Average Error: 13.2 → 0.4
Time: 5.3s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(\left(3 + \frac{\frac{2}{r}}{r}\right) - \sqrt{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}{1 - v}} \cdot \sqrt{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}{1 - v}}\right) - 4.5\]
\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{\frac{2}{r}}{r}\right) - \sqrt{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}{1 - v}} \cdot \sqrt{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \frac{\left|w \cdot r\right| \cdot \left|w \cdot r\right|}{1 - v}}\right) - 4.5
double f(double v, double w, double r) {
        double r15279 = 3.0;
        double r15280 = 2.0;
        double r15281 = r;
        double r15282 = r15281 * r15281;
        double r15283 = r15280 / r15282;
        double r15284 = r15279 + r15283;
        double r15285 = 0.125;
        double r15286 = v;
        double r15287 = r15280 * r15286;
        double r15288 = r15279 - r15287;
        double r15289 = r15285 * r15288;
        double r15290 = w;
        double r15291 = r15290 * r15290;
        double r15292 = r15291 * r15281;
        double r15293 = r15292 * r15281;
        double r15294 = r15289 * r15293;
        double r15295 = 1.0;
        double r15296 = r15295 - r15286;
        double r15297 = r15294 / r15296;
        double r15298 = r15284 - r15297;
        double r15299 = 4.5;
        double r15300 = r15298 - r15299;
        return r15300;
}
↓
double f(double v, double w, double r) {
        double r15301 = 3.0;
        double r15302 = 2.0;
        double r15303 = r;
        double r15304 = r15302 / r15303;
        double r15305 = r15304 / r15303;
        double r15306 = r15301 + r15305;
        double r15307 = 0.125;
        double r15308 = v;
        double r15309 = r15302 * r15308;
        double r15310 = r15301 - r15309;
        double r15311 = r15307 * r15310;
        double r15312 = w;
        double r15313 = r15312 * r15303;
        double r15314 = fabs(r15313);
        double r15315 = r15314 * r15314;
        double r15316 = 1.0;
        double r15317 = r15316 - r15308;
        double r15318 = r15315 / r15317;
        double r15319 = r15311 * r15318;
        double r15320 = sqrt(r15319);
        double r15321 = r15320 * r15320;
        double r15322 = r15306 - r15321;
        double r15323 = 4.5;
        double r15324 = r15322 - r15323;
        return r15324;
}

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 13.2

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

  3. Applied *-un-lft-identity13.2

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

  4. Applied times-frac8.9

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

  5. Simplified8.9

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

  7. Applied add-sqr-sqrt8.9

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

  8. Simplified8.9

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

  9. Simplified0.4

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

  11. Applied associate-/r*0.4

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

  13. Applied add-sqr-sqrt0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020020 
(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))