Rosa's TurbineBenchmark

Average Error: 12.7 → 0.5

Time: 11.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;r \leq -4.650921930387387 \cdot 10^{+34} \lor \neg \left(r \leq 2.134084632159218 \cdot 10^{+88}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(\frac{r}{1 - v} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - w \cdot \left(\left(\frac{r}{1 - v} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \left(r \cdot w\right)\right)\\ \end{array}\]

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
 (if (or (<= r -4.650921930387387e+34) (not (<= r 2.134084632159218e+88)))
   (-
    (+ (/ 2.0 (* r r)) -1.5)
    (* (* (/ r (- 1.0 v)) (+ 0.375 (* v -0.25))) (* w (* r w))))
   (-
    (+ (/ 2.0 (* r r)) -1.5)
    (* w (* (* (/ r (- 1.0 v)) (+ 0.375 (* v -0.25))) (* r w))))))

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 tmp;
	if ((r <= -4.650921930387387e+34) || !(r <= 2.134084632159218e+88)) {
		tmp = ((2.0 / (r * r)) + -1.5) - (((r / (1.0 - v)) * (0.375 + (v * -0.25))) * (w * (r * w)));
	} else {
		tmp = ((2.0 / (r * r)) + -1.5) - (w * (((r / (1.0 - v)) * (0.375 + (v * -0.25))) * (r * w)));
	}
	return tmp;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

1. Split input into 2 regimes

2. if r < -4.65092193038738699e34 or 2.134084632159218e88 < r

   1. Initial program 17.2

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

   2. Simplified 11.8

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{r \cdot \left(w \cdot w\right)}}}\]
   3. Using strategy rm

   4. Applied associate-*r*_binary64 3.8

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{\color{blue}{\left(r \cdot w\right) \cdot w}}}\]
   5. Using strategy rm

   6. Applied associate-/r/_binary64 0.3

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

   7. Applied associate-*r*_binary64 0.2

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

   8. Simplified 0.2

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

   if -4.65092193038738699e34 < r < 2.134084632159218e88

   1. Initial program 9.8

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

   2. Simplified 7.3

      \[\leadsto \color{blue}{\left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{r \cdot \left(w \cdot w\right)}}}\]
   3. Using strategy rm

   4. Applied associate-*r*_binary64 3.9

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \left(0.375 + v \cdot -0.25\right) \cdot \frac{r}{\frac{1 - v}{\color{blue}{\left(r \cdot w\right) \cdot w}}}\]
   5. Using strategy rm

   6. Applied associate-/r/_binary64 4.0

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

   7. Applied associate-*r*_binary64 4.0

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

   8. Simplified 4.0

      \[\leadsto \left(\frac{2}{r \cdot r} + -1.5\right) - \color{blue}{\left(\frac{r}{1 - v} \cdot \left(0.375 + v \cdot -0.25\right)\right)} \cdot \left(\left(r \cdot w\right) \cdot w\right)\]
   9. Using strategy rm

   10. Applied associate-*r*_binary64 0.7

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq -4.650921930387387 \cdot 10^{+34} \lor \neg \left(r \leq 2.134084632159218 \cdot 10^{+88}\right):\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - \left(\frac{r}{1 - v} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \left(w \cdot \left(r \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{r \cdot r} + -1.5\right) - w \cdot \left(\left(\frac{r}{1 - v} \cdot \left(0.375 + v \cdot -0.25\right)\right) \cdot \left(r \cdot w\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(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))