \cdot {r}^{2}}}}\right) \]

      2. unpow2 78.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot {r}^{2}}}\right) \]

      3. unpow2 78.2%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\left(w \cdot w\right) \cdot \color{blue}{\left(r \cdot r\right)}}}\right) \]

      4. swap-sqr 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

      5. unpow2 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{\left(w \cdot r\right)}^{2}}}}\right) \]

      6. *-commutative 99.7%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(r \cdot w\right)}}^{2}}}\right) \]

   9. Simplified 99.7%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. *-commutative 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(w \cdot r\right)}}^{2}}}\right) \]

       2. pow2 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       3. add-sqr-sqrt 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}}\right) \]

       4. sqrt-div 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{\sqrt{4}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       5. metadata-eval 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{\color{blue}{2}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       6. pow2 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       7. *-commutative 99.7%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       8. sqrt-pow1 77.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       9. metadata-eval 77.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{{\left(r \cdot w\right)}^{\color{blue}{1}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       10. pow1 77.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{r \cdot w}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       11. *-commutative 77.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{w \cdot r}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

       12. sqrt-div 77.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{w \cdot r} \cdot \color{blue}{\frac{\sqrt{4}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}}\right) \]

       13. metadata-eval 77.9%

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

       14. pow2 77.9%

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

       15. *-commutative 77.9%

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

       16. sqrt-pow1 99.8%

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

       17. metadata-eval 99.8%

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

       18. pow1 99.8%

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

       19. *-commutative 99.8%

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

   11. Applied egg-rr 99.8%

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

   if -5.4e13 < v < 2.50000000000000014e-33

   1. Initial program 86.4%

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

   3. Simplified 86.4%

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

      1. fma-undefine 86.4%

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

      2. *-commutative 86.4%

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

      3. +-commutative 86.4%

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

      4. associate-*r/ 86.4%

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

      5. *-commutative 86.4%

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

      6. associate-/l* 86.4%

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

      7. clear-num 86.4%

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

      8. un-div-inv 86.4%

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

      9. distribute-rgt-in 86.4%

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

      10. metadata-eval 86.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      11. *-commutative 86.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      12. associate-*l* 86.4%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

      13. metadata-eval 86.4%

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

      14. associate-*r* 82.8%

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

      15. pow2 82.8%

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

      16. pow2 82.8%

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

      17. pow-prod-down 99.8%

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

      18. *-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. *-un-lft-identity 99.8%

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

      2. unpow2 99.8%

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

      3. times-frac 99.8%

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

   8. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. associate-/r* 99.8%

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

      4. *-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in v around 0 99.8%

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

   if 2.50000000000000014e-33 < v

   1. Initial program 80.0%

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

   3. Simplified 90.2%

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

      1. associate-*l* 90.2%

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

      2. fma-undefine 90.2%

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

      3. *-commutative 90.2%

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

      4. +-commutative 90.2%

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

      5. associate-*r/ 91.5%

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

      6. *-commutative 91.5%

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

      7. associate-/l* 92.6%

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

      8. associate-*l* 92.6%

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

      9. associate-*r/ 80.0%

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

      10. clear-num 79.9%

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

      11. *-commutative 79.9%

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

      12. associate-*r* 73.9%

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

      13. pow2 73.9%

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

      14. pow2 73.9%

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

      15. pow-prod-down 86.0%

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

      16. *-commutative 86.0%

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

      17. distribute-rgt-in 86.0%

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

   6. Applied egg-rr 86.0%

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

   7. Taylor expanded in v around inf 84.3%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{r}^{2} \cdot {w}^{2}}}}\right) \]
   8. Step-by-step derivation

      1. *-commutative 84.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{w}^{2} \cdot {r}^{2}}}}\right) \]

      2. unpow2 84.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot {r}^{2}}}\right) \]

      3. unpow2 84.3%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\left(w \cdot w\right) \cdot \color{blue}{\left(r \cdot r\right)}}}\right) \]

      4. swap-sqr 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

      5. unpow2 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{\left(w \cdot r\right)}^{2}}}}\right) \]

      6. *-commutative 99.9%

         \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(r \cdot w\right)}}^{2}}}\right) \]

   9. Simplified 99.9%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{\left(r \cdot w\right)}^{2}}}}\right) \]
   10. Step-by-step derivation

       1. associate-/r/ 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{1}{4} \cdot {\left(r \cdot w\right)}^{2}}\right) \]

