Rosa's TurbineBenchmark

Percentage Accurate: 84.2% → 98.5%

Time: 15.9s

Alternatives: 9

Speedup: 1.7×

• 53.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -8.5 \cdot 10^{+159} \lor \neg \left(v \leq 2.2 \cdot 10^{-41}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{w}{\frac{1 - v}{r}} \cdot \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (or (<= v -8.5e+159) (not (<= v 2.2e-41)))
     (+ t_0 (- -1.5 (* (* (* r w) (* r w)) 0.25)))
     (+
      t_0
      (- -1.5 (* (/ w (/ (- 1.0 v) r)) (* r (* w (+ 0.375 (* v -0.25))))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if ((v <= -8.5e+159) || !(v <= 2.2e-41)) {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (r * (w * (0.375 + (v * -0.25))))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / r) / r
    if ((v <= (-8.5d+159)) .or. (.not. (v <= 2.2d-41))) then
        tmp = t_0 + ((-1.5d0) - (((r * w) * (r * w)) * 0.25d0))
    else
        tmp = t_0 + ((-1.5d0) - ((w / ((1.0d0 - v) / r)) * (r * (w * (0.375d0 + (v * (-0.25d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if ((v <= -8.5e+159) || !(v <= 2.2e-41)) {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (r * (w * (0.375 + (v * -0.25))))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if (v <= -8.5e+159) or not (v <= 2.2e-41):
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25))
	else:
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (r * (w * (0.375 + (v * -0.25))))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -8.5e+159) || !(v <= 2.2e-41))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(w / Float64(Float64(1.0 - v) / r)) * Float64(r * Float64(w * Float64(0.375 + Float64(v * -0.25)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	tmp = 0.0;
	if ((v <= -8.5e+159) || ~((v <= 2.2e-41)))
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	else
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (r * (w * (0.375 + (v * -0.25))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -8.5e+159], N[Not[LessEqual[v, 2.2e-41]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(w / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] * N[(r * N[(w * N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -8.50000000000000076e159 or 2.2e-41 < v

   1. Initial program 78.7%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in v around inf 81.2%

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

      1. *-commutative 81.2%

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

      2. *-commutative 81.2%

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

      3. unpow2 81.2%

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

      4. unpow2 81.2%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

   9. Applied egg-rr 99.8%

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

   if -8.50000000000000076e159 < v < 2.2e-41

   1. Initial program 84.5%

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

   3. Simplified 97.2%

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

      1. clear-num 97.2%

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

      2. inv-pow 97.2%

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

      3. associate-*r* 99.7%

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

      4. pow2 99.7%

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

      5. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. unpow2 99.7%

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

      3. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

       1. associate-/l/ 99.7%

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

       2. associate-/r/ 99.7%

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

       3. clear-num 99.7%

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

       4. *-commutative 99.7%

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

       5. associate-/l* 99.8%

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

       6. *-commutative 99.8%

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

   14. Applied egg-rr 99.8%

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

   15. Taylor expanded in r around 0 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -8.5 \cdot 10^{+159} \lor \neg \left(v \leq 2.2 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{w}{\frac{1 - v}{r}} \cdot \left(r \cdot \left(w \cdot \left(0.375 + v \cdot -0.25\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 97.1%

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

   1. clear-num 97.1%

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

   2. inv-pow 97.1%

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

   3. associate-*r* 99.7%

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

   4. pow2 99.7%

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

   5. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. unpow-1 99.7%

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

   2. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 98.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+231}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (<= (* w w) 4e+231)
     (+ t_0 (- -1.5 (/ (fma v -0.25 0.375) (/ (- 1.0 v) (* r (* w (* r w)))))))
     (+ t_0 (- -1.5 (* 0.375 (* (* r w) (* r w))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if ((w * w) <= 4e+231) {
		tmp = t_0 + (-1.5 - (fma(v, -0.25, 0.375) / ((1.0 - v) / (r * (w * (r * w))))));
	} else {
		tmp = t_0 + (-1.5 - (0.375 * ((r * w) * (r * w))));
	}
	return tmp;
}

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (Float64(w * w) <= 4e+231)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(fma(v, -0.25, 0.375) / Float64(Float64(1.0 - v) / Float64(r * Float64(w * Float64(r * w)))))));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * Float64(Float64(r * w) * Float64(r * w)))));
	end
	return tmp
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 4e+231], N[(t$95$0 + N[(-1.5 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(0.375 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 4.0000000000000002e231

   1. Initial program 91.0%

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

   3. Simplified 99.7%

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

   if 4.0000000000000002e231 < (*.f64 w w)

