Average Error: 0.0 → 0.0
Time: 42.4s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[e^{\log \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) + \log \left(1 - v \cdot v\right)}\]
\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 - 3 \cdot \left(v \cdot v\right)}\right) + \log \left(1 - v \cdot v\right)}
double f(double v) {
        double r430742 = 2.0;
        double r430743 = sqrt(r430742);
        double r430744 = 4.0;
        double r430745 = r430743 / r430744;
        double r430746 = 1.0;
        double r430747 = 3.0;
        double r430748 = v;
        double r430749 = r430748 * r430748;
        double r430750 = r430747 * r430749;
        double r430751 = r430746 - r430750;
        double r430752 = sqrt(r430751);
        double r430753 = r430745 * r430752;
        double r430754 = r430746 - r430749;
        double r430755 = r430753 * r430754;
        return r430755;
}
↓
double f(double v) {
        double r430756 = 2.0;
        double r430757 = sqrt(r430756);
        double r430758 = 4.0;
        double r430759 = r430757 / r430758;
        double r430760 = 1.0;
        double r430761 = 3.0;
        double r430762 = v;
        double r430763 = r430762 * r430762;
        double r430764 = r430761 * r430763;
        double r430765 = r430760 - r430764;
        double r430766 = sqrt(r430765);
        double r430767 = r430759 * r430766;
        double r430768 = log(r430767);
        double r430769 = r430760 - r430763;
        double r430770 = log(r430769);
        double r430771 = r430768 + r430770;
        double r430772 = exp(r430771);
        return r430772;
}

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-sqr-sqrt0.0

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

  4. Applied sqrt-prod0.0

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

  5. Applied associate-*r*0.0

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

  7. Applied add-exp-log0.0

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

  8. Applied add-exp-log0.0

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

  9. Applied add-exp-log0.0

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

  10. Applied add-exp-log0.0

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

  11. Applied add-exp-log0.0

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

  12. Applied div-exp0.0

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

  13. Applied prod-exp0.0

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

  14. Applied prod-exp0.0

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

  15. Applied prod-exp0.0

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

  16. Simplified0.0

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

  17. Final simplification0.0

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

Reproduce

herbie shell --seed 2019356 +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))))