Rosa's TurbineBenchmark

Average Error: 12.6 → 1.7

Time: 8.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{2}{r}}{r}\\ t_1 := \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\\ \mathbf{if}\;r \leq -3.797359757468544 \cdot 10^{-126}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot t_1, 1.5\right)\\ \mathbf{elif}\;r \leq 7.915067883874499 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \mathbf{else}:\\ \;\;\;\;t_0 - \mathsf{fma}\left(r, w \cdot \left(\left(r \cdot w\right) \cdot t_1\right), 1.5\right)\\ \end{array} \]

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 -3.797359757468544e-126)
     (- t_0 (fma r (* (* w (* r w)) t_1) 1.5))
     (if (<= r 7.915067883874499e-92)
       (- (/ 2.0 (* r r)) 1.5)
       (- t_0 (fma r (* w (* (* r 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 <= -3.797359757468544e-126) {
		tmp = t_0 - fma(r, ((w * (r * w)) * t_1), 1.5);
	} else if (r <= 7.915067883874499e-92) {
		tmp = (2.0 / (r * r)) - 1.5;
	} else {
		tmp = t_0 - fma(r, (w * ((r * 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 < -3.797359757468544e-126

   1. Initial program 12.3

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

   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 1.6

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

   4. Applied associate-/r*_binary64 1.6

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

   5. Applied *-commutative_binary64 1.6

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

   if -3.797359757468544e-126 < r < 7.91506788387449927e-92

   1. Initial program 14.9

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

   2. Simplified 14.9

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

   3. Taylor expanded in r around 0 3.6

      \[\leadsto \frac{2}{r \cdot r} - \color{blue}{1.5} \]

   if 7.91506788387449927e-92 < r

   1. Initial program 12.1

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

   2. Simplified 6.6

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

   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-/r*_binary64 0.9

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

   5. Applied pow1_binary64 0.9

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

   6. Applied pow1_binary64 0.9

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

   7. Applied pow1_binary64 0.9

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

   8. Applied pow1_binary64 0.9

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

   9. Applied pow-prod-down_binary64 0.9

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

   10. Applied pow-prod-down_binary64 0.9

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

   11. Applied pow-prod-down_binary64 0.9

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

   12. Simplified 0.9

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

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -3.797359757468544 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, \left(w \cdot \left(r \cdot w\right)\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}, 1.5\right)\\ \mathbf{elif}\;r \leq 7.915067883874499 \cdot 10^{-92}:\\ \;\;\;\;\frac{2}{r \cdot r} - 1.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(r, w \cdot \left(\left(r \cdot w\right) \cdot \frac{\mathsf{fma}\left(v, -0.25, 0.375\right)}{1 - v}\right), 1.5\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022020 
(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))