Rosa's TurbineBenchmark

Average Error: 10.1 → 0.2

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{2}{r \cdot r}\\ \mathbf{if}\;w \leq -6.408408175333973 \cdot 10^{+191} \lor \neg \left(w \leq 16826817815674.701\right):\\ \;\;\;\;t_0 - w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot \left(r \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, w \cdot \left(\left(w \cdot r\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \end{array} \]

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))))
   (if (or (<= w -6.408408175333973e+191) (not (<= w 16826817815674.701)))
     (- t_0 (* w (* r (* (/ w (- 1.0 v)) (* r (fma v -0.25 0.375))))))
     (- t_0 (fma r (* w (* (* w r) (/ (fma v -0.25 0.375) (- 1.0 v)))) 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 tmp;
	if ((w <= -6.408408175333973e+191) || !(w <= 16826817815674.701)) {
		tmp = t_0 - (w * (r * ((w / (1.0 - v)) * (r * fma(v, -0.25, 0.375)))));
	} else {
		tmp = t_0 - fma(r, (w * ((w * r) * (fma(v, -0.25, 0.375) / (1.0 - v)))), 1.5);
	}
	return tmp;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

1. Split input into 2 regimes

2. if w < -6.4084081753339727e191 or 16826817815674.7012 < w

   1. Initial program 17.5

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

   2. Simplified 16.3

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

   3. Applied associate-*r*_binary64 6.3

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

   4. Taylor expanded in r around 0 32.2

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

   5. Simplified 6.3

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

   6. Taylor expanded in r around inf 32.1

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

   7. Simplified 0.2

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

   if -6.4084081753339727e191 < w < 16826817815674.7012

   1. Initial program 6.4

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

   2. Simplified 4.1

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

   3. Applied 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 7.0

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

   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 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -6.408408175333973 \cdot 10^{+191} \lor \neg \left(w \leq 16826817815674.701\right):\\ \;\;\;\;\frac{2}{r \cdot r} - w \cdot \left(r \cdot \left(\frac{w}{1 - v} \cdot \left(r \cdot \mathsf{fma}\left(v, -0.25, 0.375\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, w \cdot \left(\left(w \cdot r\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))