Rosa's TurbineBenchmark

Average Accuracy: 38.1% → 46.5%

Time: 14.0s

Precision: binary64

Cost: 85248

Math

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

↓

\[\begin{array}{l} t_0 := \sqrt[3]{\mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)\right)}\\ t_0 \cdot {t_0}^{2} \end{array} \]

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
         (cbrt
          (fma
           (pow (* r w) 2.0)
           (/ (fma v 0.25 -0.375) (- 1.0 v))
           (fma 2.0 (pow r -2.0) -1.5)))))
   (* t_0 (pow t_0 2.0))))

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 = cbrt(fma(pow((r * w), 2.0), (fma(v, 0.25, -0.375) / (1.0 - v)), fma(2.0, pow(r, -2.0), -1.5)));
	return t_0 * pow(t_0, 2.0);
}

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 = cbrt(fma((Float64(r * w) ^ 2.0), Float64(fma(v, 0.25, -0.375) / Float64(1.0 - v)), fma(2.0, (r ^ -2.0), -1.5)))
	return Float64(t_0 * (t_0 ^ 2.0))
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[Power[N[(N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(v * 0.25 + -0.375), $MachinePrecision] / N[(1.0 - v), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[Power[r, -2.0], $MachinePrecision] + -1.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 39.0%

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

2. Simplified 35.0%

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

   [Start] 39.0	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   sub-neg [=>] 39.0	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} - 4.5 \]
   +-commutative [=>] 39.0	\[ \color{blue}{\left(\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} - 4.5 \]
   associate--l+ [=>] 39.0	\[ \color{blue}{\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right)} \]
   associate-/l* [=>] 41.4	\[ \left(-\color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}}\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) \]
   distribute-neg-frac [=>] 41.4	\[ \color{blue}{\frac{-0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}} + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) \]
   associate-/r/ [=>] 41.4	\[ \color{blue}{\frac{-0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)} + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) \]
   fma-def [=>] 41.4	\[ \color{blue}{\mathsf{fma}\left(\frac{-0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, \left(3 + \frac{2}{r \cdot r}\right) - 4.5\right)} \]
   sub-neg [=>] 41.4	\[ \mathsf{fma}\left(\frac{-0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5\right)}\right) \]

3. Applied egg-rr 46.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)\right)}} \]
   Step-by-step derivation

   [Start] 35.0

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

   add-cube-cbrt [=>] 34.4

   \[ \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right)}} \]

4. Final simplification 46.6%

   \[\leadsto \sqrt[3]{\mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)\right)} \cdot {\left(\sqrt[3]{\mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \mathsf{fma}\left(2, {r}^{-2}, -1.5\right)\right)}\right)}^{2} \]

Alternatives

Alternative 1
Accuracy	43.5%
Cost	1865

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -10200000000 \lor \neg \left(v \leq 5 \cdot 10^{-40}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(-1.5 - \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right) \cdot 0.375\right)\\ \end{array} \]

Alternative 2
Accuracy	45.6%
Cost	1600

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

Alternative 3
Accuracy	41.0%
Cost	1088

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

Alternative 4
Accuracy	32.8%
Cost	448

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

Alternative 5
Accuracy	20.0%
Cost	320

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

Reproduce

herbie shell --seed 2023157 
(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))