Average Error: 0.6 → 0.6
Time: 21.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt[3]{\frac{\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)} \cdot \mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)} \cdot \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt[3]{\frac{\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)} \cdot \mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)} \cdot \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}}\right)
double f(double v) {
        double r10017737 = 1.0;
        double r10017738 = 5.0;
        double r10017739 = v;
        double r10017740 = r10017739 * r10017739;
        double r10017741 = r10017738 * r10017740;
        double r10017742 = r10017737 - r10017741;
        double r10017743 = r10017740 - r10017737;
        double r10017744 = r10017742 / r10017743;
        double r10017745 = acos(r10017744);
        return r10017745;
}
↓
double f(double v) {
        double r10017746 = v;
        double r10017747 = -5.0;
        double r10017748 = r10017746 * r10017747;
        double r10017749 = 1.0;
        double r10017750 = fma(r10017748, r10017746, r10017749);
        double r10017751 = -1.0;
        double r10017752 = fma(r10017746, r10017746, r10017751);
        double r10017753 = r10017750 / r10017752;
        double r10017754 = r10017753 * r10017750;
        double r10017755 = r10017754 / r10017752;
        double r10017756 = r10017755 * r10017753;
        double r10017757 = cbrt(r10017756);
        double r10017758 = acos(r10017757);
        return r10017758;
}

Error

Bits error versus v

Derivation

  1. Initial program 0.6

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

  2. Simplified0.6

    \[\leadsto \color{blue}{\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)}\]
  3. Using strategy rm

  4. Applied add-cbrt-cube0.6

    \[\leadsto \cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(v, v, -1\right) \cdot \mathsf{fma}\left(v, v, -1\right)\right) \cdot \mathsf{fma}\left(v, v, -1\right)}}}\right)\]

  5. Applied add-cbrt-cube0.6

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(-5 \cdot v, v, 1\right) \cdot \mathsf{fma}\left(-5 \cdot v, v, 1\right)\right) \cdot \mathsf{fma}\left(-5 \cdot v, v, 1\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(v, v, -1\right) \cdot \mathsf{fma}\left(v, v, -1\right)\right) \cdot \mathsf{fma}\left(v, v, -1\right)}}\right)\]

  6. Applied cbrt-undiv0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\sqrt[3]{\frac{\left(\mathsf{fma}\left(-5 \cdot v, v, 1\right) \cdot \mathsf{fma}\left(-5 \cdot v, v, 1\right)\right) \cdot \mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\left(\mathsf{fma}\left(v, v, -1\right) \cdot \mathsf{fma}\left(v, v, -1\right)\right) \cdot \mathsf{fma}\left(v, v, -1\right)}}\right)}\]

  7. Simplified0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\color{blue}{\left(\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)} \cdot \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right) \cdot \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}}}\right)\]
  8. Using strategy rm

  9. Applied associate-*l/0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\color{blue}{\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right) \cdot \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}}{\mathsf{fma}\left(v, v, -1\right)}} \cdot \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}}\right)\]

  10. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\frac{\frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)} \cdot \mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)} \cdot \frac{\mathsf{fma}\left(v \cdot -5, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}}\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))