Rosa's TurbineBenchmark

Average Error: 12.8 → 1.2

Time: 17.4s

Precision: binary64

Cost: 8264

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;v \leq -6.812155230162503 \cdot 10^{+45}:\\ \;\;\;\;\left(t_0 + \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{\mathsf{fma}\left(v, -2, 3\right) \cdot -0.125}{1 - v}\right) + -4.5\\ \mathbf{elif}\;v \leq 6.484521128459391 \cdot 10^{+136}:\\ \;\;\;\;\left(t_0 - \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \end{array} \]

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)) 3.0)))
   (if (<= v -6.812155230162503e+45)
     (+
      (+ t_0 (* (* w (* r (* r w))) (/ (* (fma v -2.0 3.0) -0.125) (- 1.0 v))))
      -4.5)
     (if (<= v 6.484521128459391e+136)
       (+
        (- t_0 (* (* (fma v -0.25 0.375) (* r w)) (/ (* r w) (- 1.0 v))))
        -4.5)
       (+ (+ t_0 (* -0.25 (* (* r w) (* r w)))) -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)) + 3.0;
	double tmp;
	if (v <= -6.812155230162503e+45) {
		tmp = (t_0 + ((w * (r * (r * w))) * ((fma(v, -2.0, 3.0) * -0.125) / (1.0 - v)))) + -4.5;
	} else if (v <= 6.484521128459391e+136) {
		tmp = (t_0 - ((fma(v, -0.25, 0.375) * (r * w)) * ((r * w) / (1.0 - v)))) + -4.5;
	} else {
		tmp = (t_0 + (-0.25 * ((r * w) * (r * w)))) + -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(Float64(2.0 / Float64(r * r)) + 3.0)
	tmp = 0.0
	if (v <= -6.812155230162503e+45)
		tmp = Float64(Float64(t_0 + Float64(Float64(w * Float64(r * Float64(r * w))) * Float64(Float64(fma(v, -2.0, 3.0) * -0.125) / Float64(1.0 - v)))) + -4.5);
	elseif (v <= 6.484521128459391e+136)
		tmp = Float64(Float64(t_0 - Float64(Float64(fma(v, -0.25, 0.375) * Float64(r * w)) * Float64(Float64(r * w) / Float64(1.0 - v)))) + -4.5);
	else
		tmp = Float64(Float64(t_0 + Float64(-0.25 * Float64(Float64(r * w) * Float64(r * w)))) + -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[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + 3.0), $MachinePrecision]}, If[LessEqual[v, -6.812155230162503e+45], N[(N[(t$95$0 + N[(N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(v * -2.0 + 3.0), $MachinePrecision] * -0.125), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision], If[LessEqual[v, 6.484521128459391e+136], N[(N[(t$95$0 - N[(N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision], N[(N[(t$95$0 + N[(-0.25 * N[(N[(r * w), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if v < -6.8121552301625028e45

   1. Initial program 18.0

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

   2. Applied egg-rr 0.3

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

   3. Applied egg-rr 4.5

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

   if -6.8121552301625028e45 < v < 6.4845211284593912e136

   1. Initial program 9.5

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

   2. Taylor expanded in v around 0 17.8

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{-0.25 \cdot \left(v \cdot \left({w}^{2} \cdot {r}^{2}\right)\right) + 0.375 \cdot \left({w}^{2} \cdot {r}^{2}\right)}}{1 - v}\right) - 4.5 \]

   3. Simplified 1.4

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\color{blue}{\left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(w \cdot r\right)\right) \cdot \left(w \cdot r\right)}}{1 - v}\right) - 4.5 \]
      Proof
      (*.f64 (*.f64 (fma.f64 v -1/4 3/8) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 v -1/4) 3/8)) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1/4 v)) 3/8) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (+.f64 (*.f64 (Rewrite<= metadata-eval (*.f64 1/8 -2)) v) 3/8) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (+.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/8 (*.f64 -2 v))) 3/8) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (+.f64 (*.f64 1/8 (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) v)) 3/8) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (+.f64 (*.f64 1/8 (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 2 v)))) 3/8) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (+.f64 (*.f64 1/8 (neg.f64 (*.f64 2 v))) (Rewrite<= metadata-eval (*.f64 1/8 3))) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 1/8 (+.f64 (neg.f64 (*.f64 2 v)) 3))) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 1/8 (Rewrite<= +-commutative_binary64 (+.f64 3 (neg.f64 (*.f64 2 v))))) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (*.f64 1/8 (Rewrite<= sub-neg_binary64 (-.f64 3 (*.f64 2 v)))) (*.f64 w r)) (*.f64 w r)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r*_binary64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 w r) (*.f64 w r)))): 33 points increase in error, 46 points decrease in error
      (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 w w) (*.f64 r r)))): 72 points increase in error, 43 points decrease in error
      (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) (*.f64 r r))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (pow.f64 w 2) (Rewrite<= unpow2_binary64 (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (pow.f64 w 2) (pow.f64 r 2)) (*.f64 1/8 (-.f64 3 (*.f64 2 v))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 w 2) (pow.f64 r 2)) (*.f64 1/8 (Rewrite=> sub-neg_binary64 (+.f64 3 (neg.f64 (*.f64 2 v)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 w 2) (pow.f64 r 2)) (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 3 1/8) (*.f64 (neg.f64 (*.f64 2 v)) 1/8)))): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 w 2) (pow.f64 r 2)) (+.f64 (Rewrite=> metadata-eval 3/8) (*.f64 (neg.f64 (*.f64 2 v)) 1/8))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> distribute-rgt-in_binary64 (+.f64 (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) (*.f64 (*.f64 (neg.f64 (*.f64 2 v)) 1/8) (*.f64 (pow.f64 w 2) (pow.f64 r 2))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 1/8 (neg.f64 (*.f64 2 v)))) (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) (*.f64 (*.f64 1/8 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 2) v))) (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) (*.f64 (*.f64 1/8 (*.f64 (Rewrite=> metadata-eval -2) v)) (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) (*.f64 (Rewrite=> associate-*r*_binary64 (*.f64 (*.f64 1/8 -2) v)) (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) (*.f64 (*.f64 (Rewrite=> metadata-eval -1/4) v) (*.f64 (pow.f64 w 2) (pow.f64 r 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))) (Rewrite<= associate-*r*_binary64 (*.f64 -1/4 (*.f64 v (*.f64 (pow.f64 w 2) (pow.f64 r 2)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1/4 (*.f64 v (*.f64 (pow.f64 w 2) (pow.f64 r 2)))) (*.f64 3/8 (*.f64 (pow.f64 w 2) (pow.f64 r 2))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(w \cdot r\right)}{1} \cdot \frac{w \cdot r}{1 - v}}\right) - 4.5 \]

