Average Error: 0.6 → 0.6
Time: 23.9s
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 r7479164 = 1.0;
        double r7479165 = 5.0;
        double r7479166 = v;
        double r7479167 = r7479166 * r7479166;
        double r7479168 = r7479165 * r7479167;
        double r7479169 = r7479164 - r7479168;
        double r7479170 = r7479167 - r7479164;
        double r7479171 = r7479169 / r7479170;
        double r7479172 = acos(r7479171);
        return r7479172;
}
↓
double f(double v) {
        double r7479173 = v;
        double r7479174 = -5.0;
        double r7479175 = r7479173 * r7479174;
        double r7479176 = 1.0;
        double r7479177 = fma(r7479175, r7479173, r7479176);
        double r7479178 = -1.0;
        double r7479179 = fma(r7479173, r7479173, r7479178);
        double r7479180 = r7479177 / r7479179;
        double r7479181 = r7479180 * r7479177;
        double r7479182 = r7479181 / r7479179;
        double r7479183 = r7479182 * r7479180;
        double r7479184 = cbrt(r7479183);
        double r7479185 = acos(r7479184);
        return r7479185;
}

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))))