Average Error: 12.2 → 0.4
Time: 5.0s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\left(\sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}} \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)\]
\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(\sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}} \cdot \sqrt{\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}}\right) \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)
double f(double v, double w, double r) {
        double r16280 = 3.0;
        double r16281 = 2.0;
        double r16282 = r;
        double r16283 = r16282 * r16282;
        double r16284 = r16281 / r16283;
        double r16285 = r16280 + r16284;
        double r16286 = 0.125;
        double r16287 = v;
        double r16288 = r16281 * r16287;
        double r16289 = r16280 - r16288;
        double r16290 = r16286 * r16289;
        double r16291 = w;
        double r16292 = r16291 * r16291;
        double r16293 = r16292 * r16282;
        double r16294 = r16293 * r16282;
        double r16295 = r16290 * r16294;
        double r16296 = 1.0;
        double r16297 = r16296 - r16287;
        double r16298 = r16295 / r16297;
        double r16299 = r16285 - r16298;
        double r16300 = 4.5;
        double r16301 = r16299 - r16300;
        return r16301;
}
↓
double f(double v, double w, double r) {
        double r16302 = 2.0;
        double r16303 = r;
        double r16304 = r16302 / r16303;
        double r16305 = r16304 / r16303;
        double r16306 = 0.125;
        double r16307 = 3.0;
        double r16308 = v;
        double r16309 = r16302 * r16308;
        double r16310 = r16307 - r16309;
        double r16311 = r16306 * r16310;
        double r16312 = 1.0;
        double r16313 = r16312 - r16308;
        double r16314 = r16311 / r16313;
        double r16315 = sqrt(r16314);
        double r16316 = r16315 * r16315;
        double r16317 = w;
        double r16318 = r16317 * r16303;
        double r16319 = fabs(r16318);
        double r16320 = r16319 * r16319;
        double r16321 = r16316 * r16320;
        double r16322 = 4.5;
        double r16323 = r16322 - r16307;
        double r16324 = r16321 + r16323;
        double r16325 = r16305 - r16324;
        return r16325;
}

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 12.2

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

  2. Simplified8.3

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

  4. Applied add-sqr-sqrt8.3

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

  5. Simplified8.3

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

  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 add-sqr-sqrt0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020056 +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))