Rosa's TurbineBenchmark

Percentage Accurate: 84.5% → 99.8%

Time: 20.2s

Precision: binary64

Cost: 8516

• 53.2% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 0.002:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{1}{w}}{\left(r \cdot r\right) \cdot w}}\right)\\ \end{array} \]

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))

↓

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= (* w w) 0.002)
     (+
      t_0
      (- -1.5 (* (* r (* w (* r w))) (/ (+ 0.375 (* v -0.25)) (- 1.0 v)))))
     (+
      -4.5
      (-
       (+ 3.0 t_0)
       (*
        (fma v -2.0 3.0)
        (/ (/ 0.125 (- 1.0 v)) (/ (/ 1.0 w) (* (* r r) w)))))))))

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;
}

↓

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if ((w * w) <= 0.002) {
		tmp = t_0 + (-1.5 - ((r * (w * (r * w))) * ((0.375 + (v * -0.25)) / (1.0 - v))));
	} else {
		tmp = -4.5 + ((3.0 + t_0) - (fma(v, -2.0, 3.0) * ((0.125 / (1.0 - v)) / ((1.0 / w) / ((r * r) * w)))));
	}
	return tmp;
}

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

↓

function code(v, w, r)
	t_0 = Float64(2.0 / Float64(r * r))
	tmp = 0.0
	if (Float64(w * w) <= 0.002)
		tmp = Float64(t_0 + Float64(-1.5 - Float64(Float64(r * Float64(w * Float64(r * w))) * Float64(Float64(0.375 + Float64(v * -0.25)) / Float64(1.0 - v)))));
	else
		tmp = Float64(-4.5 + Float64(Float64(3.0 + t_0) - Float64(fma(v, -2.0, 3.0) * Float64(Float64(0.125 / Float64(1.0 - v)) / Float64(Float64(1.0 / w) / Float64(Float64(r * r) * w))))));
	end
	return tmp
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]

↓

code[v_, w_, r_] := Block[{t$95$0 = N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(w * w), $MachinePrecision], 0.002], N[(t$95$0 + N[(-1.5 - N[(N[(r * N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.375 + N[(v * -0.25), $MachinePrecision]), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-4.5 + N[(N[(3.0 + t$95$0), $MachinePrecision] - N[(N[(v * -2.0 + 3.0), $MachinePrecision] * N[(N[(0.125 / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 / w), $MachinePrecision] / N[(N[(r * r), $MachinePrecision] * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2e-3

   1. Initial program 91.7%

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

   2. Simplified 92.4%

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

      [Start] 91.7%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
      associate--l- [=>] 91.7%	\[ \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right)} \]
      +-commutative [=>] 91.7%	\[ \color{blue}{\left(\frac{2}{r \cdot r} + 3\right)} - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right) \]
      associate--l+ [=>] 91.7%	\[ \color{blue}{\frac{2}{r \cdot r} + \left(3 - \left(\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v} + 4.5\right)\right)} \]
      +-commutative [=>] 91.7%	\[ \frac{2}{r \cdot r} + \left(3 - \color{blue}{\left(4.5 + \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)}\right) \]
      associate--r+ [=>] 91.7%	\[ \frac{2}{r \cdot r} + \color{blue}{\left(\left(3 - 4.5\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} \]
      metadata-eval [=>] 91.7%	\[ \frac{2}{r \cdot r} + \left(\color{blue}{-1.5} - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
      associate-*l/ [<=] 92.4%	\[ \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}\right) \]
      *-commutative [=>] 92.4%	\[ \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \]
      *-commutative [=>] 92.4%	\[ \frac{2}{r \cdot r} + \left(-1.5 - \color{blue}{\left(r \cdot \left(\left(w \cdot w\right) \cdot r\right)\right)} \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]
      *-commutative [=>] 92.4%	\[ \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}\right) \cdot \frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}\right) \]

