Rosa's TurbineBenchmark

Average Error: 13.0 → 0.6

Time: 7.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ t_1 := \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\\ \mathbf{if}\;r \leq -2.485594501701809 \cdot 10^{-39}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot t_1, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := w \cdot t_1\\ \mathbf{if}\;r \leq 1.3374143467488828 \cdot 10^{-73}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt{\mathsf{fma}\left(\left(r \cdot r\right) \cdot w, t_2, 1.5\right)}\\ t_0 - t_3 \cdot t_3 \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, \left(r \cdot w\right) \cdot t_2, 1.5\right)\\ \end{array}\\ \end{array} \]

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 (/ 2.0 (* r r))) (t_1 (/ (fma v -0.25 0.375) (- 1.0 v))))
   (if (<= r -2.485594501701809e-39)
     (- t_0 (fma r (* (* w (* r w)) t_1) 1.5))
     (let* ((t_2 (* w t_1)))
       (if (<= r 1.3374143467488828e-73)
         (let* ((t_3 (sqrt (fma (* (* r r) w) t_2 1.5)))) (- t_0 (* t_3 t_3)))
         (- t_0 (fma r (* (* r w) t_2) 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 = 2.0 / (r * r);
	double t_1 = fma(v, -0.25, 0.375) / (1.0 - v);
	double tmp;
	if (r <= -2.485594501701809e-39) {
		tmp = t_0 - fma(r, ((w * (r * w)) * t_1), 1.5);
	} else {
		double t_2 = w * t_1;
		double tmp_1;
		if (r <= 1.3374143467488828e-73) {
			double t_3_2 = sqrt(fma(((r * r) * w), t_2, 1.5));
			tmp_1 = t_0 - (t_3_2 * t_3_2);
		} else {
			tmp_1 = t_0 - fma(r, ((r * w) * t_2), 1.5);
		}
		tmp = tmp_1;
	}
	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 < -2.48559450170180914e-39

   1. Initial program 12.5

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

   2. Simplified 6.0

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

   3. Applied associate-*r*_binary64 0.5

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

   if -2.48559450170180914e-39 < r < 1.337414346748883e-73

   1. Initial program 12.7

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

   2. Simplified 11.7

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

   3. Applied associate-*r*_binary64 6.2

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

   4. Applied associate-*l*_binary64 6.2

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

   5. Applied add-sqr-sqrt_binary64 6.2

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

   6. Simplified 6.2

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

   7. Simplified 0.3

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

   if 1.337414346748883e-73 < r

   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. Simplified 7.6

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

   3. Applied associate-*r*_binary64 0.9

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

   4. Applied associate-*l*_binary64 0.9

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -2.485594501701809 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{elif}\;r \leq 1.3374143467488828 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{r \cdot r} - \sqrt{\mathsf{fma}\left(\left(r \cdot r\right) \cdot w, w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)} \cdot \sqrt{\mathsf{fma}\left(\left(r \cdot r\right) \cdot w, w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, \left(r \cdot w\right) \cdot \left(w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022121 
(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))