Rosa's TurbineBenchmark

Average Error: 12.8 → 4.3

Time: 8.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\\ t_1 := 2 \cdot {r}^{-2}\\ \mathbf{if}\;r \leq -3.350971884985321 \cdot 10^{-49}:\\ \;\;\;\;t_1 - \mathsf{fma}\left(r, \left(r \cdot \left(w \cdot w\right)\right) \cdot t_0, 1.5\right)\\ \mathbf{elif}\;r \leq 9.21149556780841 \cdot 10^{-127}:\\ \;\;\;\;t_1 - \mathsf{fma}\left(r, t_0 \cdot \log \left({\left(e^{r}\right)}^{\left(w \cdot w\right)}\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - \mathsf{fma}\left(r, t_0 \cdot {\left(w \cdot \sqrt{r}\right)}^{2}, 1.5\right)\\ \end{array} \]

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 (/ (fma v -0.25 0.375) (- 1.0 v))) (t_1 (* 2.0 (pow r -2.0))))
   (if (<= r -3.350971884985321e-49)
     (- t_1 (fma r (* (* r (* w w)) t_0) 1.5))
     (if (<= r 9.21149556780841e-127)
       (- t_1 (fma r (* t_0 (log (pow (exp r) (* w w)))) 1.5))
       (- t_1 (fma r (* t_0 (pow (* w (sqrt 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) {
	double t_0 = fma(v, -0.25, 0.375) / (1.0 - v);
	double t_1 = 2.0 * pow(r, -2.0);
	double tmp;
	if (r <= -3.350971884985321e-49) {
		tmp = t_1 - fma(r, ((r * (w * w)) * t_0), 1.5);
	} else if (r <= 9.21149556780841e-127) {
		tmp = t_1 - fma(r, (t_0 * log(pow(exp(r), (w * w)))), 1.5);
	} else {
		tmp = t_1 - fma(r, (t_0 * pow((w * sqrt(r)), 2.0)), 1.5);
	}
	return tmp;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

1. Split input into 3 regimes

2. if r < -3.35097188498532108e-49

   1. Initial program 13.5

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

   2. Simplified 7.3

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

   3. Applied egg-rr 7.2

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

   if -3.35097188498532108e-49 < r < 9.2114955678084093e-127

   1. Initial program 12.6

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

   2. Simplified 12.1

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

   3. Applied egg-rr 11.9

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

   4. Applied egg-rr 5.1

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

   if 9.2114955678084093e-127 < r

   1. Initial program 12.4

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

   2. Simplified 7.2

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

   3. Applied egg-rr 7.1

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

   4. Applied egg-rr 1.6

      \[\leadsto 2 \cdot {r}^{-2} - \mathsf{fma}\left(r, \color{blue}{{\left(w \cdot \sqrt{r}\right)}^{2}} \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -3.350971884985321 \cdot 10^{-49}:\\ \;\;\;\;2 \cdot {r}^{-2} - \mathsf{fma}\left(r, \left(r \cdot \left(w \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{elif}\;r \leq 9.21149556780841 \cdot 10^{-127}:\\ \;\;\;\;2 \cdot {r}^{-2} - \mathsf{fma}\left(r, \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot \log \left({\left(e^{r}\right)}^{\left(w \cdot w\right)}\right), 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot {r}^{-2} - \mathsf{fma}\left(r, \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v} \cdot {\left(w \cdot \sqrt{r}\right)}^{2}, 1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))