Average Error: 0.0 → 0.0
Time: 5.8s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\sqrt{2} \cdot \frac{1 - v \cdot v}{\frac{4}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\sqrt{2} \cdot \frac{1 - v \cdot v}{\frac{4}{\sqrt{1 - 3 \cdot \left(v \cdot v\right)}}}
double f(double v) {
        double r165791 = 2.0;
        double r165792 = sqrt(r165791);
        double r165793 = 4.0;
        double r165794 = r165792 / r165793;
        double r165795 = 1.0;
        double r165796 = 3.0;
        double r165797 = v;
        double r165798 = r165797 * r165797;
        double r165799 = r165796 * r165798;
        double r165800 = r165795 - r165799;
        double r165801 = sqrt(r165800);
        double r165802 = r165794 * r165801;
        double r165803 = r165795 - r165798;
        double r165804 = r165802 * r165803;
        return r165804;
}
↓
double f(double v) {
        double r165805 = 2.0;
        double r165806 = sqrt(r165805);
        double r165807 = 1.0;
        double r165808 = v;
        double r165809 = r165808 * r165808;
        double r165810 = r165807 - r165809;
        double r165811 = 4.0;
        double r165812 = 3.0;
        double r165813 = r165812 * r165809;
        double r165814 = r165807 - r165813;
        double r165815 = sqrt(r165814);
        double r165816 = r165811 / r165815;
        double r165817 = r165810 / r165816;
        double r165818 = r165806 * r165817;
        return r165818;
}

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 *-un-lft-identity0.0

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

  4. Applied add-cube-cbrt0.0

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

  5. Applied times-frac0.0

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

  6. Applied associate-*l*0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019308 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))