Average Error: 0.0 → 0.0
Time: 10.5s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[1 \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) + \left(-v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
1 \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) + \left(-v \cdot v\right) \cdot \left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right)
double f(double v) {
        double r208796 = 2.0;
        double r208797 = sqrt(r208796);
        double r208798 = 4.0;
        double r208799 = r208797 / r208798;
        double r208800 = 1.0;
        double r208801 = 3.0;
        double r208802 = v;
        double r208803 = r208802 * r208802;
        double r208804 = r208801 * r208803;
        double r208805 = r208800 - r208804;
        double r208806 = sqrt(r208805);
        double r208807 = r208799 * r208806;
        double r208808 = r208800 - r208803;
        double r208809 = r208807 * r208808;
        return r208809;
}
↓
double f(double v) {
        double r208810 = 1.0;
        double r208811 = 2.0;
        double r208812 = sqrt(r208811);
        double r208813 = 4.0;
        double r208814 = r208812 / r208813;
        double r208815 = 3.0;
        double r208816 = v;
        double r208817 = r208816 * r208816;
        double r208818 = r208815 * r208817;
        double r208819 = r208810 - r208818;
        double r208820 = sqrt(r208819);
        double r208821 = r208814 * r208820;
        double r208822 = r208810 * r208821;
        double r208823 = -r208817;
        double r208824 = r208823 * r208821;
        double r208825 = r208822 + r208824;
        return r208825;
}

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 sub-neg0.0

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

  4. Applied distribute-lft-in0.0

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

  5. Simplified0.0

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020043 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))