Average Error: 0.0 → 0.0
Time: 4.9s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[e^{\log \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right)} \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
e^{\log \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - \log \left(e^{3 \cdot \left(v \cdot v\right)}\right)}\right)} \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r221779 = 2.0;
        double r221780 = sqrt(r221779);
        double r221781 = 4.0;
        double r221782 = r221780 / r221781;
        double r221783 = 1.0;
        double r221784 = 3.0;
        double r221785 = v;
        double r221786 = r221785 * r221785;
        double r221787 = r221784 * r221786;
        double r221788 = r221783 - r221787;
        double r221789 = sqrt(r221788);
        double r221790 = r221782 * r221789;
        double r221791 = r221783 - r221786;
        double r221792 = r221790 * r221791;
        return r221792;
}
↓
double f(double v) {
        double r221793 = 2.0;
        double r221794 = sqrt(r221793);
        double r221795 = 4.0;
        double r221796 = r221794 / r221795;
        double r221797 = 1.0;
        double r221798 = 3.0;
        double r221799 = v;
        double r221800 = r221799 * r221799;
        double r221801 = r221798 * r221800;
        double r221802 = exp(r221801);
        double r221803 = log(r221802);
        double r221804 = r221797 - r221803;
        double r221805 = sqrt(r221804);
        double r221806 = r221796 * r221805;
        double r221807 = log(r221806);
        double r221808 = exp(r221807);
        double r221809 = r221797 - r221800;
        double r221810 = r221808 * r221809;
        return r221810;
}

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

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

  10. Applied add-log-exp0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2020024 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))