Rosa's TurbineBenchmark

Average Error: 20.35% → 0.26%

Time: 17.8s

Precision: binary64

Cost: 14208

Math

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

↓

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

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

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((r ^ -2.0) * 2.0) + Float64(-1.5 - Float64(Float64(w / Float64(Float64(Float64(1.0 - v) / fma(v, -0.25, 0.375)) / r)) * Float64(r * w))))
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[(N[Power[r, -2.0], $MachinePrecision] * 2.0), $MachinePrecision] + N[(-1.5 - N[(N[(w / N[(N[(N[(1.0 - v), $MachinePrecision] / N[(v * -0.25 + 0.375), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision] * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.35

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

2. Simplified 0.41

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

   [Start] 20.35	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   sub-neg [=>] 20.35	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} - 4.5 \]
   +-commutative [=>] 20.35	\[ \color{blue}{\left(\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} - 4.5 \]
   associate--l+ [=>] 20.34	\[ \color{blue}{\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right)} \]
   +-commutative [=>] 20.34	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} \]
   sub-neg [=>] 20.34	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   +-commutative [=>] 20.34	\[ \color{blue}{\left(\left(-4.5\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   associate-+r+ [=>] 20.33	\[ \color{blue}{\left(\left(\left(-4.5\right) + 3\right) + \frac{2}{r \cdot r}\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   +-commutative [<=] 20.33	\[ \color{blue}{\left(\frac{2}{r \cdot r} + \left(\left(-4.5\right) + 3\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   associate-+r+ [<=] 20.33	\[ \color{blue}{\frac{2}{r \cdot r} + \left(\left(\left(-4.5\right) + 3\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} \]

3. Applied egg-rr 0.26

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

4. Final simplification 0.26

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

Alternatives

Alternative 1
Error	0.74%
Cost	8137

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq -4.8 \cdot 10^{+155} \lor \neg \left(r \leq 2 \cdot 10^{+66}\right):\\ \;\;\;\;\left(\left(t_0 + 3\right) + \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{-0.375 + 0.125 \cdot \left(2 \cdot v\right)}{\frac{1 - v}{r}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - w \cdot \left(r \cdot \left(r \cdot \left(\mathsf{fma}\left(-0.25, v, 0.375\right) \cdot \frac{w}{1 - v}\right)\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.41%
Cost	7872

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

Alternative 3
Error	29.5%
Cost	2260

\[\begin{array}{l} t_0 := \left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ t_1 := \frac{2}{r \cdot r}\\ t_2 := t_1 + -0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right)\\ \mathbf{if}\;w \cdot w \leq 4.5 \cdot 10^{-14}:\\ \;\;\;\;-1.5 + t_1\\ \mathbf{elif}\;w \cdot w \leq 750000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \cdot w \leq 5.2 \cdot 10^{+36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;w \cdot w \leq 1.75 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;w \cdot w \leq 6.6 \cdot 10^{+298}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	1.92%
Cost	1988

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 4.8 \cdot 10^{-278}:\\ \;\;\;\;\left(\left(t_0 + 3\right) + \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{-0.375 + 0.125 \cdot \left(2 \cdot v\right)}{\frac{1 - v}{r}}\right) + -4.5\\ \mathbf{elif}\;v \leq 10^{+60}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \end{array} \]

Alternative 5
Error	0.93%
Cost	1737

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -180000000 \lor \neg \left(v \leq 0.01\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{2.6666666666666665}{r} + \frac{v \cdot -0.8888888888888888}{r}}\right)\\ \end{array} \]

Alternative 6
Error	0.82%
Cost	1737

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -180000000 \lor \neg \left(v \leq 0.01\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r} + \frac{2}{r \cdot v}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{2.6666666666666665}{r} + \frac{v \cdot -0.8888888888888888}{r}}\right)\\ \end{array} \]

Alternative 7
Error	3.17%
Cost	1732

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 2.5 \cdot 10^{+60}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(0.375 + v \cdot -0.25\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \end{array} \]

Alternative 8
Error	6.25%
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -180000000 \lor \neg \left(v \leq 1.1 \cdot 10^{-31}\right):\\ \;\;\;\;t_0 + \left(-1.5 - w \cdot \left(r \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.375\right)\\ \end{array} \]

Alternative 9
Error	4.23%
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -180000000 \lor \neg \left(v \leq 0.01\right):\\ \;\;\;\;t_0 + \left(-1.5 - w \cdot \left(r \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	1.11%
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -180000000 \lor \neg \left(v \leq 0.01\right):\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot -0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	1.09%
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -180000000 \lor \neg \left(v \leq 0.01\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	14.94%
Cost	1348

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 2 \cdot 10^{+236}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.375\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - w \cdot \left(r \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	34.3%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;w \cdot w \leq 6.5 \cdot 10^{-16} \lor \neg \left(w \cdot w \leq 4400000000\right):\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;\left(r \cdot r\right) \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\\ \end{array} \]

Alternative 14
Error	17.03%
Cost	1088

\[\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot -0.375\right) \]

Alternative 15
Error	33.68%
Cost	448

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

Alternative 16
Error	60.46%
Cost	320

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

Alternative 17
Error	60.46%
Cost	320

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

Reproduce

herbie shell --seed 2023115 
(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))