   1. Initial program 62.2%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in v around 0 63.5%

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

      1. *-commutative 63.5%

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

      2. *-commutative 63.5%

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

      3. unpow2 63.5%

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

      4. unpow2 63.5%

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

      5. swap-sqr 98.9%

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

      6. unpow2 98.9%

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

      7. *-commutative 98.9%

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

   7. Simplified 98.9%

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

      1. unpow2 98.9%

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

   9. Applied egg-rr 98.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 4 \cdot 10^{+231}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(r \cdot w\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 97.1%

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

   1. clear-num 97.1%

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

   2. inv-pow 97.1%

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

   3. associate-*r* 99.7%

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

   4. pow2 99.7%

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

   5. *-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. unpow-1 99.7%

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

   2. *-commutative 99.7%

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

8. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. unpow2 99.7%

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

   3. times-frac 99.8%

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

10. Applied egg-rr 99.8%

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

    1. associate-*l/ 99.7%

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

    2. *-lft-identity 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 5: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;\left(3 + \frac{2}{r \cdot r}\right) + \frac{\left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \left(0.125 \cdot \left(2 \cdot v - 3\right)\right)}{1 - v} \leq -\infty:\\ \;\;\;\;t_0 + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{w}{\frac{1 - v}{r}} \cdot \left(\left(r \cdot w\right) \cdot \left(v \cdot -0.25\right) + w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (<=
        (+
         (+ 3.0 (/ 2.0 (* r r)))
         (/ (* (* r (* r (* w w))) (* 0.125 (- (* 2.0 v) 3.0))) (- 1.0 v)))
        (- INFINITY))
     (+ t_0 (- -1.5 (* (* (* r w) (* r w)) 0.25)))
     (+
      t_0
      (-
       -1.5
       (*
        (/ w (/ (- 1.0 v) r))
        (+ (* (* r w) (* v -0.25)) (* w (* r 0.375)))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (((3.0 + (2.0 / (r * r))) + (((r * (r * (w * w))) * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -((double) INFINITY)) {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (((r * w) * (v * -0.25)) + (w * (r * 0.375)))));
	}
	return tmp;
}

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if (((3.0 + (2.0 / (r * r))) + (((r * (r * (w * w))) * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -Double.POSITIVE_INFINITY) {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (((r * w) * (v * -0.25)) + (w * (r * 0.375)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if ((3.0 + (2.0 / (r * r))) + (((r * (r * (w * w))) * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -math.inf:
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25))
	else:
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (((r * w) * (v * -0.25)) + (w * (r * 0.375)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if (Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(Float64(r * Float64(r * Float64(w * w))) * Float64(0.125 * Float64(Float64(2.0 * v) - 3.0))) / Float64(1.0 - v))) <= Float64(-Inf))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(w / Float64(Float64(1.0 - v) / r)) * Float64(Float64(Float64(r * w) * Float64(v * -0.25)) + Float64(w * Float64(r * 0.375))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	tmp = 0.0;
	if (((3.0 + (2.0 / (r * r))) + (((r * (r * (w * w))) * (0.125 * ((2.0 * v) - 3.0))) / (1.0 - v))) <= -Inf)
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	else
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (((r * w) * (v * -0.25)) + (w * (r * 0.375)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))) < -inf.0

   1. Initial program 82.2%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in v around inf 87.2%

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

      1. *-commutative 87.2%

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

      2. *-commutative 87.2%

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

      3. unpow2 87.2%

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

      4. unpow2 87.2%

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

      5. swap-sqr 98.8%

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

      6. unpow2 98.8%

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

      7. *-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. unpow2 98.8%

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

   9. Applied egg-rr 98.8%

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

   if -inf.0 < (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))

