Average Error: 0.0 → 0.0
Time: 22.2s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{1 \cdot 1 - {v}^{4}}{\frac{1 + v \cdot v}{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}\]
\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}^{4}}{\frac{1 + v \cdot v}{\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}
double f(double v) {
        double r265689 = 2.0;
        double r265690 = sqrt(r265689);
        double r265691 = 4.0;
        double r265692 = r265690 / r265691;
        double r265693 = 1.0;
        double r265694 = 3.0;
        double r265695 = v;
        double r265696 = r265695 * r265695;
        double r265697 = r265694 * r265696;
        double r265698 = r265693 - r265697;
        double r265699 = sqrt(r265698);
        double r265700 = r265692 * r265699;
        double r265701 = r265693 - r265696;
        double r265702 = r265700 * r265701;
        return r265702;
}
↓
double f(double v) {
        double r265703 = 1.0;
        double r265704 = r265703 * r265703;
        double r265705 = v;
        double r265706 = 4.0;
        double r265707 = pow(r265705, r265706);
        double r265708 = r265704 - r265707;
        double r265709 = r265705 * r265705;
        double r265710 = r265703 + r265709;
        double r265711 = 2.0;
        double r265712 = sqrt(r265711);
        double r265713 = 4.0;
        double r265714 = r265712 / r265713;
        double r265715 = 3.0;
        double r265716 = r265715 * r265709;
        double r265717 = r265703 - r265716;
        double r265718 = sqrt(r265717);
        double r265719 = r265714 * r265718;
        double r265720 = r265710 / r265719;
        double r265721 = r265708 / r265720;
        return r265721;
}

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

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

  7. Applied associate-/l*0.0

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

  8. Final simplification0.0

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

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))))