Rosa's TurbineBenchmark

Average Error: 13.2 → 1.2

Time: 22.2s

Precision: binary64

Cost: 14276

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -3.2631988226918448 \cdot 10^{-24}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), w \cdot \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right), 1.5\right)\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;t_0 + \mathsf{fma}\left(r \cdot w, w \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {r}^{-2} + \left(3 + {\left(r \cdot w\right)}^{2} \cdot -0.25\right)\right) + -4.5\\ \end{array} \]

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))

↓

(FPCore (v w r)
 :precision binary64
 (let* ((t_0 (/ 2.0 (* r r))))
   (if (<= v -3.2631988226918448e-24)
     (- t_0 (fma (fma v -0.25 0.375) (* w (* r (* w (/ r (- 1.0 v))))) 1.5))
     (if (<= v 6.302705948431219e-25)
       (+ t_0 (fma (* r w) (* w (* r -0.375)) -1.5))
       (+
        (+ (* 2.0 (pow r -2.0)) (+ 3.0 (* (pow (* r w) 2.0) -0.25)))
        -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;
}

↓

double code(double v, double w, double r) {
	double t_0 = 2.0 / (r * r);
	double tmp;
	if (v <= -3.2631988226918448e-24) {
		tmp = t_0 - fma(fma(v, -0.25, 0.375), (w * (r * (w * (r / (1.0 - v))))), 1.5);
	} else if (v <= 6.302705948431219e-25) {
		tmp = t_0 + fma((r * w), (w * (r * -0.375)), -1.5);
	} else {
		tmp = ((2.0 * pow(r, -2.0)) + (3.0 + (pow((r * w), 2.0) * -0.25))) + -4.5;
	}
	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 (v <= -3.2631988226918448e-24)
		tmp = Float64(t_0 - fma(fma(v, -0.25, 0.375), Float64(w * Float64(r * Float64(w * Float64(r / Float64(1.0 - v))))), 1.5));
	elseif (v <= 6.302705948431219e-25)
		tmp = Float64(t_0 + fma(Float64(r * w), Float64(w * Float64(r * -0.375)), -1.5));
	else
		tmp = Float64(Float64(Float64(2.0 * (r ^ -2.0)) + Float64(3.0 + Float64((Float64(r * w) ^ 2.0) * -0.25))) + -4.5);
	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[v, -3.2631988226918448e-24], N[(t$95$0 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(w * N[(r * N[(w * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[v, 6.302705948431219e-25], N[(t$95$0 + N[(N[(r * w), $MachinePrecision] * N[(w * N[(r * -0.375), $MachinePrecision]), $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Power[r, -2.0], $MachinePrecision]), $MachinePrecision] + N[(3.0 + N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -3.2631988226918448e-24

   1. Initial program 17.2

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

   2. Simplified 8.2

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), \frac{w}{\frac{\frac{1 - v}{r}}{r}} \cdot w, 1.5\right)} \]
      Proof
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (fma.f64 v -1/4 3/8) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (fma.f64 v (Rewrite<= metadata-eval (*.f64 -2 1/8)) 3/8) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (fma.f64 v (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) 1/8) 3/8) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (fma.f64 v (*.f64 (neg.f64 2) 1/8) (Rewrite<= metadata-eval (*.f64 3 1/8))) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 v (*.f64 (neg.f64 2) 1/8)) (*.f64 3 1/8))) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 v (neg.f64 2)) 1/8)) (*.f64 3 1/8)) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 1 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (+.f64 (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 v 2))) 1/8) (*.f64 3 1/8)) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (+.f64 (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 v))) 1/8) (*.f64 3 1/8)) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/8 (+.f64 (neg.f64 (*.f64 2 v)) 3))) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (Rewrite<= +-commutative_binary64 (+.f64 3 (neg.f64 (*.f64 2 v))))) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (Rewrite<= sub-neg_binary64 (-.f64 3 (*.f64 2 v)))) (*.f64 (/.f64 w (/.f64 (/.f64 (-.f64 1 v) r) r)) w) 3/2)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (Rewrite<= associate-/r/_binary64 (/.f64 w (/.f64 (/.f64 (/.f64 (-.f64 1 v) r) r) w))) 3/2)): 9 points increase in error, 8 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 w (Rewrite=> associate-/l/_binary64 (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w r)))) 3/2)): 3 points increase in error, 43 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 w (Rewrite=> associate-/l/_binary64 (/.f64 (-.f64 1 v) (*.f64 (*.f64 w r) r)))) 3/2)): 27 points increase in error, 9 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 w (*.f64 (*.f64 w r) r)) (-.f64 1 v))) 3/2)): 4 points increase in error, 11 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w (*.f64 w r)) r)) (-.f64 1 v)) 3/2)): 16 points increase in error, 30 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (*.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w w) r)) r) (-.f64 1 v)) 3/2)): 41 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (-.f64 1 v)) (Rewrite<= metadata-eval (neg.f64 -3/2)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (fma.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (-.f64 1 v)) (neg.f64 (Rewrite<= metadata-eval (-.f64 3 9/2))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (-.f64 1 v))) (-.f64 3 9/2)))): 2 points increase in error, 4 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (-.f64 1 v)) (*.f64 1/8 (-.f64 3 (*.f64 2 v))))) (-.f64 3 9/2))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (*.f64 1/8 (-.f64 3 (*.f64 2 v)))) (-.f64 1 v))) (-.f64 3 9/2))): 13 points increase in error, 2 points decrease in error
      (-.f64 (/.f64 2 (*.f64 r r)) (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r))) (-.f64 1 v)) (-.f64 3 9/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 (/.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))) (-.f64 3 9/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (-.f64 (/.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))) 3) 9/2)): 4 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 3 (-.f64 (/.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))))) 9/2): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate--l+_binary64 (-.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)))) 9/2): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 1.8

