Rosa's TurbineBenchmark

Average Error: 12.9 → 0.4

Time: 9.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\\ \mathbf{if}\;r \leq -3.664195920476524 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot t_0, 1.5\right)\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \frac{2}{r \cdot r}\\ \mathbf{if}\;r \leq 0.0002711214414898074:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)}\\ t_1 - t_2 \cdot t_2 \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1 - \mathsf{fma}\left(r, w \cdot \left(\left(r \cdot w\right) \cdot t_0\right), 1.5\right)\\ \end{array}\\ \end{array} \]

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 (/ (fma v -0.25 0.375) (- 1.0 v))))
   (if (<= r -3.664195920476524e-60)
     (- (/ (/ 2.0 r) r) (fma r (* (* w (* r w)) t_0) 1.5))
     (let* ((t_1 (/ 2.0 (* r r))))
       (if (<= r 0.0002711214414898074)
         (let* ((t_2
                 (sqrt
                  (fma
                   (* w (* r r))
                   (* (fma v -0.25 0.375) (/ w (- 1.0 v)))
                   1.5))))
           (- t_1 (* t_2 t_2)))
         (- t_1 (fma r (* w (* (* r w) t_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 tmp;
	if (r <= -3.664195920476524e-60) {
		tmp = ((2.0 / r) / r) - fma(r, ((w * (r * w)) * t_0), 1.5);
	} else {
		double t_1 = 2.0 / (r * r);
		double tmp_1;
		if (r <= 0.0002711214414898074) {
			double t_2_2 = sqrt(fma((w * (r * r)), (fma(v, -0.25, 0.375) * (w / (1.0 - v))), 1.5));
			tmp_1 = t_1 - (t_2_2 * t_2_2);
		} else {
			tmp_1 = t_1 - fma(r, (w * ((r * w) * t_0)), 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 < -3.6641959204765242e-60

   1. Initial program 13.3

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

   2. 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 associate-*r*_binary64 0.8

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

   4. Applied associate-/r*_binary64 0.8

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

   5. Applied *-un-lft-identity_binary64 0.8

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

   if -3.6641959204765242e-60 < r < 2.711214414898074e-4

   1. Initial program 11.7

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

   2. Simplified 10.5

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

   3. Applied associate-*r*_binary64 4.7

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

   4. Applied associate-*l*_binary64 4.7

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

   5. Applied add-sqr-sqrt_binary64 4.7

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

   6. Simplified 4.7

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

   7. Simplified 0.3

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

   if 2.711214414898074e-4 < r

   1. Initial program 14.0

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

   2. Simplified 6.7

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

   3. Applied associate-*r*_binary64 0.2

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

   4. Taylor expanded in r around 0 9.7

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

   5. Simplified 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -3.664195920476524 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{elif}\;r \leq 0.0002711214414898074:\\ \;\;\;\;\frac{2}{r \cdot r} - \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)} \cdot \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, w \cdot \left(\left(r \cdot w\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021352 
(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))