       2. *-commutative 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{4} \cdot {\color{blue}{\left(w \cdot r\right)}}^{2}\right) \]

       3. pow2 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{4} \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       4. add-sqr-sqrt 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)} \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}\right) \]

       5. sqrt-prod 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\sqrt{\frac{1}{4}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)} \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       6. metadata-eval 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\sqrt{\color{blue}{0.25}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       7. metadata-eval 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{0.5} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       8. pow2 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       9. *-commutative 99.9%

          \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       10. sqrt-pow1 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       11. metadata-eval 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot {\left(r \cdot w\right)}^{\color{blue}{1}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       12. pow1 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       13. *-commutative 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

       14. sqrt-prod 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{4}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)}\right) \]

       15. metadata-eval 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(\sqrt{\color{blue}{0.25}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)\right) \]

       16. metadata-eval 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(\color{blue}{0.5} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)\right) \]

       17. pow2 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}\right)\right) \]

       18. *-commutative 81.6%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}\right)\right) \]

       19. sqrt-pow1 99.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}\right)\right) \]

       20. metadata-eval 99.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot {\left(r \cdot w\right)}^{\color{blue}{1}}\right)\right) \]

       21. pow1 99.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(r \cdot w\right)}\right)\right) \]

       22. *-commutative 99.9%

           \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \]

   11. Applied egg-rr 99.9%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \left(w \cdot r\right)\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -54000000000000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{-1}{\frac{2}{r \cdot w} \cdot \frac{2}{r \cdot w}}\right)\\ \mathbf{elif}\;v \leq 2.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \frac{0.375 + v \cdot -0.25}{\frac{\frac{-1}{r \cdot w}}{r \cdot w}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot 0.5\right) \cdot \left(\left(r \cdot w\right) \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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) + ((0.375d0 + (v * (-0.25d0))) / (((1.0d0 - v) / (r * w)) * (((-1.0d0) / r) / w))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 88.3%

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

   1. fma-undefine 88.3%

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

   2. *-commutative 88.3%

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

   3. +-commutative 88.3%

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

   4. associate-*r/ 88.7%

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

   5. *-commutative 88.7%

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

   6. associate-/l* 89.1%

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

   7. clear-num 89.1%

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

   8. un-div-inv 89.1%

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

   9. distribute-rgt-in 89.1%

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

   10. metadata-eval 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 89.1%

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

   14. associate-*r* 82.2%

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

   15. pow2 82.2%

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

   16. pow2 82.2%

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

   17. pow-prod-down 99.8%

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

   18. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

   1. associate-/r* 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-/l/ 98.4%

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

   4. div-inv 98.4%

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

   5. *-commutative 98.4%

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

   6. times-frac 98.3%

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

   7. associate-/l/ 99.8%

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

   8. *-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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) + ((0.375d0 + (v * (-0.25d0))) / ((v + (-1.0d0)) / ((r * w) * (r * w)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

3. Simplified 88.3%

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

   1. fma-undefine 88.3%

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

   2. *-commutative 88.3%

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

   3. +-commutative 88.3%

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

   4. associate-*r/ 88.7%

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

   5. *-commutative 88.7%

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

   6. associate-/l* 89.1%

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

   7. clear-num 89.1%

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

   8. un-div-inv 89.1%

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

   9. distribute-rgt-in 89.1%

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

   10. metadata-eval 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{\color{blue}{0.375} + \left(-2 \cdot v\right) \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   11. *-commutative 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{\left(v \cdot -2\right)} \cdot 0.125}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   12. associate-*l* 89.1%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{0.375 + \color{blue}{v \cdot \left(-2 \cdot 0.125\right)}}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) \]

   13. metadata-eval 89.1%

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

   14. associate-*r* 82.2%

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

   15. pow2 82.2%

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

   16. pow2 82.2%

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

   17. pow-prod-down 99.8%

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

   18. *-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

   1. unpow2 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 5: 93.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{r \cdot w}\\ \frac{2}{r \cdot r} + \left(-1.5 + \frac{-1}{t\_0 \cdot t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r w))))
   (+ (/ 2.0 (* r r)) (+ -1.5 (/ -1.0 (* t_0 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * w);
	return (2.0 / (r * r)) + (-1.5 + (-1.0 / (t_0 * t_0)));
}

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
    t_0 = 2.0d0 / (r * w)
    code = (2.0d0 / (r * r)) + ((-1.5d0) + ((-1.0d0) / (t_0 * t_0)))
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * w);
	return (2.0 / (r * r)) + (-1.5 + (-1.0 / (t_0 * t_0)));
}

