Average Error: 12.7 → 0.4
Time: 6.7s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{\frac{2}{r}}{r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \left|w \cdot r\right| \cdot \left|w \cdot r\right|, 4.5\right) - 3\right)\]
\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 r16134 = 3.0;
        double r16135 = 2.0;
        double r16136 = r;
        double r16137 = r16136 * r16136;
        double r16138 = r16135 / r16137;
        double r16139 = r16134 + r16138;
        double r16140 = 0.125;
        double r16141 = v;
        double r16142 = r16135 * r16141;
        double r16143 = r16134 - r16142;
        double r16144 = r16140 * r16143;
        double r16145 = w;
        double r16146 = r16145 * r16145;
        double r16147 = r16146 * r16136;
        double r16148 = r16147 * r16136;
        double r16149 = r16144 * r16148;
        double r16150 = 1.0;
        double r16151 = r16150 - r16141;
        double r16152 = r16149 / r16151;
        double r16153 = r16139 - r16152;
        double r16154 = 4.5;
        double r16155 = r16153 - r16154;
        return r16155;
}
↓
double f(double v, double w, double r) {
        double r16156 = 2.0;
        double r16157 = r;
        double r16158 = r16156 / r16157;
        double r16159 = r16158 / r16157;
        double r16160 = 0.125;
        double r16161 = 3.0;
        double r16162 = v;
        double r16163 = r16156 * r16162;
        double r16164 = r16161 - r16163;
        double r16165 = r16160 * r16164;
        double r16166 = 1.0;
        double r16167 = r16166 - r16162;
        double r16168 = r16165 / r16167;
        double r16169 = w;
        double r16170 = r16169 * r16157;
        double r16171 = fabs(r16170);
        double r16172 = r16171 * r16171;
        double r16173 = 4.5;
        double r16174 = fma(r16168, r16172, r16173);
        double r16175 = r16174 - r16161;
        double r16176 = r16159 - r16175;
        return r16176;
}

Error

Bits error versus v

Bits error versus w

Bits error versus r

Derivation

  1. Initial program 12.7

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

  2. Simplified8.5

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

  4. Applied add-sqr-sqrt8.6

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

  5. Simplified8.6

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

  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. Using strategy rm

  8. Applied associate-/r*0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020034 +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))