   1. Initial program 82.1%

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

   3. Simplified 97.5%

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

      1. clear-num 97.5%

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

      2. inv-pow 97.5%

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

      3. associate-*r* 99.7%

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

      4. pow2 99.7%

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

      5. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. unpow2 99.7%

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

      3. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.7%

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

       2. *-lft-identity 99.7%

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

   12. Simplified 99.7%

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

       1. associate-/l/ 99.1%

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

       2. associate-/r/ 99.1%

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

       3. clear-num 99.1%

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

       4. *-commutative 99.1%

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

       5. associate-/l* 99.2%

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

       6. *-commutative 99.2%

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

   14. Applied egg-rr 99.2%

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

       1. fma-udef 99.2%

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

       2. distribute-lft-in 99.2%

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

       3. *-commutative 99.2%

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

       4. *-commutative 99.2%

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

       5. associate-*l* 99.2%

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

   16. Applied egg-rr 99.2%

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

4. Final simplification 99.1%

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

Alternative 6: 99.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -10000000000 \lor \neg \left(v \leq 2.2 \cdot 10^{-41}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \frac{w}{\frac{1 - v}{r}} \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)))
   (if (or (<= v -10000000000.0) (not (<= v 2.2e-41)))
     (+ t_0 (- -1.5 (* (* (* r w) (* r w)) 0.25)))
     (+ t_0 (- -1.5 (* (/ w (/ (- 1.0 v) r)) (* w (* r 0.375))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if ((v <= -10000000000.0) || !(v <= 2.2e-41)) {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (w * (r * 0.375))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (2.0d0 / r) / r
    if ((v <= (-10000000000.0d0)) .or. (.not. (v <= 2.2d-41))) then
        tmp = t_0 + ((-1.5d0) - (((r * w) * (r * w)) * 0.25d0))
    else
        tmp = t_0 + ((-1.5d0) - ((w / ((1.0d0 - v) / r)) * (w * (r * 0.375d0))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double tmp;
	if ((v <= -10000000000.0) || !(v <= 2.2e-41)) {
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	} else {
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (w * (r * 0.375))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	tmp = 0
	if (v <= -10000000000.0) or not (v <= 2.2e-41):
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25))
	else:
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (w * (r * 0.375))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -10000000000.0) || !(v <= 2.2e-41))
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(Float64(r * w) * Float64(r * w)) * 0.25)));
	else
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(w / Float64(Float64(1.0 - v) / r)) * Float64(w * Float64(r * 0.375)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	tmp = 0.0;
	if ((v <= -10000000000.0) || ~((v <= 2.2e-41)))
		tmp = t_0 + (-1.5 - (((r * w) * (r * w)) * 0.25));
	else
		tmp = t_0 + (-1.5 - ((w / ((1.0 - v) / r)) * (w * (r * 0.375))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -10000000000.0], N[Not[LessEqual[v, 2.2e-41]], $MachinePrecision]], N[(t$95$0 + N[(-1.5 - N[(N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.5 - N[(N[(w / N[(N[(1.0 - v), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] * N[(w * N[(r * 0.375), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if v < -1e10 or 2.2e-41 < v

   1. Initial program 79.9%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in v around inf 81.4%

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

      1. *-commutative 81.4%

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

      2. *-commutative 81.4%

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

      3. unpow2 81.4%

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

      4. unpow2 81.4%

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

      5. swap-sqr 99.7%

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

      6. unpow2 99.7%

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

      7. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. unpow2 99.7%

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

   9. Applied egg-rr 99.7%

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

   if -1e10 < v < 2.2e-41

   1. Initial program 85.0%

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

   3. Simplified 97.2%

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

      1. clear-num 97.1%

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

      2. inv-pow 97.1%

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

      3. associate-*r* 99.7%

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

      4. pow2 99.7%

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

      5. *-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. unpow-1 99.7%

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

      2. *-commutative 99.7%

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

   8. Simplified 99.7%

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

      1. *-un-lft-identity 99.7%

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

      2. unpow2 99.7%

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

      3. times-frac 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

   12. Simplified 99.8%

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

       1. associate-/l/ 99.8%

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

       2. associate-/r/ 99.8%

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

       3. clear-num 99.8%

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

       4. *-commutative 99.8%

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

       5. associate-/l* 99.8%

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

       6. *-commutative 99.8%

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

   14. Applied egg-rr 99.8%

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

   15. Taylor expanded in v around 0 99.8%

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

       1. *-commutative 99.8%

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

       2. *-commutative 99.8%

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

       3. associate-*r* 99.8%

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

   17. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -10000000000 \lor \neg \left(v \leq 2.2 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \frac{w}{\frac{1 - v}{r}} \cdot \left(w \cdot \left(r \cdot 0.375\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ t_1 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ \mathbf{if}\;v \leq -64000000000:\\ \;\;\;\;t_0 + \left(-1.5 - t_1 \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 2.4 \cdot 10^{+43}:\\ \;\;\;\;t_0 + \left(-1.5 - 0.375 \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + \left(-1.5 - 0.25 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ (/ 2.0 r) r)) (t_1 (* (* r w) (* r w))))
   (if (<= v -64000000000.0)
     (+ t_0 (- -1.5 (* t_1 0.25)))
     (if (<= v 2.4e+43)
       (+ t_0 (- -1.5 (* 0.375 t_1)))
       (+ (/ 1.0 (* r (/ r 2.0))) (- -1.5 (* 0.25 (* r (* w (* r w))))))))))