   3. Taylor expanded in r around 0 92.4%

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

   4. Simplified 99.8%

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

      [Start] 92.4%	\[ \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left({w}^{2} \cdot r\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
      unpow2 [=>] 92.4%	\[ \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(\color{blue}{\left(w \cdot w\right)} \cdot r\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]
      associate-*l* [=>] 99.8%	\[ \frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right) \]

   if 2e-3 < (*.f64 w w)

   1. Initial program 78.0%

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

   2. Simplified 80.3%

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

      [Start] 78.0%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
      sub-neg [=>] 78.0%	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(-4.5\right)} \]
      associate-/l* [=>] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}}\right) + \left(-4.5\right) \]
      cancel-sign-sub-inv [=>] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \color{blue}{\left(3 + \left(-2\right) \cdot v\right)}}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}\right) + \left(-4.5\right) \]
      metadata-eval [=>] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + \color{blue}{-2} \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}\right) + \left(-4.5\right) \]
      *-commutative [=>] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{\color{blue}{r \cdot \left(\left(w \cdot w\right) \cdot r\right)}}}\right) + \left(-4.5\right) \]
      *-commutative [=>] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \color{blue}{\left(r \cdot \left(w \cdot w\right)\right)}}}\right) + \left(-4.5\right) \]
      metadata-eval [=>] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) + \color{blue}{-4.5} \]

   3. Taylor expanded in v around 0 65.5%

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

   4. Simplified 80.4%

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

      [Start] 65.5%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{{w}^{2} \cdot {r}^{2}} + -1 \cdot \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      *-commutative [=>] 65.5%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{\color{blue}{{r}^{2} \cdot {w}^{2}}} + -1 \cdot \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      unpow2 [=>] 65.5%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{\color{blue}{\left(r \cdot r\right)} \cdot {w}^{2}} + -1 \cdot \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      associate-*r* [<=] 67.2%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{\color{blue}{r \cdot \left(r \cdot {w}^{2}\right)}} + -1 \cdot \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      unpow2 [=>] 67.2%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{r \cdot \left(r \cdot \color{blue}{\left(w \cdot w\right)}\right)} + -1 \cdot \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      mul-1-neg [=>] 67.2%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)} + \color{blue}{\left(-\frac{v}{{w}^{2} \cdot {r}^{2}}\right)}}\right) + -4.5 \]
      sub-neg [<=] 67.2%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\color{blue}{\frac{1}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)} - \frac{v}{{w}^{2} \cdot {r}^{2}}}}\right) + -4.5 \]
      unpow2 [<=] 67.2%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{r \cdot \left(r \cdot \color{blue}{{w}^{2}}\right)} - \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      associate-*r* [=>] 65.5%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{\color{blue}{\left(r \cdot r\right) \cdot {w}^{2}}} - \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      unpow2 [<=] 65.5%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{\color{blue}{{r}^{2}} \cdot {w}^{2}} - \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      *-commutative [<=] 65.5%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1}{\color{blue}{{w}^{2} \cdot {r}^{2}}} - \frac{v}{{w}^{2} \cdot {r}^{2}}}\right) + -4.5 \]
      div-sub [<=] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\color{blue}{\frac{1 - v}{{w}^{2} \cdot {r}^{2}}}}\right) + -4.5 \]
      associate-/r* [=>] 80.4%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\color{blue}{\frac{\frac{1 - v}{{w}^{2}}}{{r}^{2}}}}\right) + -4.5 \]
      unpow2 [=>] 80.4%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1 - v}{{w}^{2}}}{\color{blue}{r \cdot r}}}\right) + -4.5 \]
      unpow2 [=>] 80.4%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1 - v}{\color{blue}{w \cdot w}}}{r \cdot r}}\right) + -4.5 \]

   5. Applied egg-rr 98.5%

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.125}{\frac{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}{\mathsf{fma}\left(v, -2, 3\right)}}\right)} - 1\right)}\right) + -4.5 \]
      Step-by-step derivation

