Average Error: 12.8 → 0.4
Time: 22.1s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot {\left(\left|w \cdot r\right|\right)}^{2}} \cdot \left(\sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}} \cdot \left|\left|w \cdot r\right|\right|\right)\right) - 4.5\]
\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
↓
\left(\left(3 + \frac{2}{r \cdot r}\right) - \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v} \cdot {\left(\left|w \cdot r\right|\right)}^{2}} \cdot \left(\sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}} \cdot \left|\left|w \cdot r\right|\right|\right)\right) - 4.5
double f(double v, double w, double r) {
        double r27328 = 3.0;
        double r27329 = 2.0;
        double r27330 = r;
        double r27331 = r27330 * r27330;
        double r27332 = r27329 / r27331;
        double r27333 = r27328 + r27332;
        double r27334 = 0.125;
        double r27335 = v;
        double r27336 = r27329 * r27335;
        double r27337 = r27328 - r27336;
        double r27338 = r27334 * r27337;
        double r27339 = w;
        double r27340 = r27339 * r27339;
        double r27341 = r27340 * r27330;
        double r27342 = r27341 * r27330;
        double r27343 = r27338 * r27342;
        double r27344 = 1.0;
        double r27345 = r27344 - r27335;
        double r27346 = r27343 / r27345;
        double r27347 = r27333 - r27346;
        double r27348 = 4.5;
        double r27349 = r27347 - r27348;
        return r27349;
}
↓
double f(double v, double w, double r) {
        double r27350 = 3.0;
        double r27351 = 2.0;
        double r27352 = r;
        double r27353 = r27352 * r27352;
        double r27354 = r27351 / r27353;
        double r27355 = r27350 + r27354;
        double r27356 = 0.125;
        double r27357 = v;
        double r27358 = r27351 * r27357;
        double r27359 = r27350 - r27358;
        double r27360 = r27356 * r27359;
        double r27361 = 1.0;
        double r27362 = r27361 - r27357;
        double r27363 = r27360 / r27362;
        double r27364 = w;
        double r27365 = r27364 * r27352;
        double r27366 = fabs(r27365);
        double r27367 = 2.0;
        double r27368 = pow(r27366, r27367);
        double r27369 = r27363 * r27368;
        double r27370 = sqrt(r27369);
        double r27371 = sqrt(r27363);
        double r27372 = fabs(r27366);
        double r27373 = r27371 * r27372;
        double r27374 = r27370 * r27373;
        double r27375 = r27355 - r27374;
        double r27376 = 4.5;
        double r27377 = r27375 - r27376;
        return r27377;
}

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

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

  4. Applied add-sqr-sqrt8.6

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

  5. Simplified8.5

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

  6. Simplified0.4

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

  8. Applied fma-udef0.4

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

  9. Applied associate--r+0.4

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

  10. Simplified0.4

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

  12. Applied add-sqr-sqrt0.4

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

  14. Applied sqrt-prod0.4

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2019326 +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))