C

double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double t_1 = (r * w) * (r * w);
	double tmp;
	if (v <= -64000000000.0) {
		tmp = t_0 + (-1.5 - (t_1 * 0.25));
	} else if (v <= 2.4e+43) {
		tmp = t_0 + (-1.5 - (0.375 * t_1));
	} else {
		tmp = (1.0 / (r * (r / 2.0))) + (-1.5 - (0.25 * (r * (w * (r * w)))));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (2.0d0 / r) / r
    t_1 = (r * w) * (r * w)
    if (v <= (-64000000000.0d0)) then
        tmp = t_0 + ((-1.5d0) - (t_1 * 0.25d0))
    else if (v <= 2.4d+43) then
        tmp = t_0 + ((-1.5d0) - (0.375d0 * t_1))
    else
        tmp = (1.0d0 / (r * (r / 2.0d0))) + ((-1.5d0) - (0.25d0 * (r * (w * (r * w)))))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (2.0 / r) / r;
	double t_1 = (r * w) * (r * w);
	double tmp;
	if (v <= -64000000000.0) {
		tmp = t_0 + (-1.5 - (t_1 * 0.25));
	} else if (v <= 2.4e+43) {
		tmp = t_0 + (-1.5 - (0.375 * t_1));
	} else {
		tmp = (1.0 / (r * (r / 2.0))) + (-1.5 - (0.25 * (r * (w * (r * w)))));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (2.0 / r) / r
	t_1 = (r * w) * (r * w)
	tmp = 0
	if v <= -64000000000.0:
		tmp = t_0 + (-1.5 - (t_1 * 0.25))
	elif v <= 2.4e+43:
		tmp = t_0 + (-1.5 - (0.375 * t_1))
	else:
		tmp = (1.0 / (r * (r / 2.0))) + (-1.5 - (0.25 * (r * (w * (r * w)))))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(2.0 / r) / r)
	t_1 = Float64(Float64(r * w) * Float64(r * w))
	tmp = 0.0
	if (v <= -64000000000.0)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(t_1 * 0.25)));
	elseif (v <= 2.4e+43)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(0.375 * t_1)));
	else
		tmp = Float64(Float64(1.0 / Float64(r * Float64(r / 2.0))) + Float64(-1.5 - Float64(0.25 * Float64(r * Float64(w * Float64(r * w))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (2.0 / r) / r;
	t_1 = (r * w) * (r * w);
	tmp = 0.0;
	if (v <= -64000000000.0)
		tmp = t_0 + (-1.5 - (t_1 * 0.25));
	elseif (v <= 2.4e+43)
		tmp = t_0 + (-1.5 - (0.375 * t_1));
	else
		tmp = (1.0 / (r * (r / 2.0))) + (-1.5 - (0.25 * (r * (w * (r * w)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, Block[{t$95$1 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[v, -64000000000.0], N[(t$95$0 + N[(-1.5 - N[(t$95$1 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 2.4e+43], N[(t$95$0 + N[(-1.5 - N[(0.375 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(r * N[(r / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(0.25 * N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.4e10

   1. Initial program 76.5%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in v around inf 77.0%

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

      1. *-commutative 77.0%

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

      2. *-commutative 77.0%

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

      3. unpow2 77.0%

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

      4. unpow2 77.0%

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

      5. swap-sqr 99.6%

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

      6. unpow2 99.6%

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

      7. *-commutative 99.6%

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

   7. Simplified 99.6%

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

      1. unpow2 99.6%

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

   9. Applied egg-rr 99.6%

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

   if -6.4e10 < v < 2.40000000000000023e43

   1. Initial program 84.3%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in v around 0 78.1%

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

      1. *-commutative 78.1%

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

      2. *-commutative 78.1%

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

      3. unpow2 78.1%

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

      4. unpow2 78.1%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 89.0%

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

   9. Applied egg-rr 99.8%

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

   if 2.40000000000000023e43 < v

   1. Initial program 85.4%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in v around inf 89.1%

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

      1. *-commutative 89.1%

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

      2. *-commutative 89.1%

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

      3. unpow2 89.1%

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

      4. unpow2 89.1%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 99.8%

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

      2. *-commutative 99.8%

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

      3. associate-*r* 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. clear-num 99.7%