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

   if -3.2631988226918448e-24 < v < 6.30270594843121875e-25

   1. Initial program 8.4

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

   2. Applied egg-rr 2.2

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

   3. Taylor expanded in v around 0 16.9

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) - 4.5 \]

   4. Simplified 16.9

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot 0.375\right)}\right) - 4.5 \]
      Proof
      (*.f64 (*.f64 w w) (*.f64 (*.f64 r r) 3/8)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) (*.f64 (*.f64 r r) 3/8)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 w 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 r 2)) 3/8)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 w 2) (pow.f64 r 2)) 3/8)): 16 points increase in error, 21 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in r around 0 16.9

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

   6. Simplified 0.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(w \cdot r, w \cdot \left(r \cdot -0.375\right), -1.5\right) - \frac{-2}{r \cdot r}} \]
      Proof
      (-.f64 (fma.f64 (*.f64 w r) (*.f64 w (*.f64 r -3/8)) -3/2) (/.f64 -2 (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (-.f64 (fma.f64 (*.f64 w r) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w r) -3/8)) -3/2) (/.f64 -2 (*.f64 r r))): 11 points increase in error, 12 points decrease in error
      (-.f64 (fma.f64 (*.f64 w r) (*.f64 (*.f64 w r) -3/8) (Rewrite<= metadata-eval (neg.f64 3/2))) (/.f64 -2 (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (*.f64 w r) (*.f64 (*.f64 w r) -3/8)) 3/2)) (/.f64 -2 (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 w r) (*.f64 w r)) -3/8)) 3/2) (/.f64 -2 (*.f64 r r))): 15 points increase in error, 18 points decrease in error
      (-.f64 (-.f64 (*.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 w w) (*.f64 r r))) -3/8) 3/2) (/.f64 -2 (*.f64 r r))): 82 points increase in error, 8 points decrease in error
      (-.f64 (-.f64 (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) (*.f64 r r)) -3/8) 3/2) (/.f64 -2 (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (*.f64 (*.f64 (pow.f64 w 2) (Rewrite<= unpow2_binary64 (pow.f64 r 2))) -3/8) 3/2) (/.f64 -2 (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2)))) 3/2) (/.f64 -2 (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2) (/.f64 (Rewrite<= metadata-eval (*.f64 -2 1)) (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2) (/.f64 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) 1) (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2) (/.f64 (*.f64 (neg.f64 2) 1) (Rewrite<= unpow2_binary64 (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2) (Rewrite<= associate-*r/_binary64 (*.f64 (neg.f64 2) (/.f64 1 (pow.f64 r 2))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2) (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 (/.f64 1 (pow.f64 r 2)))))): 0 points increase in error, 0 points decrease in error
      (-.f64 (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2) (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 2 (/.f64 1 (pow.f64 r 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate--r-_binary64 (+.f64 (-.f64 (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2) 0) (*.f64 2 (/.f64 1 (pow.f64 r 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> --rgt-identity_binary64 (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2)) (*.f64 2 (/.f64 1 (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 r 2))) (-.f64 (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) 3/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 2 (/.f64 1 (pow.f64 r 2))) (*.f64 -3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2)))) 3/2)): 0 points increase in error, 0 points decrease in error

   if 6.30270594843121875e-25 < v

   1. Initial program 17.1

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

   2. Taylor expanded in v around inf 18.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{0.25 \cdot \left({w}^{2} \cdot {r}^{2}\right)}\right) - 4.5 \]