      [Start] 80.4%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1 - v}{w \cdot w}}{r \cdot r}}\right) + -4.5 \]
      expm1-log1p-u [=>] 79.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1 - v}{w \cdot w}}{r \cdot r}}\right)\right)}\right) + -4.5 \]
      expm1-udef [=>] 79.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{\frac{1 - v}{w \cdot w}}{r \cdot r}}\right)} - 1\right)}\right) + -4.5 \]

   6. Simplified 99.8%

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

      [Start] 98.5%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \left(e^{\mathsf{log1p}\left(\frac{0.125}{\frac{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}{\mathsf{fma}\left(v, -2, 3\right)}}\right)} - 1\right)\right) + -4.5 \]
      expm1-def [=>] 98.5%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{\frac{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}{\mathsf{fma}\left(v, -2, 3\right)}}\right)\right)}\right) + -4.5 \]
      expm1-log1p [=>] 99.9%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125}{\frac{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}{\mathsf{fma}\left(v, -2, 3\right)}}}\right) + -4.5 \]
      associate-/r/ [=>] 99.8%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{0.125}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}} \cdot \mathsf{fma}\left(v, -2, 3\right)}\right) + -4.5 \]
      *-commutative [=>] 99.8%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\mathsf{fma}\left(v, -2, 3\right) \cdot \frac{0.125}{\left(1 - v\right) \cdot {\left(r \cdot w\right)}^{-2}}}\right) + -4.5 \]
      associate-/r* [=>] 99.8%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \color{blue}{\frac{\frac{0.125}{1 - v}}{{\left(r \cdot w\right)}^{-2}}}\right) + -4.5 \]
      *-commutative [=>] 99.8%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{{\color{blue}{\left(w \cdot r\right)}}^{-2}}\right) + -4.5 \]

   7. Applied egg-rr 99.9%

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

      [Start] 99.8%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{{\left(w \cdot r\right)}^{-2}}\right) + -4.5 \]
      metadata-eval [<=] 99.8%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{{\left(w \cdot r\right)}^{\color{blue}{\left(-1 + -1\right)}}}\right) + -4.5 \]
      pow-prod-up [<=] 99.8%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\color{blue}{{\left(w \cdot r\right)}^{-1} \cdot {\left(w \cdot r\right)}^{-1}}}\right) + -4.5 \]
      pow-prod-down [=>] 99.8%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\color{blue}{{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)}^{-1}}}\right) + -4.5 \]
      unswap-sqr [<=] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{{\color{blue}{\left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right)}}^{-1}}\right) + -4.5 \]
      inv-pow [<=] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\color{blue}{\frac{1}{\left(w \cdot w\right) \cdot \left(r \cdot r\right)}}}\right) + -4.5 \]
      metadata-eval [<=] 80.3%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\color{blue}{1 \cdot 1}}{\left(w \cdot w\right) \cdot \left(r \cdot r\right)}}\right) + -4.5 \]
      frac-times [<=] 80.4%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\color{blue}{\frac{1}{w \cdot w} \cdot \frac{1}{r \cdot r}}}\right) + -4.5 \]
      associate-/r* [=>] 82.6%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\color{blue}{\frac{\frac{1}{w}}{w}} \cdot \frac{1}{r \cdot r}}\right) + -4.5 \]
      metadata-eval [<=] 82.6%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{\color{blue}{\sqrt{1}}}{w}}{w} \cdot \frac{1}{r \cdot r}}\right) + -4.5 \]
      add-sqr-sqrt [=>] 41.4%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{\sqrt{1}}{\color{blue}{\sqrt{w} \cdot \sqrt{w}}}}{w} \cdot \frac{1}{r \cdot r}}\right) + -4.5 \]
      sqrt-prod [<=] 52.4%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{\sqrt{1}}{\color{blue}{\sqrt{w \cdot w}}}}{w} \cdot \frac{1}{r \cdot r}}\right) + -4.5 \]
      sqrt-div [<=] 52.4%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\color{blue}{\sqrt{\frac{1}{w \cdot w}}}}{w} \cdot \frac{1}{r \cdot r}}\right) + -4.5 \]
      frac-times [=>] 53.2%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\color{blue}{\frac{\sqrt{\frac{1}{w \cdot w}} \cdot 1}{w \cdot \left(r \cdot r\right)}}}\right) + -4.5 \]
      sqrt-div [=>] 53.2%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\color{blue}{\frac{\sqrt{1}}{\sqrt{w \cdot w}}} \cdot 1}{w \cdot \left(r \cdot r\right)}}\right) + -4.5 \]
      metadata-eval [=>] 53.2%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{\color{blue}{1}}{\sqrt{w \cdot w}} \cdot 1}{w \cdot \left(r \cdot r\right)}}\right) + -4.5 \]
      sqrt-prod [=>] 51.1%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{1}{\color{blue}{\sqrt{w} \cdot \sqrt{w}}} \cdot 1}{w \cdot \left(r \cdot r\right)}}\right) + -4.5 \]
      add-sqr-sqrt [<=] 99.9%	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{1}{\color{blue}{w}} \cdot 1}{w \cdot \left(r \cdot r\right)}}\right) + -4.5 \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 0.002:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + \frac{2}{r \cdot r}\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{1}{w}}{\left(r \cdot r\right) \cdot w}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	8516

