Rosa's TurbineBenchmark

Average Error: 12.5 → 0.2

Time: 18.4s

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 12.5

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

2. Simplified 0.3

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

   [Start] 12.5	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   sub-neg [=>] 12.5	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} - 4.5 \]
   +-commutative [=>] 12.5	\[ \color{blue}{\left(\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} - 4.5 \]
   associate--l+ [=>] 12.5	\[ \color{blue}{\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right)} \]
   +-commutative [=>] 12.5	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} \]
   sub-neg [=>] 12.5	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   +-commutative [=>] 12.5	\[ \color{blue}{\left(\left(-4.5\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   associate-+r+ [=>] 12.4	\[ \color{blue}{\left(\left(\left(-4.5\right) + 3\right) + \frac{2}{r \cdot r}\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   +-commutative [<=] 12.4	\[ \color{blue}{\left(\frac{2}{r \cdot r} + \left(\left(-4.5\right) + 3\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   associate-+r+ [<=] 12.4	\[ \color{blue}{\frac{2}{r \cdot r} + \left(\left(\left(-4.5\right) + 3\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} \]

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.3
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 2
Error	11.0
Cost	2130

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 10^{-160} \lor \neg \left(w \cdot w \leq 2 \cdot 10^{-83}\right) \land \left(w \cdot w \leq 2 \cdot 10^{+62} \lor \neg \left(w \cdot w \leq 10^{+242}\right)\right):\\ \;\;\;\;t_0 + \left(-1.5 + r \cdot \left(-0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\\ \end{array} \]

Alternative 3
Error	0.3
Cost	2121

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq -3.8 \cdot 10^{+29} \lor \neg \left(r \leq 5 \cdot 10^{+40}\right):\\ \;\;\;\;\left(\left(t_0 + 3\right) + \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot \frac{\left(3 + -2 \cdot v\right) \cdot -0.125}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - w \cdot \left(\left(r \cdot w\right) \cdot \frac{r}{\frac{1 - v}{0.375 + v \cdot -0.25}}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.3
Cost	1864

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -4 \cdot 10^{+49}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{elif}\;v \leq 2 \cdot 10^{+23}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{1 - v}{r \cdot \left(0.375 + v \cdot -0.25\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	1.0
Cost	1737

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6 \lor \neg \left(v \leq 8 \cdot 10^{-34}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r} + \frac{2}{r \cdot v}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{2.6666666666666665}{r} + -0.8888888888888888 \cdot \frac{v}{r}}\right)\\ \end{array} \]

Alternative 6
Error	1.0
Cost	1736

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-34}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{2.6666666666666665}{r} + -0.8888888888888888 \cdot \frac{v}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	15.2
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := -1.5 + t_0\\ t_2 := t_0 + \left(-1.5 + \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -2 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 10^{-80}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 3.1 \cdot 10^{+149}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	1.0
Cost	1608

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-34}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot \left(-0.375 + v \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	2.1
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6.5 \lor \neg \left(v \leq 8 \cdot 10^{-34}\right):\\ \;\;\;\;t_0 + \left(-1.5 + r \cdot \left(-0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \end{array} \]

Alternative 10
Error	1.1
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6 \lor \neg \left(v \leq 8 \cdot 10^{-34}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right)\\ \end{array} \]

Alternative 11
Error	1.1
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6 \lor \neg \left(v \leq 8 \cdot 10^{-34}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	1.1
Cost	1352

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -6:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{elif}\;v \leq 8 \cdot 10^{-34}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	21.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;w \cdot w \leq 2.5 \cdot 10^{+218}:\\ \;\;\;\;-1.5 + \frac{2}{r \cdot r}\\ \mathbf{elif}\;w \cdot w \leq 4.9 \cdot 10^{+302}:\\ \;\;\;\;\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot -0.375\\ \mathbf{else}:\\ \;\;\;\;-1.5 + \frac{\frac{2}{r}}{r}\\ \end{array} \]

Alternative 14
Error	20.7
Cost	448

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

Alternative 15
Error	37.8
Cost	320

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

Reproduce

herbie shell --seed 2023187 
(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))