   3. Simplified 18.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(w \cdot w\right) \cdot \left(\left(r \cdot r\right) \cdot 0.25\right)}\right) - 4.5 \]
      Proof
      (*.f64 (*.f64 w w) (*.f64 (*.f64 r r) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) (*.f64 (*.f64 r r) 1/4)): 0 points increase in error, 0 points decrease in error
      (*.f64 (pow.f64 w 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 r 2)) 1/4)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 w 2) (pow.f64 r 2)) 1/4)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 1/4 (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 2.1

      \[\leadsto \color{blue}{\left(2 \cdot {r}^{-2} + \left(3 - 0.25 \cdot {\left(w \cdot r\right)}^{2}\right)\right)} - 4.5 \]
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -3.2631988226918448 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), w \cdot \left(r \cdot \left(w \cdot \frac{r}{1 - v}\right)\right), 1.5\right)\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{r \cdot r} + \mathsf{fma}\left(r \cdot w, w \cdot \left(r \cdot -0.375\right), -1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot {r}^{-2} + \left(3 + {\left(r \cdot w\right)}^{2} \cdot -0.25\right)\right) + -4.5\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	27200

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

Alternative 2
Error	1.0
Cost	14024

\[\begin{array}{l} t_0 := {\left(r \cdot w\right)}^{2}\\ t_1 := \left(2 \cdot {r}^{-2} + \left(3 + t_0 \cdot -0.25\right)\right) + -4.5\\ \mathbf{if}\;v \leq -2.9516536836600776 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{t_0 \cdot \left(\left(3 - 2 \cdot v\right) \cdot -0.125\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.4
Cost	8392

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{w \cdot \left(r \cdot w\right)}{\frac{8}{\mathsf{fma}\left(v, -2, 3\right)} \cdot \frac{1 - v}{r}}\right) + -4.5\\ \mathbf{if}\;r \leq -1 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 10^{+151}:\\ \;\;\;\;\left(t_0 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	0.5
Cost	8392

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{\frac{r}{1 - v} \cdot \left(w \cdot \left(r \cdot w\right)\right)}{\frac{8}{\mathsf{fma}\left(v, -2, 3\right)}}\right) + -4.5\\ \mathbf{if}\;r \leq -1 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 10^{+155}:\\ \;\;\;\;\left(t_0 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	1.1
Cost	8328

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{if}\;v \leq -2.9516536836600776 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;\left(t_0 + \frac{{\left(r \cdot w\right)}^{2} \cdot \left(\left(3 - 2 \cdot v\right) \cdot -0.125\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.0
Cost	8264

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{if}\;v \leq -2.9516536836600776 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;\left(t_0 - \frac{\left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	1.0
Cost	1992

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 + -0.25 \cdot t_0\right) + -4.5\\ \mathbf{if}\;v \leq -7352299.727616896:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;\left(t_1 - t_0 \cdot \left(0.375 + v \cdot \left(0.125 + 0.125 \cdot v\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	1.0
Cost	1736

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 + -0.25 \cdot t_0\right) + -4.5\\ \mathbf{if}\;v \leq -7352299.727616896:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;\left(t_1 + t_0 \cdot \left(-0.375 + v \cdot -0.125\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	16.5
Cost	1496

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ t_1 := -1.5 + w \cdot \left(w \cdot \left(r \cdot \left(r \cdot -0.25\right)\right)\right)\\ \mathbf{if}\;r \leq -4.5 \cdot 10^{+222}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq -1.35 \cdot 10^{+175}:\\ \;\;\;\;-1.5\\ \mathbf{elif}\;r \leq -1 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq -8.6 \cdot 10^{+18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 400000:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{elif}\;r \leq 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	9.3
Cost	1480

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{w \cdot \left(r \cdot w\right)}{\frac{2.6666666666666665}{r}}\right) + -4.5\\ \mathbf{if}\;r \leq -1 \cdot 10^{+127}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 10^{+151}:\\ \;\;\;\;\left(t_0 - w \cdot \left(w \cdot \left(r \cdot \left(r \cdot 0.375\right)\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	3.0
Cost	1480

\[\begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 - \frac{t_0}{\frac{4}{r}}\right) + -4.5\\ \mathbf{if}\;v \leq -7352299.727616896:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;\left(t_1 - \frac{t_0}{\frac{2.6666666666666665}{r}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	1.9
Cost	1480

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{if}\;v \leq -7352299.727616896:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 6.302705948431219 \cdot 10^{-25}:\\ \;\;\;\;\left(t_0 - \frac{w \cdot \left(r \cdot w\right)}{\frac{2.6666666666666665}{r}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	10.8
Cost	1216

\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{w \cdot \left(r \cdot w\right)}{\frac{2.6666666666666665}{r}}\right) + -4.5 \]

Alternative 14
Error	20.0
Cost	840

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ \mathbf{if}\;r \leq -4.5 \cdot 10^{+222}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 2 \cdot 10^{+139}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	21.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;r \leq -34000:\\ \;\;\;\;-1.5\\ \mathbf{elif}\;r \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]

Alternative 16
Error	21.0
Cost	448

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

Alternative 17
Error	46.4
Cost	64

\[-1.5 \]

Reproduce

herbie shell --seed 2022294 
(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))