Average Error: 0.5 → 0.6
Time: 11.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)
double f(double v) {
        double r340647 = 1.0;
        double r340648 = 5.0;
        double r340649 = v;
        double r340650 = r340649 * r340649;
        double r340651 = r340648 * r340650;
        double r340652 = r340647 - r340651;
        double r340653 = r340650 - r340647;
        double r340654 = r340652 / r340653;
        double r340655 = acos(r340654);
        return r340655;
}
↓
double f(double v) {
        double r340656 = 1.0;
        double r340657 = 5.0;
        double r340658 = v;
        double r340659 = r340658 * r340658;
        double r340660 = r340657 * r340659;
        double r340661 = r340656 - r340660;
        double r340662 = sqrt(r340661);
        double r340663 = r340659 - r340656;
        double r340664 = r340662 / r340663;
        double r340665 = r340662 * r340664;
        double r340666 = acos(r340665);
        return r340666;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.5

    \[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.5

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{1 \cdot \left(v \cdot v - 1\right)}}\right)\]

  4. Applied add-sqr-sqrt0.6

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \sqrt{1 - 5 \cdot \left(v \cdot v\right)}}}{1 \cdot \left(v \cdot v - 1\right)}\right)\]

  5. Applied times-frac0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{1} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)}\]

  6. Simplified0.6

    \[\leadsto \cos^{-1} \left(\color{blue}{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]

  7. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\sqrt{1 - 5 \cdot \left(v \cdot v\right)} \cdot \frac{\sqrt{1 - 5 \cdot \left(v \cdot v\right)}}{v \cdot v - 1}\right)\]

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))