Average Error: 0.0 → 0.0
Time: 19.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(1 + 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\left(\frac{\sqrt{2}}{4} \cdot \frac{\sqrt{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}}{\sqrt{1 \cdot 1 + \left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(1 + 3 \cdot \left(v \cdot v\right)\right)}}\right) \cdot \left(1 - v \cdot v\right)
double f(double v) {
        double r141688 = 2.0;
        double r141689 = sqrt(r141688);
        double r141690 = 4.0;
        double r141691 = r141689 / r141690;
        double r141692 = 1.0;
        double r141693 = 3.0;
        double r141694 = v;
        double r141695 = r141694 * r141694;
        double r141696 = r141693 * r141695;
        double r141697 = r141692 - r141696;
        double r141698 = sqrt(r141697);
        double r141699 = r141691 * r141698;
        double r141700 = r141692 - r141695;
        double r141701 = r141699 * r141700;
        return r141701;
}
↓
double f(double v) {
        double r141702 = 2.0;
        double r141703 = sqrt(r141702);
        double r141704 = 4.0;
        double r141705 = r141703 / r141704;
        double r141706 = 1.0;
        double r141707 = 3.0;
        double r141708 = pow(r141706, r141707);
        double r141709 = 3.0;
        double r141710 = v;
        double r141711 = r141710 * r141710;
        double r141712 = r141709 * r141711;
        double r141713 = pow(r141712, r141707);
        double r141714 = r141708 - r141713;
        double r141715 = sqrt(r141714);
        double r141716 = r141706 * r141706;
        double r141717 = r141706 + r141712;
        double r141718 = r141712 * r141717;
        double r141719 = r141716 + r141718;
        double r141720 = sqrt(r141719);
        double r141721 = r141715 / r141720;
        double r141722 = r141705 * r141721;
        double r141723 = r141706 - r141711;
        double r141724 = r141722 * r141723;
        return r141724;
}

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. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))