Average Error: 0.0 → 0.0
Time: 24.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{1 \cdot 1 - v \cdot {v}^{3}}{\frac{1 + v \cdot v}{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{1 \cdot 1 - v \cdot {v}^{3}}{\frac{1 + v \cdot v}{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}
double f(double v) {
        double r226885 = 2.0;
        double r226886 = sqrt(r226885);
        double r226887 = 4.0;
        double r226888 = r226886 / r226887;
        double r226889 = 1.0;
        double r226890 = 3.0;
        double r226891 = v;
        double r226892 = r226891 * r226891;
        double r226893 = r226890 * r226892;
        double r226894 = r226889 - r226893;
        double r226895 = sqrt(r226894);
        double r226896 = r226888 * r226895;
        double r226897 = r226889 - r226892;
        double r226898 = r226896 * r226897;
        return r226898;
}
↓
double f(double v) {
        double r226899 = 1.0;
        double r226900 = r226899 * r226899;
        double r226901 = v;
        double r226902 = 3.0;
        double r226903 = pow(r226901, r226902);
        double r226904 = r226901 * r226903;
        double r226905 = r226900 - r226904;
        double r226906 = r226901 * r226901;
        double r226907 = r226899 + r226906;
        double r226908 = 2.0;
        double r226909 = sqrt(r226908);
        double r226910 = 4.0;
        double r226911 = r226909 / r226910;
        double r226912 = 3.0;
        double r226913 = r226912 * r226906;
        double r226914 = r226899 - r226913;
        double r226915 = sqrt(r226914);
        double r226916 = r226911 * r226915;
        double r226917 = r226907 / r226916;
        double r226918 = r226905 / r226917;
        return r226918;
}

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 flip--0.0

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

  4. Applied associate-*r/0.0

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

  5. Simplified0.0

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

  7. Applied associate-/l*0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))