   if 6.4845211284593912e136 < v

   1. Initial program 20.0

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

   2. Applied egg-rr 0.5

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

   3. Taylor expanded in v around inf 18.0

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

   4. Simplified 0.3

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(\left(w \cdot r\right) \cdot \left(w \cdot r\right)\right) \cdot 0.25}\right) - 4.5 \]
      Proof
      (*.f64 (*.f64 (*.f64 w r) (*.f64 w r)) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite<= unswap-sqr_binary64 (*.f64 (*.f64 w w) (*.f64 r r))) 1/4): 90 points increase in error, 51 points decrease in error
      (*.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) (*.f64 r r)) 1/4): 0 points increase in error, 0 points decrease in error
      (*.f64 (*.f64 (pow.f64 w 2) (Rewrite<= unpow2_binary64 (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
3. Recombined 3 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;v \leq -6.812155230162503 \cdot 10^{+45}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + \left(w \cdot \left(r \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{\mathsf{fma}\left(v, -2, 3\right) \cdot -0.125}{1 - v}\right) + -4.5\\ \mathbf{elif}\;v \leq 6.484521128459391 \cdot 10^{+136}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) - \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{2}{r \cdot r} + 3\right) + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	14144

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

Alternative 2
Error	0.3
Cost	8264

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ t_1 := \left(t_0 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{if}\;v \leq -6.812155230162503 \cdot 10^{+45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 6.484521128459391 \cdot 10^{+136}:\\ \;\;\;\;\left(t_0 - \left(\mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \left(r \cdot w\right)\right) \cdot \frac{r \cdot w}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	0.4
Cost	1992

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ t_1 := \left(t_0 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{if}\;v \leq -2.941038584983021 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 7.600128050144566 \cdot 10^{+38}:\\ \;\;\;\;\left(t_0 + \frac{\left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot \left(-0.375 - v \cdot -0.25\right)\right)\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.2
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ t_2 := t_0 + \left(-1.5 - \left(w \cdot w\right) \cdot \left(r \cdot \left(r \cdot 0.375\right)\right)\right)\\ \mathbf{if}\;r \leq -4.2 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -3.2967741832024085 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 2.554604916287782 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 9.8 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.8
Cost	1480

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ \mathbf{if}\;r \leq -4.2 \cdot 10^{+147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 9.8 \cdot 10^{+153}:\\ \;\;\;\;\left(\left(t_0 + 3\right) + \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right) \cdot -0.375\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	14.1
Cost	1480

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + 3\\ \mathbf{if}\;w \leq -1.4539584604098923 \cdot 10^{-97}:\\ \;\;\;\;\left(t_1 + w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right)\right)\right) + -4.5\\ \mathbf{elif}\;w \leq 1.891281458564282 \cdot 10^{-99}:\\ \;\;\;\;t_0 + -1.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right) \cdot -0.375\right) + -4.5\\ \end{array} \]

Alternative 7
Error	11.7
Cost	1480

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;w \leq -1 \cdot 10^{+50}:\\ \;\;\;\;\left(t_0 + w \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot -0.375\right)\right)\right) + -4.5\\ \mathbf{elif}\;w \leq 10^{-75}:\\ \;\;\;\;\left(t_0 + r \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_0 + \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right) \cdot -0.375\right) + -4.5\\ \end{array} \]

Alternative 8
Error	5.3
Cost	1480

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ t_1 := \left(t_0 + -0.25 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{if}\;v \leq -107.72499127418773:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 9.000517397336851 \cdot 10^{-39}:\\ \;\;\;\;\left(t_0 + r \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	1.3
Cost	1480

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r} + 3\\ t_2 := \left(t_1 + -0.25 \cdot t_0\right) + -4.5\\ \mathbf{if}\;v \leq -107.72499127418773:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 9.000517397336851 \cdot 10^{-39}:\\ \;\;\;\;\left(t_1 - 0.375 \cdot t_0\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	20.5
Cost	448

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

Alternative 11
Error	37.9
Cost	320

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

Reproduce

herbie shell --seed 2022317 
(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))