Average Error: 0.0 → 0.0
Time: 22.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt[3]{{\left(\frac{\sqrt{2}}{\frac{4}{\mathsf{fma}\left(v, -v, 1\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}\right)}^{3}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt[3]{{\left(\frac{\sqrt{2}}{\frac{4}{\mathsf{fma}\left(v, -v, 1\right) \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}\right)}^{3}}
double f(double v) {
        double r212931 = 2.0;
        double r212932 = sqrt(r212931);
        double r212933 = 4.0;
        double r212934 = r212932 / r212933;
        double r212935 = 1.0;
        double r212936 = 3.0;
        double r212937 = v;
        double r212938 = r212937 * r212937;
        double r212939 = r212936 * r212938;
        double r212940 = r212935 - r212939;
        double r212941 = sqrt(r212940);
        double r212942 = r212934 * r212941;
        double r212943 = r212935 - r212938;
        double r212944 = r212942 * r212943;
        return r212944;
}
↓
double f(double v) {
        double r212945 = 2.0;
        double r212946 = sqrt(r212945);
        double r212947 = 4.0;
        double r212948 = v;
        double r212949 = -r212948;
        double r212950 = 1.0;
        double r212951 = fma(r212948, r212949, r212950);
        double r212952 = 3.0;
        double r212953 = r212948 * r212948;
        double r212954 = r212952 * r212953;
        double r212955 = r212950 - r212954;
        double r212956 = sqrt(r212955);
        double r212957 = r212951 * r212956;
        double r212958 = r212947 / r212957;
        double r212959 = r212946 / r212958;
        double r212960 = 3.0;
        double r212961 = pow(r212959, r212960);
        double r212962 = cbrt(r212961);
        return r212962;
}

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 add-log-exp0.0

    \[\leadsto \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \color{blue}{\log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}}\right) \cdot \left(1 - v \cdot v\right)\]
  4. Using strategy rm

  5. Applied add-cbrt-cube0.0

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

  6. Applied add-cbrt-cube0.0

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

  7. Applied add-cbrt-cube0.0

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

  8. Applied cbrt-unprod0.0

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

  9. Applied cbrt-unprod0.0

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

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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))))