Rosa's TurbineBenchmark

Average Accuracy: 80.6% → 99.6%

Time: 18.3s

Precision: binary64

Cost: 26688

Math

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

↓

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

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

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

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

Derivation

1. Initial program 80.6%

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

2. Simplified 73.5%

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

   [Start] 80.6	\[ \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5 \]
   sub-neg [=>] 80.6	\[ \color{blue}{\left(\left(3 + \frac{2}{r \cdot r}\right) + \left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right)\right)} - 4.5 \]
   +-commutative [=>] 80.6	\[ \color{blue}{\left(\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(3 + \frac{2}{r \cdot r}\right)\right)} - 4.5 \]
   associate--l+ [=>] 80.6	\[ \color{blue}{\left(-\frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right)} \]
   associate-/l* [=>] 87.3	\[ \left(-\color{blue}{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}}\right) + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) \]
   distribute-neg-frac [=>] 87.3	\[ \color{blue}{\frac{-0.125 \cdot \left(3 - 2 \cdot v\right)}{\frac{1 - v}{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}} + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) \]
   associate-/r/ [=>] 87.4	\[ \color{blue}{\frac{-0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)} + \left(\left(3 + \frac{2}{r \cdot r}\right) - 4.5\right) \]
   fma-def [=>] 87.4	\[ \color{blue}{\mathsf{fma}\left(\frac{-0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, \left(3 + \frac{2}{r \cdot r}\right) - 4.5\right)} \]
   sub-neg [=>] 87.4	\[ \mathsf{fma}\left(\frac{-0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, \color{blue}{\left(3 + \frac{2}{r \cdot r}\right) + \left(-4.5\right)}\right) \]

3. Applied egg-rr 99.6%

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

   [Start] 73.5	\[ \mathsf{fma}\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{2}{r \cdot r} + -1.5\right) \]
   fma-udef [=>] 73.5	\[ \color{blue}{\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \left(\frac{2}{r \cdot r} + -1.5\right)} \]
   associate-+r+ [=>] 73.5	\[ \color{blue}{\left(\frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v} \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) + \frac{2}{r \cdot r}\right) + -1.5} \]
   *-commutative [=>] 73.5	\[ \left(\color{blue}{\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}} + \frac{2}{r \cdot r}\right) + -1.5 \]
   fma-def [=>] 73.5	\[ \color{blue}{\mathsf{fma}\left(\left(r \cdot r\right) \cdot \left(w \cdot w\right), \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \frac{2}{r \cdot r}\right)} + -1.5 \]
   unswap-sqr [=>] 99.5	\[ \mathsf{fma}\left(\color{blue}{\left(r \cdot w\right) \cdot \left(r \cdot w\right)}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \frac{2}{r \cdot r}\right) + -1.5 \]
   pow2 [=>] 99.5	\[ \mathsf{fma}\left(\color{blue}{{\left(r \cdot w\right)}^{2}}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \frac{2}{r \cdot r}\right) + -1.5 \]
   div-inv [=>] 99.5	\[ \mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, \color{blue}{2 \cdot \frac{1}{r \cdot r}}\right) + -1.5 \]
   pow2 [=>] 99.5	\[ \mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, 2 \cdot \frac{1}{\color{blue}{{r}^{2}}}\right) + -1.5 \]
   pow-flip [=>] 99.6	\[ \mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, 2 \cdot \color{blue}{{r}^{\left(-2\right)}}\right) + -1.5 \]
   metadata-eval [=>] 99.6	\[ \mathsf{fma}\left({\left(r \cdot w\right)}^{2}, \frac{\mathsf{fma}\left(v, 0.25, -0.375\right)}{1 - v}, 2 \cdot {r}^{\color{blue}{-2}}\right) + -1.5 \]

4. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	97.5%
Cost	1865

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq -2 \cdot 10^{-65} \lor \neg \left(r \leq 5 \cdot 10^{-143}\right):\\ \;\;\;\;t_0 + \left(-1.5 - \frac{0.375 + v \cdot -0.25}{1 - v} \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(t_0 + 3\right) - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right) + -4.5\\ \end{array} \]

Alternative 2
Accuracy	76.0%
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := -1.5 + t_0\\ t_2 := t_0 + \left(-0.375 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -2.6 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 4 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	75.9%
Cost	1616

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := -1.5 + t_0\\ t_2 := t_0 + \left(-0.25 \cdot \left(\left(r \cdot r\right) \cdot \left(w \cdot w\right)\right) - 1.5\right)\\ \mathbf{if}\;r \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq -7 \cdot 10^{-34}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 4.7 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{elif}\;r \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	99.5%
Cost	1600

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

Alternative 5
Accuracy	91.2%
Cost	1481

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -30000000 \lor \neg \left(v \leq 1.45\right):\\ \;\;\;\;\left(-1.5 + t_0\right) - 0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(\left(t_0 + 3\right) - 0.375 \cdot \left(r \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	95.8%
Cost	1481

\[\begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;v \leq -30000000 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;-4.5 + \left(t_1 - r \cdot \left(0.25 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(t_1 - 0.375 \cdot \left(r \cdot t_0\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	96.2%
Cost	1481

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;v \leq -30000000 \lor \neg \left(v \leq 3 \cdot 10^{+71}\right):\\ \;\;\;\;-4.5 + \left(t_0 - r \cdot \left(0.25 \cdot \left(w \cdot \left(r \cdot w\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(t_0 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	98.9%
Cost	1481

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r} + 3\\ \mathbf{if}\;v \leq -30000000 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;\left(t_0 - \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot 0.25\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;-4.5 + \left(t_0 - \left(r \cdot w\right) \cdot \left(r \cdot \left(w \cdot 0.375\right)\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	79.5%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;r \leq -2 \cdot 10^{-33} \lor \neg \left(r \leq 7 \cdot 10^{-108}\right):\\ \;\;\;\;\left(-1.5 + \frac{2}{r \cdot r}\right) - 0.25 \cdot \left(r \cdot \left(r \cdot \left(w \cdot w\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \end{array} \]

Alternative 10
Accuracy	86.9%
Cost	1353

\[\begin{array}{l} t_0 := r \cdot \left(r \cdot \left(w \cdot w\right)\right)\\ t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -30000000 \lor \neg \left(v \leq 1\right):\\ \;\;\;\;\left(-1.5 + t_1\right) - 0.25 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 - \left(0.375 \cdot t_0 + 1.5\right)\\ \end{array} \]

Alternative 11
Accuracy	67.3%
Cost	448

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

Alternative 12
Accuracy	39.6%
Cost	320

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

Alternative 13
Accuracy	39.6%
Cost	320

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

Reproduce

herbie shell --seed 2023147 
(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))