Python

def code(v, w, r):
	t_0 = 2.0 / (r * w)
	return (2.0 / (r * r)) + (-1.5 + (-1.0 / (t_0 * t_0)))

Julia

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * w))
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 + Float64(-1.0 / Float64(t_0 * t_0))))
end

MATLAB

function tmp = code(v, w, r)
	t_0 = 2.0 / (r * w);
	tmp = (2.0 / (r * r)) + (-1.5 + (-1.0 / (t_0 * t_0)));
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * w), $MachinePrecision]), $MachinePrecision]}, N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 + N[(-1.0 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.5%

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

3. Simplified 88.3%

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

   1. associate-*l* 88.3%

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

   2. fma-undefine 88.3%

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

   3. *-commutative 88.3%

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

   4. +-commutative 88.3%

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

   5. associate-*r/ 88.7%

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

   6. *-commutative 88.7%

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

   7. associate-/l* 89.1%

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

   8. associate-*l* 89.1%

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

   9. associate-*r/ 82.5%

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

   10. clear-num 82.5%

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

   11. *-commutative 82.5%

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

   12. associate-*r* 77.5%

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

   13. pow2 77.5%

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

   14. pow2 77.5%

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

   15. pow-prod-down 91.7%

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

   16. *-commutative 91.7%

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

   17. distribute-rgt-in 91.7%

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

6. Applied egg-rr 91.7%

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

7. Taylor expanded in v around inf 78.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{r}^{2} \cdot {w}^{2}}}}\right) \]
8. Step-by-step derivation

   1. *-commutative 78.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{w}^{2} \cdot {r}^{2}}}}\right) \]

   2. unpow2 78.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot {r}^{2}}}\right) \]

   3. unpow2 78.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\left(w \cdot w\right) \cdot \color{blue}{\left(r \cdot r\right)}}}\right) \]

   4. swap-sqr 94.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

   5. unpow2 94.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{\left(w \cdot r\right)}^{2}}}}\right) \]

   6. *-commutative 94.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(r \cdot w\right)}}^{2}}}\right) \]

9. Simplified 94.2%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{\left(r \cdot w\right)}^{2}}}}\right) \]
10. Step-by-step derivation

    1. *-commutative 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(w \cdot r\right)}}^{2}}}\right) \]

    2. pow2 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    3. add-sqr-sqrt 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}}\right) \]

    4. sqrt-div 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{\sqrt{4}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    5. metadata-eval 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{\color{blue}{2}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    6. pow2 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    7. *-commutative 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    8. sqrt-pow1 75.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    9. metadata-eval 75.7%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{{\left(r \cdot w\right)}^{\color{blue}{1}}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    10. pow1 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{r \cdot w}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    11. *-commutative 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{\color{blue}{w \cdot r}} \cdot \sqrt{\frac{4}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

    12. sqrt-div 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{2}{w \cdot r} \cdot \color{blue}{\frac{\sqrt{4}}{\sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}}\right) \]

    13. metadata-eval 75.7%

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

    14. pow2 75.7%

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

    15. *-commutative 75.7%

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

    16. sqrt-pow1 94.2%

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

    17. metadata-eval 94.2%

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

    18. pow1 94.2%

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

    19. *-commutative 94.2%

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

11. Applied egg-rr 94.2%

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

12. Final simplification 94.2%

    \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 + \frac{-1}{\frac{2}{r \cdot w} \cdot \frac{2}{r \cdot w}}\right) \]
13. Add Preprocessing

Alternative 6: 93.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot 0.5\\ \frac{2}{r \cdot r} + \left(-1.5 - t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) 0.5))) (+ (/ 2.0 (* r r)) (- -1.5 (* t_0 t_0)))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * 0.5;
	return (2.0 / (r * r)) + (-1.5 - (t_0 * t_0));
}

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
    t_0 = (r * w) * 0.5d0
    code = (2.0d0 / (r * r)) + ((-1.5d0) - (t_0 * t_0))
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * 0.5;
	return (2.0 / (r * r)) + (-1.5 - (t_0 * t_0));
}

Python

def code(v, w, r):
	t_0 = (r * w) * 0.5
	return (2.0 / (r * r)) + (-1.5 - (t_0 * t_0))

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * 0.5)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(t_0 * t_0)))
end

