Average Error: 12.9 → 0.3
Time: 5.6s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
\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(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)
double f(double v, double w, double r) {
        double r19242 = 3.0;
        double r19243 = 2.0;
        double r19244 = r;
        double r19245 = r19244 * r19244;
        double r19246 = r19243 / r19245;
        double r19247 = r19242 + r19246;
        double r19248 = 0.125;
        double r19249 = v;
        double r19250 = r19243 * r19249;
        double r19251 = r19242 - r19250;
        double r19252 = r19248 * r19251;
        double r19253 = w;
        double r19254 = r19253 * r19253;
        double r19255 = r19254 * r19244;
        double r19256 = r19255 * r19244;
        double r19257 = r19252 * r19256;
        double r19258 = 1.0;
        double r19259 = r19258 - r19249;
        double r19260 = r19257 / r19259;
        double r19261 = r19247 - r19260;
        double r19262 = 4.5;
        double r19263 = r19261 - r19262;
        return r19263;
}
↓
double f(double v, double w, double r) {
        double r19264 = 2.0;
        double r19265 = r;
        double r19266 = r19264 / r19265;
        double r19267 = r19266 / r19265;
        double r19268 = 0.125;
        double r19269 = 3.0;
        double r19270 = v;
        double r19271 = r19264 * r19270;
        double r19272 = r19269 - r19271;
        double r19273 = r19268 * r19272;
        double r19274 = 1.0;
        double r19275 = r19274 - r19270;
        double r19276 = r19273 / r19275;
        double r19277 = w;
        double r19278 = r19277 * r19265;
        double r19279 = fabs(r19278);
        double r19280 = r19279 * r19279;
        double r19281 = 4.5;
        double r19282 = fma(r19276, r19280, r19281);
        double r19283 = r19282 - r19269;
        double r19284 = r19267 - r19283;
        return r19284;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.9

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

  2. Simplified8.7

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

  4. Applied add-sqr-sqrt8.7

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

  5. Simplified8.7

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

  6. Simplified0.3

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

  8. Applied associate-/r*0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2020064 +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))