Average Error: 12.9 → 0.4
Time: 12.7s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\left(3 + \frac{\frac{2}{r}}{r}\right) - \mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right)\]
\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(3 + \frac{\frac{2}{r}}{r}\right) - \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)
double f(double v, double w, double r) {
        double r17107 = 3.0;
        double r17108 = 2.0;
        double r17109 = r;
        double r17110 = r17109 * r17109;
        double r17111 = r17108 / r17110;
        double r17112 = r17107 + r17111;
        double r17113 = 0.125;
        double r17114 = v;
        double r17115 = r17108 * r17114;
        double r17116 = r17107 - r17115;
        double r17117 = r17113 * r17116;
        double r17118 = w;
        double r17119 = r17118 * r17118;
        double r17120 = r17119 * r17109;
        double r17121 = r17120 * r17109;
        double r17122 = r17117 * r17121;
        double r17123 = 1.0;
        double r17124 = r17123 - r17114;
        double r17125 = r17122 / r17124;
        double r17126 = r17112 - r17125;
        double r17127 = 4.5;
        double r17128 = r17126 - r17127;
        return r17128;
}
↓
double f(double v, double w, double r) {
        double r17129 = 3.0;
        double r17130 = 2.0;
        double r17131 = r;
        double r17132 = r17130 / r17131;
        double r17133 = r17132 / r17131;
        double r17134 = r17129 + r17133;
        double r17135 = 0.125;
        double r17136 = v;
        double r17137 = r17130 * r17136;
        double r17138 = r17129 - r17137;
        double r17139 = r17135 * r17138;
        double r17140 = 1.0;
        double r17141 = r17140 - r17136;
        double r17142 = r17139 / r17141;
        double r17143 = w;
        double r17144 = r17143 * r17131;
        double r17145 = fabs(r17144);
        double r17146 = r17145 * r17145;
        double r17147 = 4.5;
        double r17148 = fma(r17142, r17146, r17147);
        double r17149 = r17134 - r17148;
        return r17149;
}

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. Simplified8.4

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

  4. Applied add-sqr-sqrt8.4

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

  5. Simplified8.4

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019351 +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))