Average Error: 12.5 → 0.3
Time: 6.2s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\frac{0.375 - 0.25 \cdot v}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)\]
\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(\frac{0.375 - 0.25 \cdot v}{1 - v} \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 r15278 = 3.0;
        double r15279 = 2.0;
        double r15280 = r;
        double r15281 = r15280 * r15280;
        double r15282 = r15279 / r15281;
        double r15283 = r15278 + r15282;
        double r15284 = 0.125;
        double r15285 = v;
        double r15286 = r15279 * r15285;
        double r15287 = r15278 - r15286;
        double r15288 = r15284 * r15287;
        double r15289 = w;
        double r15290 = r15289 * r15289;
        double r15291 = r15290 * r15280;
        double r15292 = r15291 * r15280;
        double r15293 = r15288 * r15292;
        double r15294 = 1.0;
        double r15295 = r15294 - r15285;
        double r15296 = r15293 / r15295;
        double r15297 = r15283 - r15296;
        double r15298 = 4.5;
        double r15299 = r15297 - r15298;
        return r15299;
}
↓
double f(double v, double w, double r) {
        double r15300 = 2.0;
        double r15301 = r;
        double r15302 = r15300 / r15301;
        double r15303 = r15302 / r15301;
        double r15304 = 0.375;
        double r15305 = 0.25;
        double r15306 = v;
        double r15307 = r15305 * r15306;
        double r15308 = r15304 - r15307;
        double r15309 = 1.0;
        double r15310 = r15309 - r15306;
        double r15311 = r15308 / r15310;
        double r15312 = w;
        double r15313 = r15312 * r15301;
        double r15314 = fabs(r15313);
        double r15315 = r15314 * r15314;
        double r15316 = r15311 * r15315;
        double r15317 = 4.5;
        double r15318 = 3.0;
        double r15319 = r15317 - r15318;
        double r15320 = r15316 + r15319;
        double r15321 = r15303 - r15320;
        return r15321;
}

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 12.5

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

  2. Simplified8.2

    \[\leadsto \color{blue}{\frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left(\left(w \cdot w\right) \cdot r\right) \cdot r, 4.5\right) - 3\right)}\]
  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. Taylor expanded around 0 0.3

    \[\leadsto \frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{\color{blue}{0.375 - 0.25 \cdot v}}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
  8. Using strategy rm

  9. Applied fma-udef0.3

    \[\leadsto \frac{2}{r \cdot r} - \left(\color{blue}{\left(\frac{0.375 - 0.25 \cdot v}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + 4.5\right)} - 3\right)\]

  10. Applied associate--l+0.3

    \[\leadsto \frac{2}{r \cdot r} - \color{blue}{\left(\frac{0.375 - 0.25 \cdot v}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)}\]
  11. Using strategy rm

  12. Applied associate-/r*0.3

    \[\leadsto \color{blue}{\frac{\frac{2}{r}}{r}} - \left(\frac{0.375 - 0.25 \cdot v}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)\]

  13. Final simplification0.3

    \[\leadsto \frac{\frac{2}{r}}{r} - \left(\frac{0.375 - 0.25 \cdot v}{1 - v} \cdot \left(\left|w \cdot r\right| \cdot \left|w \cdot r\right|\right) + \left(4.5 - 3\right)\right)\]

Reproduce

herbie shell --seed 2020081 +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))