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

       2. inv-pow 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

       2. associate-/r/ 99.8%

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

   13. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -64000000000:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{elif}\;v \leq 2.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{r \cdot \frac{r}{2}} + \left(-1.5 - 0.25 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;v \leq -64000000000 \lor \neg \left(v \leq 2.2 \cdot 10^{-41}\right):\\ \;\;\;\;t_1 + \left(-1.5 - t_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - 0.375 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (* (* r w) (* r w))) (t_1 (/ (/ 2.0 r) r)))
   (if (or (<= v -64000000000.0) (not (<= v 2.2e-41)))
     (+ t_1 (- -1.5 (* t_0 0.25)))
     (+ t_1 (- -1.5 (* 0.375 t_0))))))

C

double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -64000000000.0) || !(v <= 2.2e-41)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (r * w) * (r * w)
    t_1 = (2.0d0 / r) / r
    if ((v <= (-64000000000.0d0)) .or. (.not. (v <= 2.2d-41))) then
        tmp = t_1 + ((-1.5d0) - (t_0 * 0.25d0))
    else
        tmp = t_1 + ((-1.5d0) - (0.375d0 * t_0))
    end if
    code = tmp
end function

Java

public static double code(double v, double w, double r) {
	double t_0 = (r * w) * (r * w);
	double t_1 = (2.0 / r) / r;
	double tmp;
	if ((v <= -64000000000.0) || !(v <= 2.2e-41)) {
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	} else {
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	}
	return tmp;
}

Python

def code(v, w, r):
	t_0 = (r * w) * (r * w)
	t_1 = (2.0 / r) / r
	tmp = 0
	if (v <= -64000000000.0) or not (v <= 2.2e-41):
		tmp = t_1 + (-1.5 - (t_0 * 0.25))
	else:
		tmp = t_1 + (-1.5 - (0.375 * t_0))
	return tmp

Julia

function code(v, w, r)
	t_0 = Float64(Float64(r * w) * Float64(r * w))
	t_1 = Float64(Float64(2.0 / r) / r)
	tmp = 0.0
	if ((v <= -64000000000.0) || !(v <= 2.2e-41))
		tmp = Float64(t_1 + Float64(-1.5 - Float64(t_0 * 0.25)));
	else
		tmp = Float64(t_1 + Float64(-1.5 - Float64(0.375 * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(v, w, r)
	t_0 = (r * w) * (r * w);
	t_1 = (2.0 / r) / r;
	tmp = 0.0;
	if ((v <= -64000000000.0) || ~((v <= 2.2e-41)))
		tmp = t_1 + (-1.5 - (t_0 * 0.25));
	else
		tmp = t_1 + (-1.5 - (0.375 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[v_, w_, r_] := Block[{t$95$0 = N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(2.0 / r), $MachinePrecision] / r), $MachinePrecision]}, If[Or[LessEqual[v, -64000000000.0], N[Not[LessEqual[v, 2.2e-41]], $MachinePrecision]], N[(t$95$1 + N[(-1.5 - N[(t$95$0 * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(-1.5 - N[(0.375 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if v < -6.4e10 or 2.2e-41 < v

   1. Initial program 79.6%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in v around inf 81.1%

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

      1. *-commutative 81.1%

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

      2. *-commutative 81.1%

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

      3. unpow2 81.1%

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

      4. unpow2 81.1%

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

      5. swap-sqr 99.7%

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

      6. unpow2 99.7%

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

      7. *-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. unpow2 99.7%

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

   9. Applied egg-rr 99.7%

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

   if -6.4e10 < v < 2.2e-41

   1. Initial program 85.3%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in v around 0 78.0%

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

      1. *-commutative 78.0%

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

      2. *-commutative 78.0%

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

      3. unpow2 78.0%

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

      4. unpow2 78.0%

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

      5. swap-sqr 99.8%

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

      6. unpow2 99.8%

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

      7. *-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. unpow2 87.2%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -64000000000 \lor \neg \left(v \leq 2.2 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} + \left(-1.5 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(v, w, r)
    real(8), intent (in) :: v
    real(8), intent (in) :: w
    real(8), intent (in) :: r
    code = ((2.0d0 / r) / r) + ((-1.5d0) - (((r * w) * (r * w)) * 0.25d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.1%

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

3. Simplified 97.1%

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

5. Taylor expanded in v around inf 76.4%

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

   1. *-commutative 76.4%

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

   2. *-commutative 76.4%

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

   3. unpow2 76.4%

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

   4. unpow2 76.4%

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

   5. swap-sqr 94.2%

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

   6. unpow2 94.2%

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

   7. *-commutative 94.2%

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

7. Simplified 94.2%

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

   1. unpow2 94.2%

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

9. Applied egg-rr 94.2%

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

10. Final simplification 94.2%

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

Reproduce

herbie shell --seed 2024024 
(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))