Average Error: 0.0 → 0.0
Time: 4.0s
Precision: 64
\[\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)\]
↓
\[\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot 1 + \sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(-v \cdot v\right)\right)\]
\left(\frac{\sqrt{2}}{4} \cdot \sqrt{1 - 3 \cdot \left(v \cdot v\right)}\right) \cdot \left(1 - v \cdot v\right)
↓
\frac{\sqrt{2}}{4} \cdot \left(\sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot 1 + \sqrt{1 - 3 \cdot \left(v \cdot v\right)} \cdot \left(-v \cdot v\right)\right)
double f(double v) {
        double r173785 = 2.0;
        double r173786 = sqrt(r173785);
        double r173787 = 4.0;
        double r173788 = r173786 / r173787;
        double r173789 = 1.0;
        double r173790 = 3.0;
        double r173791 = v;
        double r173792 = r173791 * r173791;
        double r173793 = r173790 * r173792;
        double r173794 = r173789 - r173793;
        double r173795 = sqrt(r173794);
        double r173796 = r173788 * r173795;
        double r173797 = r173789 - r173792;
        double r173798 = r173796 * r173797;
        return r173798;
}
↓
double f(double v) {
        double r173799 = 2.0;
        double r173800 = sqrt(r173799);
        double r173801 = 4.0;
        double r173802 = r173800 / r173801;
        double r173803 = 1.0;
        double r173804 = 3.0;
        double r173805 = v;
        double r173806 = r173805 * r173805;
        double r173807 = r173804 * r173806;
        double r173808 = r173803 - r173807;
        double r173809 = sqrt(r173808);
        double r173810 = r173809 * r173803;
        double r173811 = -r173806;
        double r173812 = r173809 * r173811;
        double r173813 = r173810 + r173812;
        double r173814 = r173802 * r173813;
        return r173814;
}

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 associate-*l*0.0

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

  5. Applied sub-neg0.0

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

  6. Applied distribute-lft-in0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020002 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 2"
  :precision binary64
  (* (* (/ (sqrt 2) 4) (sqrt (- 1 (* 3 (* v v))))) (- 1 (* v v))))