MATLAB

function tmp = code(v, w, r)
	t_0 = (r * w) * 0.5;
	tmp = (2.0 / (r * r)) + (-1.5 - (t_0 * t_0));
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * 0.5), $MachinePrecision]}, N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.5%

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

3. Simplified 88.3%

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

   1. associate-*l* 88.3%

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

   2. fma-undefine 88.3%

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

   3. *-commutative 88.3%

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

   4. +-commutative 88.3%

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

   5. associate-*r/ 88.7%

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

   6. *-commutative 88.7%

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

   7. associate-/l* 89.1%

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

   8. associate-*l* 89.1%

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

   9. associate-*r/ 82.5%

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

   10. clear-num 82.5%

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

   11. *-commutative 82.5%

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

   12. associate-*r* 77.5%

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

   13. pow2 77.5%

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

   14. pow2 77.5%

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

   15. pow-prod-down 91.7%

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

   16. *-commutative 91.7%

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

   17. distribute-rgt-in 91.7%

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

6. Applied egg-rr 91.7%

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

7. Taylor expanded in v around inf 78.8%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{r}^{2} \cdot {w}^{2}}}}\right) \]
8. Step-by-step derivation

   1. *-commutative 78.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{w}^{2} \cdot {r}^{2}}}}\right) \]

   2. unpow2 78.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot w\right)} \cdot {r}^{2}}}\right) \]

   3. unpow2 78.8%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\left(w \cdot w\right) \cdot \color{blue}{\left(r \cdot r\right)}}}\right) \]

   4. swap-sqr 94.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}}}\right) \]

   5. unpow2 94.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{\color{blue}{{\left(w \cdot r\right)}^{2}}}}\right) \]

   6. *-commutative 94.2%

      \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\frac{4}{{\color{blue}{\left(r \cdot w\right)}}^{2}}}\right) \]

9. Simplified 94.2%

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{\color{blue}{\frac{4}{{\left(r \cdot w\right)}^{2}}}}\right) \]
10. Step-by-step derivation

    1. associate-/r/ 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{1}{4} \cdot {\left(r \cdot w\right)}^{2}}\right) \]

    2. *-commutative 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{4} \cdot {\color{blue}{\left(w \cdot r\right)}}^{2}\right) \]

    3. pow2 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \frac{1}{4} \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    4. add-sqr-sqrt 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)} \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}}\right) \]

    5. sqrt-prod 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\sqrt{\frac{1}{4}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)} \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    6. metadata-eval 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\sqrt{\color{blue}{0.25}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    7. metadata-eval 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\color{blue}{0.5} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    8. pow2 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    9. *-commutative 94.2%

       \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    10. sqrt-pow1 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    11. metadata-eval 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot {\left(r \cdot w\right)}^{\color{blue}{1}}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    12. pow1 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{\left(r \cdot w\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    13. *-commutative 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \color{blue}{\left(w \cdot r\right)}\right) \cdot \sqrt{\frac{1}{4} \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}\right) \]

    14. sqrt-prod 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \color{blue}{\left(\sqrt{\frac{1}{4}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)}\right) \]

    15. metadata-eval 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(\sqrt{\color{blue}{0.25}} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)\right) \]

    16. metadata-eval 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(\color{blue}{0.5} \cdot \sqrt{\left(w \cdot r\right) \cdot \left(w \cdot r\right)}\right)\right) \]

    17. pow2 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \sqrt{\color{blue}{{\left(w \cdot r\right)}^{2}}}\right)\right) \]

    18. *-commutative 75.7%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \sqrt{{\color{blue}{\left(r \cdot w\right)}}^{2}}\right)\right) \]

    19. sqrt-pow1 94.2%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{{\left(r \cdot w\right)}^{\left(\frac{2}{2}\right)}}\right)\right) \]

    20. metadata-eval 94.2%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot {\left(r \cdot w\right)}^{\color{blue}{1}}\right)\right) \]

    21. pow1 94.2%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(r \cdot w\right)}\right)\right) \]

    22. *-commutative 94.2%

        \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \color{blue}{\left(w \cdot r\right)}\right)\right) \]

11. Applied egg-rr 94.2%

    \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(0.5 \cdot \left(w \cdot r\right)\right) \cdot \left(0.5 \cdot \left(w \cdot r\right)\right)}\right) \]

12. Final simplification 94.2%

    \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot 0.5\right) \cdot \left(\left(r \cdot w\right) \cdot 0.5\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024105 
(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))