Rosa's TurbineBenchmark

Average Error: 12.8 → 0.3

Time: 15.7s

Precision: binary64

Cost: 7872

Math

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

↓

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

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
 (+
  (/ 2.0 (* r r))
  (- -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 (2.0 / (r * r)) + (-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(2.0 / Float64(r * r)) + 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[(2.0 / N[(r * r), $MachinePrecision]), $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 12.8

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

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] 12.8	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   sub-neg [=>] 12.8	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} - 4.5 \]
   +-commutative [=>] 12.8	\[ \color{blue}{\left(\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} - 4.5 \]
   associate--l+ [=>] 12.8	\[ \color{blue}{\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right)} \]
   +-commutative [=>] 12.8	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)} \]
   sub-neg [=>] 12.8	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   +-commutative [=>] 12.8	\[ \color{blue}{\left(\left(-4.5\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   associate-+r+ [=>] 12.8	\[ \color{blue}{\left(\left(\left(-4.5\right) + 3\right) + \frac{2}{r \cdot r}\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   +-commutative [<=] 12.8	\[ \color{blue}{\left(\frac{2}{r \cdot r} + \left(\left(-4.5\right) + 3\right)\right)} + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) \]
   associate-+r+ [<=] 12.8	\[ \color{blue}{\frac{2}{r \cdot r} + \left(\left(\left(-4.5\right) + 3\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} \]

3. Final simplification 0.3

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

Alternatives

Alternative 1
Error	2.0
Cost	7556

\[\begin{array}{l} \mathbf{if}\;v \leq 540000:\\ \;\;\;\;\frac{2}{r \cdot r} + \left(-1.5 + \left(r \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(-0.375 + v \cdot 0.25\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {r}^{-2} + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \end{array} \]

Alternative 2
Error	2.4
Cost	2376

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \cdot w \leq 0:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(-0.375 + v \cdot 0.25\right)}{1 - v}\right)\\ \mathbf{elif}\;w \cdot w \leq 2 \cdot 10^{+256}:\\ \;\;\;\;\left(\left(t_0 + 3\right) + \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right) \cdot \frac{0.125 \cdot \left(2 \cdot v + -3\right)}{1 - v}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r} + \frac{2}{r \cdot v}}\right)\\ \end{array} \]

Alternative 3
Error	0.5
Cost	1736

\[\begin{array}{l} t_0 := 4 + \frac{2}{v}\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -15.8:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r}{\frac{t_0}{w}}\right)\\ \mathbf{elif}\;v \leq 0.0175:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{2.6666666666666665}{r} + \frac{v \cdot -0.8888888888888888}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{t_0}{r}}\right)\\ \end{array} \]

Alternative 4
Error	2.0
Cost	1732

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq 540000:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \frac{\left(r \cdot w\right) \cdot \left(-0.375 + v \cdot 0.25\right)}{1 - v}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	1609

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -15.8 \lor \neg \left(v \leq 0.0175\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r}{\frac{4 + \frac{2}{v}}{w}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	0.6
Cost	1608

\[\begin{array}{l} t_0 := 4 + \frac{2}{v}\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -15.8:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r}{\frac{t_0}{w}}\right)\\ \mathbf{elif}\;v \leq 0.0175:\\ \;\;\;\;t_1 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{t_0}{r}}\right)\\ \end{array} \]

Alternative 7
Error	0.5
Cost	1608

\[\begin{array}{l} t_0 := 4 + \frac{2}{v}\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -15.8:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r}{\frac{t_0}{w}}\right)\\ \mathbf{elif}\;v \leq 0.0175:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{r \cdot w}{2.6666666666666665 + v \cdot -0.8888888888888888}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{t_0}{r}}\right)\\ \end{array} \]

Alternative 8
Error	0.7
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -15.8 \lor \neg \left(v \leq 0.0175\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.25\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	0.7
Cost	1353

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -15.8 \lor \neg \left(v \leq 0.0175\right):\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot w\right) \cdot \frac{w}{\frac{4}{r}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 + \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	10.4
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 11
Error	8.8
Cost	1088

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

Alternative 12
Error	8.8
Cost	1088

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

Reproduce

herbie shell --seed 2023039 
(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))