Average Error: 12.9 → 0.4
Time: 4.9s
Precision: 64
\[\left(\left(3 + \frac{2}{r \cdot r}\right) - \frac{\left(0.125 \cdot \left(3 - 2 \cdot v\right)\right) \cdot \left(\left(\left(w \cdot w\right) \cdot r\right) \cdot r\right)}{1 - v}\right) - 4.5\]
↓
\[\frac{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \sqrt{\left|w \cdot r\right|} \cdot {\left(\left|w \cdot r\right|\right)}^{\frac{3}{2}}, 4.5\right) - 3\right)\]
\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{2}{r \cdot r} - \left(\mathsf{fma}\left(\frac{0.125 \cdot \left(3 - 2 \cdot v\right)}{1 - v}, \sqrt{\left|w \cdot r\right|} \cdot {\left(\left|w \cdot r\right|\right)}^{\frac{3}{2}}, 4.5\right) - 3\right)
double f(double v, double w, double r) {
        double r14981 = 3.0;
        double r14982 = 2.0;
        double r14983 = r;
        double r14984 = r14983 * r14983;
        double r14985 = r14982 / r14984;
        double r14986 = r14981 + r14985;
        double r14987 = 0.125;
        double r14988 = v;
        double r14989 = r14982 * r14988;
        double r14990 = r14981 - r14989;
        double r14991 = r14987 * r14990;
        double r14992 = w;
        double r14993 = r14992 * r14992;
        double r14994 = r14993 * r14983;
        double r14995 = r14994 * r14983;
        double r14996 = r14991 * r14995;
        double r14997 = 1.0;
        double r14998 = r14997 - r14988;
        double r14999 = r14996 / r14998;
        double r15000 = r14986 - r14999;
        double r15001 = 4.5;
        double r15002 = r15000 - r15001;
        return r15002;
}
↓
double f(double v, double w, double r) {
        double r15003 = 2.0;
        double r15004 = r;
        double r15005 = r15004 * r15004;
        double r15006 = r15003 / r15005;
        double r15007 = 0.125;
        double r15008 = 3.0;
        double r15009 = v;
        double r15010 = r15003 * r15009;
        double r15011 = r15008 - r15010;
        double r15012 = r15007 * r15011;
        double r15013 = 1.0;
        double r15014 = r15013 - r15009;
        double r15015 = r15012 / r15014;
        double r15016 = w;
        double r15017 = r15016 * r15004;
        double r15018 = fabs(r15017);
        double r15019 = sqrt(r15018);
        double r15020 = 1.5;
        double r15021 = pow(r15018, r15020);
        double r15022 = r15019 * r15021;
        double r15023 = 4.5;
        double r15024 = fma(r15015, r15022, r15023);
        double r15025 = r15024 - r15008;
        double r15026 = r15006 - r15025;
        return r15026;
}

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. Simplified8.5

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

  4. Applied add-sqr-sqrt8.5

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

  5. Simplified8.5

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

  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 add-sqr-sqrt0.4

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

  9. Applied associate-*l*0.4

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

  10. Simplified0.5

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

  12. Applied pow1/20.5

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

  13. Applied pow-pow0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2020039 +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))