Average Error: 12.8 → 0.4
Time: 4.8s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|w \cdot r\right|\right) \cdot \left|w \cdot r\right| + \left(4.5 - 3\right)\right)\]
\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
↓
\frac{\frac{2}{r}}{r} - \left(\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left|w \cdot r\right|\right) \cdot \left|w \cdot r\right| + \left(4.5 - 3\right)\right)
double f(double v, double w, double r) {
        double r14396 = 3.0;
        double r14397 = 2.0;
        double r14398 = r;
        double r14399 = r14398 * r14398;
        double r14400 = r14397 / r14399;
        double r14401 = r14396 + r14400;
        double r14402 = 0.125;
        double r14403 = v;
        double r14404 = r14397 * r14403;
        double r14405 = r14396 - r14404;
        double r14406 = r14402 * r14405;
        double r14407 = w;
        double r14408 = r14407 * r14407;
        double r14409 = r14408 * r14398;
        double r14410 = r14409 * r14398;
        double r14411 = r14406 * r14410;
        double r14412 = 1.0;
        double r14413 = r14412 - r14403;
        double r14414 = r14411 / r14413;
        double r14415 = r14401 - r14414;
        double r14416 = 4.5;
        double r14417 = r14415 - r14416;
        return r14417;
}
↓
double f(double v, double w, double r) {
        double r14418 = 2.0;
        double r14419 = r;
        double r14420 = r14418 / r14419;
        double r14421 = r14420 / r14419;
        double r14422 = 0.125;
        double r14423 = 3.0;
        double r14424 = v;
        double r14425 = r14418 * r14424;
        double r14426 = r14423 - r14425;
        double r14427 = r14422 * r14426;
        double r14428 = 1.0;
        double r14429 = r14428 - r14424;
        double r14430 = r14427 / r14429;
        double r14431 = w;
        double r14432 = r14431 * r14419;
        double r14433 = fabs(r14432);
        double r14434 = r14430 * r14433;
        double r14435 = r14434 * r14433;
        double r14436 = 4.5;
        double r14437 = r14436 - r14423;
        double r14438 = r14435 + r14437;
        double r14439 = r14421 - r14438;
        return r14439;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 12.8

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

  2. Simplified8.5

    \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, 4.5\right) - 3\right)}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt8.6

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \color{blue}{\sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}}, 4.5\right) - 3\right)\]

  5. Simplified8.6

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \color{blue}{\left|w \cdot r\right|} \cdot \sqrt{\left(\left(w \cdot w\right) \cdot r\right) \cdot r}, 4.5\right) - 3\right)\]

  6. Simplified0.4

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \color{blue}{\left|w \cdot r\right|}, 4.5\right) - 3\right)\]
  7. Using strategy rm

  8. Applied fma-udef0.4

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

  9. Applied associate--l+0.4

    \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)}\]
  10. Using strategy rm

  11. Applied associate-/r*0.4

    \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} - \left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)\]
  12. Using strategy rm

  13. Applied associate-*r*0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (v w r)
  :name "Rosa's TurbineBenchmark"
  :precision binary64
  (- (- (+ 3 (/ 2 (* r r))) (/ (* (* 0.125 (- 3 (* 2 v))) (* (* (* w w) r) r)) (- 1 v))) 4.5))