Rosa's TurbineBenchmark

Average Error: 12.3 → 0.4

Time: 13.7s

Precision: binary64

Cost: 14464

Math

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

↓

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

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
 (+
  (+
   (+ 3.0 (/ 2.0 (* r r)))
   (* (* 0.125 (fma v -2.0 3.0)) (/ -1.0 (/ (- 1.0 v) (pow (* r w) 2.0)))))
  -4.5))

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 ((3.0 + (2.0 / (r * r))) + ((0.125 * fma(v, -2.0, 3.0)) * (-1.0 / ((1.0 - v) / pow((r * w), 2.0))))) + -4.5;
}

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(Float64(Float64(3.0 + Float64(2.0 / Float64(r * r))) + Float64(Float64(0.125 * fma(v, -2.0, 3.0)) * Float64(-1.0 / Float64(Float64(1.0 - v) / (Float64(r * w) ^ 2.0))))) + -4.5)
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[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.125 * N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(N[(1.0 - v), $MachinePrecision] / N[Power[N[(r * w), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]

Derivation

1. Initial program 12.3

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

2. Applied egg-rr 0.4

   \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\left(0.125 \cdot \mathsf{fma}\left(v, -2, 3\right)\right) \cdot \frac{1}{\frac{1 - v}{{\left(w \cdot r\right)}^{2}}}}\right) - 4.5 \]

3. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.5
Cost	8264

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := w \cdot \left(\left(r \cdot w\right) \cdot -0.125\right)\\ t_2 := \left(t_0 + \frac{r}{1 - v} \cdot \left(\left(v \cdot -2\right) \cdot t_1 + 3 \cdot t_1\right)\right) + -4.5\\ \mathbf{if}\;r \leq -1 \cdot 10^{+150}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;r \leq 10^{+140}:\\ \;\;\;\;\left(t_0 - \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \left(w \cdot \left(\left(r \cdot r\right) \cdot w\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	2.5
Cost	8128

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

Alternative 3
Error	2.7
Cost	2120

\[\begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ t_2 := \left(t_1 - \frac{t_0}{\frac{4}{r}}\right) + -4.5\\ \mathbf{if}\;v \leq -1.8179282493338328 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;v \leq 0.6207891505361259:\\ \;\;\;\;\left(t_1 - \frac{t_0}{8 \cdot \frac{1 - v}{r \cdot \left(3 + v \cdot -2\right)}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	1.7
Cost	1992

\[\begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1613816883.9497578:\\ \;\;\;\;\left(t_1 - \frac{t_0}{\frac{4}{r} + \frac{\frac{2}{v}}{r}}\right) + -4.5\\ \mathbf{elif}\;v \leq 0.6207891505361259:\\ \;\;\;\;\left(t_1 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(-0.375 + v \cdot \left(-0.125 + v \cdot -0.041666666666666664\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \frac{t_0}{\frac{4}{r}}\right) + -4.5\\ \end{array} \]

Alternative 5
Error	1.8
Cost	1736

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{w \cdot \left(r \cdot w\right)}{\frac{4}{r}}\right) + -4.5\\ \mathbf{if}\;v \leq -1613816883.9497578:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 0.6207891505361259:\\ \;\;\;\;\left(t_0 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(-0.375 + v \cdot -0.125\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	1.7
Cost	1736

\[\begin{array}{l} t_0 := w \cdot \left(r \cdot w\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -1613816883.9497578:\\ \;\;\;\;\left(t_1 - \frac{t_0}{\frac{4}{r} + \frac{\frac{2}{v}}{r}}\right) + -4.5\\ \mathbf{elif}\;v \leq 0.6207891505361259:\\ \;\;\;\;\left(t_1 + \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right) \cdot \left(-0.375 + v \cdot -0.125\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 - \frac{t_0}{\frac{4}{r}}\right) + -4.5\\ \end{array} \]

Alternative 7
Error	12.5
Cost	1480

\[\begin{array}{l} t_0 := \left(\left(3 + \frac{2}{r \cdot r}\right) + r \cdot \left(\left(r \cdot \left(w \cdot w\right)\right) \cdot -0.375\right)\right) + -4.5\\ \mathbf{if}\;r \leq -3.0843587467304782 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 3.4170412273655313 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	1.8
Cost	1480

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{w \cdot \left(r \cdot w\right)}{\frac{4}{r}}\right) + -4.5\\ \mathbf{if}\;v \leq -1613816883.9497578:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 0.6207891505361259:\\ \;\;\;\;\left(t_0 - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	9.4
Cost	1216

\[\left(\left(3 + \frac{2}{r \cdot r}\right) - 0.375 \cdot \left(\left(r \cdot w\right) \cdot \left(r \cdot w\right)\right)\right) + -4.5 \]

Alternative 10
Error	20.0
Cost	840

\[\begin{array}{l} t_0 := \left(r \cdot w\right) \cdot \left(w \cdot \left(r \cdot -0.375\right)\right)\\ \mathbf{if}\;r \leq -1.75 \cdot 10^{+210}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 6.6 \cdot 10^{+174}:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	21.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;r \leq -54.62777205185027:\\ \;\;\;\;-1.5\\ \mathbf{elif}\;r \leq 0.00047428484603962307:\\ \;\;\;\;\frac{2}{r \cdot r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]

Alternative 12
Error	21.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;r \leq -54.62777205185027:\\ \;\;\;\;-1.5\\ \mathbf{elif}\;r \leq 0.00047428484603962307:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;-1.5\\ \end{array} \]

Alternative 13
Error	21.1
Cost	448

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

Alternative 14
Error	45.8
Cost	64

\[-1.5 \]

Reproduce

herbie shell --seed 2022325 
(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))