Rosa's TurbineBenchmark

Average Error: 12.4 → 0.3

Time: 14.5s

Precision: binary64

Cost: 7872

Math

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

↓

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

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
 (+
  (/ 2.0 (* r r))
  (- -1.5 (* (fma v -0.25 0.375) (* (* r w) (/ (* r w) (- 1.0 v)))))))

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

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)
	return Float64(Float64(2.0 / Float64(r * r)) + Float64(-1.5 - Float64(fma(v, -0.25, 0.375) * Float64(Float64(r * w) * Float64(Float64(r * w) / Float64(1.0 - v))))))
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_] := N[(N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision] + N[(-1.5 - N[(N[(v * -0.25 + 0.375), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] * N[(N[(r * w), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 12.4

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

2. Simplified 8.5

   \[\leadsto \color{blue}{\frac{2}{r \cdot r} + \left(-1.5 - \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{r}{\frac{\frac{1 - v}{r}}{w \cdot w}}\right)} \]
   Proof
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 -3/2 (*.f64 (fma.f64 v -1/4 3/8) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (Rewrite<= metadata-eval (-.f64 3 9/2)) (*.f64 (fma.f64 v -1/4 3/8) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (fma.f64 v (Rewrite<= metadata-eval (*.f64 -2 1/8)) 3/8) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (fma.f64 v (*.f64 (Rewrite<= metadata-eval (neg.f64 2)) 1/8) 3/8) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (fma.f64 v (*.f64 (neg.f64 2) 1/8) (Rewrite<= metadata-eval (*.f64 3 1/8))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 v (*.f64 (neg.f64 2) 1/8)) (*.f64 3 1/8))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 v (neg.f64 2)) 1/8)) (*.f64 3 1/8)) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (+.f64 (*.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 v 2))) 1/8) (*.f64 3 1/8)) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (+.f64 (*.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 2 v))) 1/8) (*.f64 3 1/8)) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (Rewrite<= distribute-rgt-in_binary64 (*.f64 1/8 (+.f64 (neg.f64 (*.f64 2 v)) 3))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (Rewrite<= +-commutative_binary64 (+.f64 3 (neg.f64 (*.f64 2 v))))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (Rewrite<= sub-neg_binary64 (-.f64 3 (*.f64 2 v)))) (/.f64 r (/.f64 (/.f64 (-.f64 1 v) r) (*.f64 w w)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 r (Rewrite=> associate-/l/_binary64 (/.f64 (-.f64 1 v) (*.f64 (*.f64 w w) r))))))): 8 points increase in error, 5 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 r (*.f64 (*.f64 w w) r)) (-.f64 1 v)))))): 8 points increase in error, 14 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (-.f64 1 v)) (*.f64 1/8 (-.f64 3 (*.f64 2 v))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 (*.f64 (*.f64 (*.f64 w w) r) r) (*.f64 1/8 (-.f64 3 (*.f64 2 v)))) (-.f64 1 v))))): 15 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 (-.f64 3 9/2) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r))) (-.f64 1 v)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (Rewrite<= associate--r+_binary64 (-.f64 3 (+.f64 9/2 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)))))): 1 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 2 (*.f64 r r)) (-.f64 3 (Rewrite<= +-commutative_binary64 (+.f64 (/.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
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 2 (*.f64 r r)) 3) (+.f64 (/.f64 (*.f64 (*.f64 1/8 (-.f64 3 (*.f64 2 v))) (*.f64 (*.f64 (*.f64 w w) r) r)) (-.f64 1 v)) 9/2))): 2 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 3 (/.f64 2 (*.f64 r r)))) (+.f64 (/.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
   (Rewrite<= associate--l-_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 17.1

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

4. Simplified 0.3

   \[\leadsto \frac{2}{r \cdot r} + \left(-1.5 - \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \color{blue}{\left(\left(w \cdot r\right) \cdot \frac{w \cdot r}{1 - v}\right)}\right) \]
   Proof
   (*.f64 (*.f64 w r) (/.f64 (*.f64 w r) (-.f64 1 v))): 0 points increase in error, 0 points decrease in error
   (*.f64 (*.f64 w r) (Rewrite<= associate-*r/_binary64 (*.f64 w (/.f64 r (-.f64 1 v))))): 4 points increase in error, 19 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 w r) w) (/.f64 r (-.f64 1 v)))): 40 points increase in error, 31 points decrease in error
   (*.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 r w)) w) (/.f64 r (-.f64 1 v))): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= associate-*r*_binary64 (*.f64 r (*.f64 w w))) (/.f64 r (-.f64 1 v))): 36 points increase in error, 34 points decrease in error
   (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (*.f64 r (*.f64 w w)) r) (-.f64 1 v))): 5 points increase in error, 8 points decrease in error
   (/.f64 (*.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 w w) r)) r) (-.f64 1 v)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite=> associate-*l*_binary64 (*.f64 (*.f64 w w) (*.f64 r r))) (-.f64 1 v)): 66 points increase in error, 18 points decrease in error
   (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 w 2)) (*.f64 r r)) (-.f64 1 v)): 0 points increase in error, 0 points decrease in error
   (/.f64 (*.f64 (pow.f64 w 2) (Rewrite<= unpow2_binary64 (pow.f64 r 2))) (-.f64 1 v)): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.3

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

Alternatives

Alternative 1
Error	22.0
Cost	2136

\[\begin{array}{l} t_0 := \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ t_1 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;w \cdot w \leq 1.3 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \cdot w \leq 5.8 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \cdot w \leq 3.8 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \cdot w \leq 6.8 \cdot 10^{+222}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \cdot w \leq 6 \cdot 10^{+268}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;w \cdot w \leq 4.5 \cdot 10^{+300}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \]

Alternative 2
Error	0.4
Cost	1984

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

Alternative 3
Error	15.1
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ t_2 := t_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -3.5 \cdot 10^{-37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.1
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -3.5 \cdot 10^{-37}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot r\right) \cdot \left(w \cdot \left(0.375 \cdot w\right)\right)\right)\\ \mathbf{elif}\;r \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.1
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -3.6 \cdot 10^{-37}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot r\right) \cdot \left(w \cdot \left(0.375 \cdot w\right)\right)\right)\\ \mathbf{elif}\;r \leq 2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(w \cdot \left(-0.25 \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.1
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + -1.5\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -5.8 \cdot 10^{-37}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{elif}\;r \leq 2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot r\right) \cdot \left(w \cdot \left(-0.25 \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	9.4
Cost	1352

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := t_0 + \left(-1.5 - \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot 0.25\right)\\ \mathbf{if}\;v \leq -7 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq -1.95 \cdot 10^{-126}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot r\right) \cdot \left(w \cdot \left(0.375 \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	1.6
Cost	1352

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \left(r \cdot w\right) \cdot \left(r \cdot w\right)\\ t_2 := t_0 + \left(-1.5 - t_1 \cdot 0.25\right)\\ \mathbf{if}\;v \leq -3.6 \cdot 10^{+37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 1.42 \cdot 10^{-33}:\\ \;\;\;\;t_0 + \left(-1.5 + t_1 \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	21.7
Cost	840

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + -1.5\\ \mathbf{if}\;w \leq 26000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	20.9
Cost	448

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

Alternative 11
Error	37.9
Cost	320

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

Alternative 12
Error	37.9
Cost	320

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

Reproduce

herbie shell --seed 2022330 
(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))