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 0.002:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \mathsf{fma}\left(v, -2, 3\right) \cdot \frac{\frac{0.125}{1 - v}}{\frac{\frac{1}{w}}{\left(r \cdot r\right) \cdot w}}\right)\\ \end{array} \]

Alternative 2
Accuracy	99.7%
Cost	14336

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

Alternative 3
Accuracy	99.1%
Cost	1860

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+301}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{0.375 + v \cdot -0.25}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \end{array} \]

Alternative 4
Accuracy	97.8%
Cost	1481

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -0.9 \lor \neg \left(v \leq 10^{-41}\right):\\ \;\;\;\;-4.5 + \left(\left(3 + t_0\right) - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot 0.375\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	99.1%
Cost	1481

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1.15 \lor \neg \left(v \leq 10^{-41}\right):\\ \;\;\;\;-4.5 + \left(t_1 - t_0 \cdot 0.25\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(t_1 - 0.375 \cdot t_0\right)\\ \end{array} \]

Alternative 6
Accuracy	90.2%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;r \leq -9.5 \cdot 10^{-101} \lor \neg \left(r \leq 2.6 \cdot 10^{-93}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - r \cdot \left(0.25 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \]

Alternative 7
Accuracy	90.2%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;r \leq -1.42 \cdot 10^{-100} \lor \neg \left(r \leq 5 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 - r \cdot \left(0.375 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \]

Alternative 8
Accuracy	91.1%
Cost	1220

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6 \cdot 10^{+20}:\\ \;\;\;\;t_0 + \left(-1.5 - r \cdot \left(0.25 \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - r \cdot \left(\left(w \cdot \left(r \cdot w\right)\right) \cdot 0.375\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	81.7%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;r \leq -1.52 \cdot 10^{-6} \lor \neg \left(r \leq 42\right):\\ \;\;\;\;-4.5 + \left(3 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]

Alternative 10
Accuracy	70.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;r \leq -1.05 \cdot 10^{+24} \lor \neg \left(r \leq 1.95 \cdot 10^{+189}\right):\\ \;\;\;\;-4.5 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \end{array} \]

Alternative 11
Accuracy	46.6%
Cost	448

\[\frac{2}{r \cdot r} + -4.5 \]

Alternative 12
Accuracy	56.9%
Cost	448

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

Reproduce

herbie shell --seed 2023271 
(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))