Average Error: 12.9 → 0.3
Time: 5.2s
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 r15384 = 3.0;
        double r15385 = 2.0;
        double r15386 = r;
        double r15387 = r15386 * r15386;
        double r15388 = r15385 / r15387;
        double r15389 = r15384 + r15388;
        double r15390 = 0.125;
        double r15391 = v;
        double r15392 = r15385 * r15391;
        double r15393 = r15384 - r15392;
        double r15394 = r15390 * r15393;
        double r15395 = w;
        double r15396 = r15395 * r15395;
        double r15397 = r15396 * r15386;
        double r15398 = r15397 * r15386;
        double r15399 = r15394 * r15398;
        double r15400 = 1.0;
        double r15401 = r15400 - r15391;
        double r15402 = r15399 / r15401;
        double r15403 = r15389 - r15402;
        double r15404 = 4.5;
        double r15405 = r15403 - r15404;
        return r15405;
}
↓
double f(double v, double w, double r) {
        double r15406 = 2.0;
        double r15407 = r;
        double r15408 = r15406 / r15407;
        double r15409 = r15408 / r15407;
        double r15410 = 0.125;
        double r15411 = 3.0;
        double r15412 = v;
        double r15413 = r15406 * r15412;
        double r15414 = r15411 - r15413;
        double r15415 = r15410 * r15414;
        double r15416 = 1.0;
        double r15417 = r15416 - r15412;
        double r15418 = r15415 / r15417;
        double r15419 = w;
        double r15420 = r15419 * r15407;
        double r15421 = fabs(r15420);
        double r15422 = r15418 * r15421;
        double r15423 = r15422 * r15421;
        double r15424 = 4.5;
        double r15425 = r15424 - r15411;
        double r15426 = r15423 + r15425;
        double r15427 = r15409 - r15426;
        return r15427;
}

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 12.9

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

  2. Simplified8.4

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

  4. Applied add-sqr-sqrt8.4

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

  5. Simplified8.4

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

  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*0.3

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

  13. Applied associate-/r*0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019356 +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))