Rosa's TurbineBenchmark

Average Error: 12.7 → 1.3

Time: 12.2s

Precision: binary64

Cost: 8392

Math

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

↓

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - \frac{\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}}{\frac{8}{\mathsf{fma}\left(v, -2, 3\right)}}\right) + -4.5\\ \mathbf{if}\;r \leq -2.6508925819619255 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;r \leq 3.7339630255213245 \cdot 10^{-92}:\\ \;\;\;\;\left(t_0 + \frac{-0.125}{\frac{0.3333333333333333}{w \cdot \left(r \cdot \left(r \cdot w\right)\right)}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (+ 3.0 (/ 2.0 (* r r))))
        (t_1
         (+
          (-
           t_0
           (/ (* (* w (* r w)) (/ r (- 1.0 v))) (/ 8.0 (fma v -2.0 3.0))))
          -4.5)))
   (if (<= r -2.6508925819619255e-60)
     t_1
     (if (<= r 3.7339630255213245e-92)
       (+ (+ t_0 (/ -0.125 (/ 0.3333333333333333 (* w (* r (* r w)))))) -4.5)
       t_1))))

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 = 3.0 + (2.0 / (r * r));
	double t_1 = (t_0 - (((w * (r * w)) * (r / (1.0 - v))) / (8.0 / fma(v, -2.0, 3.0)))) + -4.5;
	double tmp;
	if (r <= -2.6508925819619255e-60) {
		tmp = t_1;
	} else if (r <= 3.7339630255213245e-92) {
		tmp = (t_0 + (-0.125 / (0.3333333333333333 / (w * (r * (r * w)))))) + -4.5;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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 = Float64(3.0 + Float64(2.0 / Float64(r * r)))
	t_1 = Float64(Float64(t_0 - Float64(Float64(Float64(w * Float64(r * w)) * Float64(r / Float64(1.0 - v))) / Float64(8.0 / fma(v, -2.0, 3.0)))) + -4.5)
	tmp = 0.0
	if (r <= -2.6508925819619255e-60)
		tmp = t_1;
	elseif (r <= 3.7339630255213245e-92)
		tmp = Float64(Float64(t_0 + Float64(-0.125 / Float64(0.3333333333333333 / Float64(w * Float64(r * Float64(r * w)))))) + -4.5);
	else
		tmp = t_1;
	end
	return tmp
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[(3.0 + N[(2.0 / N[(r * r), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 - N[(N[(N[(w * N[(r * w), $MachinePrecision]), $MachinePrecision] * N[(r / N[(1.0 - v), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(8.0 / N[(v * -2.0 + 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision]}, If[LessEqual[r, -2.6508925819619255e-60], t$95$1, If[LessEqual[r, 3.7339630255213245e-92], N[(N[(t$95$0 + N[(-0.125 / N[(0.3333333333333333 / N[(w * N[(r * N[(r * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -4.5), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if r < -2.6508925819619255e-60 or 3.7339630255213245e-92 < r

   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 3.8

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

   3. Applied egg-rr 1.0

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \color{blue}{\frac{\left(w \cdot \left(w \cdot r\right)\right) \cdot \frac{r}{1 - v}}{\frac{8}{\mathsf{fma}\left(v, -2, 3\right)}}}\right) - 4.5 \]

   if -2.6508925819619255e-60 < r < 3.7339630255213245e-92

   1. Initial program 13.7

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

   2. Applied egg-rr 0.4

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

   3. Taylor expanded in w around 0 13.7

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

   4. Simplified 0.4

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

   5. Taylor expanded in v around 0 2.1

      \[\leadsto \left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{0.125}{\frac{\color{blue}{0.3333333333333333}}{w \cdot \left(r \cdot \left(w \cdot r\right)\right)}}\right) - 4.5 \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.6508925819619255 \cdot 10^{-60}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}}{\frac{8}{\mathsf{fma}\left(v, -2, 3\right)}}\right) + -4.5\\ \mathbf{elif}\;r \leq 3.7339630255213245 \cdot 10^{-92}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) + \frac{-0.125}{\frac{0.3333333333333333}{w \cdot \left(r \cdot \left(r \cdot w\right)\right)}}\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{r}{1 - v}}{\frac{8}{\mathsf{fma}\left(v, -2, 3\right)}}\right) + -4.5\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	7688

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 - {\left(w \cdot \left(r \cdot 0.5\right)\right)}^{2}\right) + -4.5\\ \mathbf{if}\;v \leq -13926.996856193944:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 1.9548672116058826 \cdot 10^{-16}:\\ \;\;\;\;\left(t_0 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	2.7
Cost	1732

\[\begin{array}{l} t_0 := w \cdot \left(r \cdot \left(r \cdot w\right)\right)\\ t_1 := 3 + \frac{2}{r \cdot r}\\ \mathbf{if}\;v \leq -13926.996856193944:\\ \;\;\;\;\left(t_1 + \frac{-0.125}{\frac{0.5 + \frac{0.25}{v}}{t_0}}\right) + -4.5\\ \mathbf{elif}\;v \leq 1.9548672116058826 \cdot 10^{-16}:\\ \;\;\;\;\left(t_1 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 + \frac{-0.125}{\frac{0.5}{t_0}}\right) + -4.5\\ \end{array} \]

Alternative 3
Error	2.7
Cost	1608

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + \frac{-0.125}{\frac{0.5}{w \cdot \left(r \cdot \left(r \cdot w\right)\right)}}\right) + -4.5\\ \mathbf{if}\;v \leq -13926.996856193944:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 1.9548672116058826 \cdot 10^{-16}:\\ \;\;\;\;\left(t_0 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	4.8
Cost	1480

\[\begin{array}{l} t_0 := 3 + \frac{2}{r \cdot r}\\ t_1 := \left(t_0 + \left(r \cdot \left(w \cdot w\right)\right) \cdot \left(r \cdot -0.25\right)\right) + -4.5\\ \mathbf{if}\;v \leq -39314485958367730:\\ \;\;\;\;t_1\\ \mathbf{elif}\;v \leq 1.9548672116058826 \cdot 10^{-16}:\\ \;\;\;\;\left(t_0 + \left(r \cdot w\right) \cdot \left(\left(r \cdot w\right) \cdot -0.375\right)\right) + -4.5\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.3
Cost	1352

\[\begin{array}{l} t_0 := \left(\frac{2}{r \cdot r} + r \cdot \left(r \cdot \left(\left(w \cdot w\right) \cdot -0.375\right)\right)\right) + -1.5\\ \mathbf{if}\;r \leq -4.2874445631080376 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 3.025952168455046 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{2}{r}}{r}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	8.9
Cost	1216

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

Alternative 7
Error	14.0
Cost	968

\[\begin{array}{l} t_0 := -1.5 + r \cdot \left(r \cdot \left(w \cdot \left(w \cdot -0.375\right)\right)\right)\\ \mathbf{if}\;r \leq -2.1 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;r \leq 22849.36530185334:\\ \;\;\;\;\frac{2}{r \cdot r} + -1.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	21.0
Cost	448

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

Alternative 9
Error	38.5
Cost	320

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

Reproduce

herbie shell --seed 2022228 
(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))