Average Error: 0.0 → 0.0
Time: 13.5s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[e^{\left(\log \left(\sqrt{1 - \left(v \cdot 3\right) \cdot v}\right) + \log \left(\sqrt{2}\right)\right) - \log 4} \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
e^{\left(\log \left(\sqrt{1 - \left(v \cdot 3\right) \cdot v}\right) + \log \left(\sqrt{2}\right)\right) - \log 4} \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r141025 = 2.0;
        double r141026 = sqrt(r141025);
        double r141027 = 4.0;
        double r141028 = r141026 / r141027;
        double r141029 = 1.0;
        double r141030 = 3.0;
        double r141031 = v;
        double r141032 = r141031 * r141031;
        double r141033 = r141030 * r141032;
        double r141034 = r141029 - r141033;
        double r141035 = sqrt(r141034);
        double r141036 = r141028 * r141035;
        double r141037 = r141029 - r141032;
        double r141038 = r141036 * r141037;
        return r141038;
}
↓
double f(double v) {
        double r141039 = 1.0;
        double r141040 = v;
        double r141041 = 3.0;
        double r141042 = r141040 * r141041;
        double r141043 = r141042 * r141040;
        double r141044 = r141039 - r141043;
        double r141045 = sqrt(r141044);
        double r141046 = log(r141045);
        double r141047 = 2.0;
        double r141048 = sqrt(r141047);
        double r141049 = log(r141048);
        double r141050 = r141046 + r141049;
        double r141051 = 4.0;
        double r141052 = log(r141051);
        double r141053 = r141050 - r141052;
        double r141054 = exp(r141053);
        double r141055 = r141040 * r141040;
        double r141056 = r141039 - r141055;
        double r141057 = r141054 * r141056;
        return r141057;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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-exp-log0.0

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

  4. Applied add-exp-log0.0

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

  5. Applied add-exp-log0.0

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

  6. Applied div-exp0.0

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

  7. Applied prod-exp0.0

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

  8. Simplified0.0

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

  10. Applied add-exp-log0.0

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

  11. Applied add-exp-log0.0

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

  12. Applied add-exp-log0.0

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

  13. Applied prod-exp0.0

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

  14. Applied div-exp0.0

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

  15. Applied rem-log-exp0.0

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

  16. Final simplification0.0

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

Reproduce

herbie shell --seed 2019194 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  (* (* (/ (sqrt 2.0) 4.0) (sqrt (- 1.0 (* 3.0 (* v v))))) (- 1.0 (* v v))))