Average Error: 0.0 → 0.0
Time: 13.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left(1 - v \cdot v\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\left(\frac{\sqrt{2}}{4} \cdot \sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}\right) \cdot \left(1 - v \cdot v\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}
double f(double v) {
        double r193906 = 2.0;
        double r193907 = sqrt(r193906);
        double r193908 = 4.0;
        double r193909 = r193907 / r193908;
        double r193910 = 1.0;
        double r193911 = 3.0;
        double r193912 = v;
        double r193913 = r193912 * r193912;
        double r193914 = r193911 * r193913;
        double r193915 = r193910 - r193914;
        double r193916 = sqrt(r193915);
        double r193917 = r193909 * r193916;
        double r193918 = r193910 - r193913;
        double r193919 = r193917 * r193918;
        return r193919;
}
↓
double f(double v) {
        double r193920 = 2.0;
        double r193921 = sqrt(r193920);
        double r193922 = 4.0;
        double r193923 = r193921 / r193922;
        double r193924 = 1.0;
        double r193925 = 3.0;
        double r193926 = pow(r193924, r193925);
        double r193927 = 3.0;
        double r193928 = v;
        double r193929 = r193928 * r193928;
        double r193930 = r193927 * r193929;
        double r193931 = pow(r193930, r193925);
        double r193932 = r193926 - r193931;
        double r193933 = sqrt(r193932);
        double r193934 = r193923 * r193933;
        double r193935 = r193924 - r193929;
        double r193936 = r193934 * r193935;
        double r193937 = r193924 * r193924;
        double r193938 = r193930 * r193930;
        double r193939 = r193924 * r193930;
        double r193940 = r193938 + r193939;
        double r193941 = r193937 + r193940;
        double r193942 = sqrt(r193941);
        double r193943 = r193936 / r193942;
        return r193943;
}

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 flip3--0.0

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

  4. Applied sqrt-div0.0

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

  5. Applied associate-*r/0.0

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

  6. Applied associate-*l/0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019350 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))