Average Error: 0.0 → 0.0
Time: 4.1s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(v \cdot v, v \cdot v, 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right)}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(v \cdot v, v \cdot v, 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right)}
double f(double v) {
        double r267964 = 2.0;
        double r267965 = sqrt(r267964);
        double r267966 = 4.0;
        double r267967 = r267965 / r267966;
        double r267968 = 1.0;
        double r267969 = 3.0;
        double r267970 = v;
        double r267971 = r267970 * r267970;
        double r267972 = r267969 * r267971;
        double r267973 = r267968 - r267972;
        double r267974 = sqrt(r267973);
        double r267975 = r267967 * r267974;
        double r267976 = r267968 - r267971;
        double r267977 = r267975 * r267976;
        return r267977;
}
↓
double f(double v) {
        double r267978 = 2.0;
        double r267979 = sqrt(r267978);
        double r267980 = 1.0;
        double r267981 = r267980 * r267980;
        double r267982 = 3.0;
        double r267983 = v;
        double r267984 = r267983 * r267983;
        double r267985 = r267982 * r267984;
        double r267986 = r267985 * r267985;
        double r267987 = r267981 - r267986;
        double r267988 = sqrt(r267987);
        double r267989 = r267979 * r267988;
        double r267990 = 3.0;
        double r267991 = pow(r267980, r267990);
        double r267992 = pow(r267984, r267990);
        double r267993 = r267991 - r267992;
        double r267994 = r267989 * r267993;
        double r267995 = r267980 * r267984;
        double r267996 = fma(r267984, r267984, r267995);
        double r267997 = fma(r267980, r267980, r267996);
        double r267998 = 4.0;
        double r267999 = r267980 + r267985;
        double r268000 = sqrt(r267999);
        double r268001 = r267998 * r268000;
        double r268002 = r267997 * r268001;
        double r268003 = r267994 / r268002;
        return r268003;
}

Error

Bits error versus v

Derivation

  1. Initial program 0.0

    \[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
  2. Using strategy rm

  3. Applied flip3--0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \color{blue}{\frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}}\]

  4. Applied flip--0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{\color{blue}{\frac{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}\]

  5. Applied sqrt-div0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \color{blue}{\frac{\sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}{\sqrt{1 + 3 \cdot \left(v \cdot v\right)}}}\right) \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}\]

  6. Applied frac-times0.0

    \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}}{4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}}} \cdot \frac{{1}^{3} - {\left(v \cdot v\right)}^{3}}{1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)}\]

  7. Applied frac-times0.0

    \[\leadsto \color{blue}{\frac{\left(\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\left(4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)}}\]

  8. Simplified0.0

    \[\leadsto \frac{\left(\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\color{blue}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(v \cdot v, v \cdot v, 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right)}}\]

  9. Final simplification0.0

    \[\leadsto \frac{\left(\sqrt{2} \cdot \sqrt{1 \cdot 1 - \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left({1}^{3} - {\left(v \cdot v\right)}^{3}\right)}{\mathsf{fma}\left(1, 1, \mathsf{fma}\left(v \cdot v, v \cdot v, 1 \cdot \left(v \cdot v\right)\right)\right) \cdot \left(4 \cdot \sqrt{1 + 3 \cdot \left(v \cdot v\right)}\right)}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))