Rosa's TurbineBenchmark

Average Error: 12.5 → 0.3

Time: 8.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ t_1 := \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\\ \mathbf{if}\;r \leq -9.065172876585645 \cdot 10^{-21}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, w \cdot \left(\left(r \cdot w\right) \cdot t_1\right), 1.5\right)\\ \mathbf{elif}\;r \leq 5.929669123769266 \cdot 10^{-41}:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)}\\ t_0 - t_2 \cdot t_2 \end{array}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, \left(r \cdot w\right) \cdot \left(w \cdot t_1\right), 1.5\right)\\ \end{array} \]

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)) (t_1 (/ (fma v -0.25 0.375) (- 1.0 v))))
   (if (<= r -9.065172876585645e-21)
     (- t_0 (fma r (* w (* (* r w) t_1)) 1.5))
     (if (<= r 5.929669123769266e-41)
       (let* ((t_2
               (sqrt
                (fma
                 (* w (* r r))
                 (* (fma v -0.25 0.375) (/ w (- 1.0 v)))
                 1.5))))
         (- t_0 (* t_2 t_2)))
       (- (/ 2.0 (* r r)) (fma r (* (* r w) (* w t_1)) 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 <= -9.065172876585645e-21) {
		tmp = t_0 - fma(r, (w * ((r * w) * t_1)), 1.5);
	} else if (r <= 5.929669123769266e-41) {
		double t_2 = sqrt(fma((w * (r * r)), (fma(v, -0.25, 0.375) * (w / (1.0 - v))), 1.5));
		tmp = t_0 - (t_2 * t_2);
	} else {
		tmp = (2.0 / (r * r)) - fma(r, ((r * w) * (w * t_1)), 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 < -9.06517287658564543e-21

   1. Initial program 13.6

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

   2. Simplified 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 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. Applied associate-/r*_binary64 0.2

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

   5. Taylor expanded in r around 0 9.6

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

   6. Simplified 0.2

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

   if -9.06517287658564543e-21 < r < 5.92966912376926576e-41

   1. Initial program 11.4

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

   2. Simplified 10.4

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

   3. Applied associate-*r*_binary64 5.8

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

   4. Applied associate-/r*_binary64 5.8

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

   5. Applied associate-*l*_binary64 5.8

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

   6. Applied add-sqr-sqrt_binary64 5.8

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

   7. Simplified 5.8

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

   8. Simplified 0.3

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

   if 5.92966912376926576e-41 < r

   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 7.0

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

   3. Applied associate-*r*_binary64 0.5

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

   4. Applied associate-/r*_binary64 0.5

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

   5. Applied associate-*l*_binary64 0.5

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

   6. Applied *-un-lft-identity_binary64 0.5

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

   7. Applied *-un-lft-identity_binary64 0.5

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

   8. Applied *-un-lft-identity_binary64 0.5

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

   9. Applied times-frac_binary64 0.5

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

   10. Applied times-frac_binary64 0.5

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

   11. Simplified 0.5

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

   12. Simplified 0.5

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -9.065172876585645 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, w \cdot \left(\left(r \cdot w\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \mathbf{elif}\;r \leq 5.929669123769266 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)} \cdot \sqrt{\mathsf{fma}\left(w \cdot \left(r \cdot r\right), \mathsf{fma}\left(v, -0.25, 0.375\right) \cdot \frac{w}{1 - v}, 1.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{r \cdot r} - \mathsf{fma}\left(r, \left(r \cdot w\right) \cdot \left(w \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022081 
(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))