Average Error: 0.0 → 0.0
Time: 4.7s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - v \cdot v}\right) \cdot \sqrt{1 - v \cdot v}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \sqrt{1 - v \cdot v}\right) \cdot \sqrt{1 - v \cdot v}
double f(double v) {
        double r184864 = 2.0;
        double r184865 = sqrt(r184864);
        double r184866 = 4.0;
        double r184867 = r184865 / r184866;
        double r184868 = 1.0;
        double r184869 = 3.0;
        double r184870 = v;
        double r184871 = r184870 * r184870;
        double r184872 = r184869 * r184871;
        double r184873 = r184868 - r184872;
        double r184874 = sqrt(r184873);
        double r184875 = r184867 * r184874;
        double r184876 = r184868 - r184871;
        double r184877 = r184875 * r184876;
        return r184877;
}
↓
double f(double v) {
        double r184878 = 2.0;
        double r184879 = sqrt(r184878);
        double r184880 = 4.0;
        double r184881 = r184879 / r184880;
        double r184882 = 1.0;
        double r184883 = 3.0;
        double r184884 = v;
        double r184885 = r184884 * r184884;
        double r184886 = r184883 * r184885;
        double r184887 = r184882 - r184886;
        double r184888 = sqrt(r184887);
        double r184889 = r184881 * r184888;
        double r184890 = r184882 - r184885;
        double r184891 = sqrt(r184890);
        double r184892 = r184889 * r184891;
        double r184893 = r184892 * r184891;
        return r184893;
}

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

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

  4. Applied associate-*r*0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020100 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))