Rosa's TurbineBenchmark

Average Error: 12.9 → 0.3

Time: 13.5s

Precision: binary64

Cost: 8004

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(\left(3 + t_0\right) + \frac{0.125 \cdot \left(-3 + 2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_0 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \end{array} \]

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) 2e+266)
     (+
      (+
       (+ 3.0 t_0)
       (/ (* 0.125 (+ -3.0 (* 2.0 v))) (/ (- 1.0 v) (* r (* w (* w r))))))
      -4.5)
     (- t_0 (* (/ (fma v -0.25 0.375) (- 1.0 v)) (* w (* w (* r r))))))))

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) <= 2e+266) {
		tmp = ((3.0 + t_0) + ((0.125 * (-3.0 + (2.0 * v))) / ((1.0 - v) / (r * (w * (w * r)))))) + -4.5;
	} else {
		tmp = t_0 - ((fma(v, -0.25, 0.375) / (1.0 - v)) * (w * (w * (r * r))));
	}
	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) <= 2e+266)
		tmp = Float64(Float64(Float64(3.0 + t_0) + Float64(Float64(0.125 * Float64(-3.0 + Float64(2.0 * v))) / Float64(Float64(1.0 - v) / Float64(r * Float64(w * Float64(w * r)))))) + -4.5);
	else
		tmp = Float64(t_0 - Float64(Float64(fma(v, -0.25, 0.375) / Float64(1.0 - v)) * Float64(w * Float64(w * Float64(r * r)))));
	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], 2e+266], N[(N[(N[(3.0 + t$95$0), $MachinePrecision] + N[(N[(0.125 * N[(-3.0 + N[(2.0 * v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(1.0 - v), $MachinePrecision] / N[(r * N[(w * N[(w * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision], N[(t$95$0 - N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] * N[(w * N[(w * N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 w w) < 2.0000000000000001e266

   1. Initial program 9.2

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

   2. Simplified 4.6

      \[\leadsto \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(r \cdot \left(w \cdot w\right)\right)}}\right) + -4.5} \]
      Proof
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (+.f64 3 (*.f64 -2 v))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (+.f64 3 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) v))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 3 (*.f64 2 v)))) (/.f64 (-.f64 1 v) (*.f64 r (*.f64 r (*.f64 w w)))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (-.f64 1 v) (*.f64 r (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 w w) r)))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (/.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (-.f64 1 v) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 w w) r) r))))) -9/2): 0 points increase in error, 0 points decrease in error
      (+.f64 (-.f64 (+.f64 3 (/.f64 2 (*.f64 r r))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))) -9/2): 17 points increase in error, 10 points decrease in error
      (+.f64 (-.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))) (Rewrite<= metadata-eval (neg.f64 9/2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= sub-neg_binary64 (-.f64 (-.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. Taylor expanded in r around 0 4.6

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

   4. Simplified 0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125 \cdot \left(3 + -2 \cdot v\right)}{\frac{1 - v}{r \cdot \color{blue}{\left(w \cdot \left(w \cdot r\right)\right)}}}\right) + -4.5 \]
      Proof
      (*.f64 w (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w w) r)): 57 points increase in error, 41 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) r): 1 points increase in error, 0 points decrease in error

   if 2.0000000000000001e266 < (*.f64 w w)

   1. Initial program 51.9

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

   2. Simplified 7.8

      \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \mathsf{fma}\left(\mathsf{fma}\left(v, -0.25, 0.375\right), \frac{w}{\frac{\frac{1 - v}{r}}{r}} \cdot w, 1.5\right)} \]
      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)): 0 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)): 7 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))) (/.f64 w (Rewrite=> associate-/l/_binary64 (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w r)))) 3/2)): 6 points increase in error, 39 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)): 16 points increase in error, 12 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)): 6 points increase in error, 7 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)): 20 points increase in error, 21 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)): 36 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))) (/.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)))): 3 points increase in error, 2 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))): 14 points increase in error, 1 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)): 1 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. Taylor expanded in w around inf 51.9

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

   4. Simplified 0.5

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)} \]
      Proof
      (*.f64 (/.f64 (fma.f64 v -1/4 3/8) (-.f64 1 v)) (*.f64 w (*.f64 w (*.f64 r r)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (fma.f64 v -1/4 3/8) (-.f64 1 v)) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 w w) (*.f64 r r)))): 40 points increase in error, 32 points decrease in error
      (*.f64 (/.f64 (fma.f64 v -1/4 3/8) (-.f64 1 v)) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) (*.f64 r r))): 1 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 (fma.f64 v -1/4 3/8) (-.f64 1 v)) (*.f64 (pow.f64 w 2) (Rewrite<= unpow2_binary64 (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 (fma.f64 v -1/4 3/8) (/.f64 (-.f64 1 v) (*.f64 (pow.f64 w 2) (pow.f64 r 2))))): 20 points increase in error, 13 points decrease in error
      (/.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 v -1/4) 3/8)) (/.f64 (-.f64 1 v) (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/4 v)) 3/8) (/.f64 (-.f64 1 v) (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 (*.f64 -1/4 v) 3/8) (*.f64 (pow.f64 w 2) (pow.f64 r 2))) (-.f64 1 v))): 19 points increase in error, 20 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+266}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{0.125 \cdot \left(-3 + 2 \cdot v\right)}{\frac{1 - v}{r \cdot \left(w \cdot \left(w \cdot r\right)\right)}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(w \cdot \left(w \cdot \left(r \cdot r\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	15.5
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ t_2 := t_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{if}\;r \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -8.2 \cdot 10^{-47}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 10^{-69}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 3 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	15.5
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ t_2 := t_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)\right)\\ \mathbf{if}\;r \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -9.8 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 3 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 3 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	1.2
Cost	1608

\[\begin{array}{l} t_0 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := t_1 + \left(-1.5 + -0.25 \cdot t_0\right)\\ \mathbf{if}\;v \leq -3.6 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 6.2 \cdot 10^{-14}:\\ \;\;\;\;t_1 + \left(-1.5 + t_0 \cdot \left(-0.375 + v \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	17.3
Cost	1488

\[\begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := t_1 + \left(\left(w \cdot w\right) \cdot \left(r \cdot r\right)\right) \cdot -0.375\\ \mathbf{if}\;w \leq -3.5 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq -3.2 \cdot 10^{-99}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;w \leq 9 \cdot 10^{-7}:\\ \;\;\;\;t_1 + -1.5\\ \mathbf{elif}\;w \leq 6 \cdot 10^{+146}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	12.8
Cost	1352

\[\begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ \mathbf{if}\;w \leq -3.5 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq 6 \cdot 10^{+146}:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	5.3
Cost	1352

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + \left(-1.5 + -0.25 \cdot \left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right)\right)\\ \mathbf{if}\;v \leq -3.6 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 1.7 \cdot 10^{-17}:\\ \;\;\;\;t_0 + \left(-1.5 + r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	1.2
Cost	1352

\[\begin{array}{l} t_0 := \left(w \cdot r\right) \cdot \left(w \cdot r\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := t_1 + \left(-1.5 + -0.25 \cdot t_0\right)\\ \mathbf{if}\;v \leq -3.6 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 6.2 \cdot 10^{-14}:\\ \;\;\;\;t_1 + \left(-1.5 - 0.375 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	20.9
Cost	448

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

Alternative 9
Error	38.3
Cost	320

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

Alternative 10
Error	38.3
Cost	320

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

Reproduce

herbie shell --seed 2022332 
(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))