Average Error: 0.5 → 0.6
Time: 5.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\left(\sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}} \cdot \sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right) \cdot \sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\left(\sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}} \cdot \sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right) \cdot \sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right)
double f(double v) {
        double r235660 = 1.0;
        double r235661 = 5.0;
        double r235662 = v;
        double r235663 = r235662 * r235662;
        double r235664 = r235661 * r235663;
        double r235665 = r235660 - r235664;
        double r235666 = r235663 - r235660;
        double r235667 = r235665 / r235666;
        double r235668 = acos(r235667);
        return r235668;
}
↓
double f(double v) {
        double r235669 = atan2(1.0, 0.0);
        double r235670 = 2.0;
        double r235671 = r235669 / r235670;
        double r235672 = 1.0;
        double r235673 = 5.0;
        double r235674 = v;
        double r235675 = r235674 * r235674;
        double r235676 = r235673 * r235675;
        double r235677 = r235672 - r235676;
        double r235678 = r235675 - r235672;
        double r235679 = r235677 / r235678;
        double r235680 = cbrt(r235679);
        double r235681 = r235680 * r235680;
        double r235682 = r235681 * r235680;
        double r235683 = asin(r235682);
        double r235684 = r235671 - r235683;
        return r235684;
}

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 acos-asin0.5

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.6

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \color{blue}{\left(\left(\sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}} \cdot \sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right) \cdot \sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right)}\]

  6. Final simplification0.6

    \[\leadsto \frac{\pi}{2} - \sin^{-1} \left(\left(\sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}} \cdot \sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right) \cdot \sqrt[3]{\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}}\right)\]

Reproduce

herbie shell --seed 2019344 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))