Average Error: 12.6 → 0.4
Time: 8.5s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[3 + \left(\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(\frac{0.375 - 0.25 \cdot v}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\right)\]
\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
↓
3 + \left(\frac{\frac{2}{r}}{r} - \mathsf{fma}\left(\frac{0.375 - 0.25 \cdot v}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\right)
double f(double v, double w, double r) {
        double r16250 = 3.0;
        double r16251 = 2.0;
        double r16252 = r;
        double r16253 = r16252 * r16252;
        double r16254 = r16251 / r16253;
        double r16255 = r16250 + r16254;
        double r16256 = 0.125;
        double r16257 = v;
        double r16258 = r16251 * r16257;
        double r16259 = r16250 - r16258;
        double r16260 = r16256 * r16259;
        double r16261 = w;
        double r16262 = r16261 * r16261;
        double r16263 = r16262 * r16252;
        double r16264 = r16263 * r16252;
        double r16265 = r16260 * r16264;
        double r16266 = 1.0;
        double r16267 = r16266 - r16257;
        double r16268 = r16265 / r16267;
        double r16269 = r16255 - r16268;
        double r16270 = 4.5;
        double r16271 = r16269 - r16270;
        return r16271;
}
↓
double f(double v, double w, double r) {
        double r16272 = 3.0;
        double r16273 = 2.0;
        double r16274 = r;
        double r16275 = r16273 / r16274;
        double r16276 = r16275 / r16274;
        double r16277 = 0.375;
        double r16278 = 0.25;
        double r16279 = v;
        double r16280 = r16278 * r16279;
        double r16281 = r16277 - r16280;
        double r16282 = 1.0;
        double r16283 = r16282 - r16279;
        double r16284 = r16281 / r16283;
        double r16285 = w;
        double r16286 = r16285 * r16274;
        double r16287 = fabs(r16286);
        double r16288 = r16287 * r16287;
        double r16289 = 4.5;
        double r16290 = fma(r16284, r16288, r16289);
        double r16291 = r16276 - r16290;
        double r16292 = r16272 + r16291;
        return r16292;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.6

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

  2. Simplified8.4

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

  4. Applied associate-*l*2.4

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

  6. Applied add-sqr-sqrt2.5

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

  7. Simplified2.5

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

  8. Simplified0.4

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

  9. Taylor expanded around 0 0.4

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

  11. Applied associate-/r*0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2020045 +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))