Average Error: 0.6 → 0.6
Time: 26.4s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{\left(1 - 5 \cdot {v}^{2}\right) \cdot \mathsf{fma}\left(1, \mathsf{fma}\left(v, v, 1\right), {v}^{4}\right)}{{v}^{6} - {1}^{3}}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)}\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{\left(1 - 5 \cdot {v}^{2}\right) \cdot \mathsf{fma}\left(1, \mathsf{fma}\left(v, v, 1\right), {v}^{4}\right)}{{v}^{6} - {1}^{3}}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)}\right)}
double f(double v) {
        double r143192 = 1.0;
        double r143193 = 5.0;
        double r143194 = v;
        double r143195 = r143194 * r143194;
        double r143196 = r143193 * r143195;
        double r143197 = r143192 - r143196;
        double r143198 = r143195 - r143192;
        double r143199 = r143197 / r143198;
        double r143200 = acos(r143199);
        return r143200;
}
↓
double f(double v) {
        double r143201 = 1.0;
        double r143202 = 5.0;
        double r143203 = v;
        double r143204 = 2.0;
        double r143205 = pow(r143203, r143204);
        double r143206 = r143202 * r143205;
        double r143207 = r143201 - r143206;
        double r143208 = fma(r143203, r143203, r143201);
        double r143209 = 4.0;
        double r143210 = pow(r143203, r143209);
        double r143211 = fma(r143201, r143208, r143210);
        double r143212 = r143207 * r143211;
        double r143213 = 6.0;
        double r143214 = pow(r143203, r143213);
        double r143215 = 3.0;
        double r143216 = pow(r143201, r143215);
        double r143217 = r143214 - r143216;
        double r143218 = r143212 / r143217;
        double r143219 = acos(r143218);
        double r143220 = log(r143219);
        double r143221 = sqrt(r143220);
        double r143222 = exp(r143221);
        double r143223 = r143207 / r143217;
        double r143224 = r143203 * r143203;
        double r143225 = r143201 + r143224;
        double r143226 = fma(r143201, r143225, r143210);
        double r143227 = r143223 * r143226;
        double r143228 = acos(r143227);
        double r143229 = log(r143228);
        double r143230 = sqrt(r143229);
        double r143231 = pow(r143222, r143230);
        return r143231;
}

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. Using strategy rm

  3. Applied flip3--0.6

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

  4. Applied associate-/r/0.6

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

  5. Simplified0.6

    \[\leadsto \cos^{-1} \left(\color{blue}{\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\]
  6. Using strategy rm

  7. Applied add-exp-log0.6

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)}}\]

  8. Simplified0.6

    \[\leadsto e^{\color{blue}{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)}}\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt0.6

    \[\leadsto e^{\color{blue}{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)}}}\]

  11. Applied exp-prod0.6

    \[\leadsto \color{blue}{{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)}\right)}}\]

  12. Simplified0.6

    \[\leadsto {\color{blue}{\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{\left(1 - 5 \cdot {v}^{2}\right) \cdot \mathsf{fma}\left(1, \mathsf{fma}\left(v, v, 1\right), {v}^{4}\right)}{{v}^{6} - {1}^{3}}\right)\right)}}\right)}}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)}\right)}\]

  13. Final simplification0.6

    \[\leadsto {\left(e^{\sqrt{\log \left(\cos^{-1} \left(\frac{\left(1 - 5 \cdot {v}^{2}\right) \cdot \mathsf{fma}\left(1, \mathsf{fma}\left(v, v, 1\right), {v}^{4}\right)}{{v}^{6} - {1}^{3}}\right)\right)}}\right)}^{\left(\sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \mathsf{fma}\left(1, 1 + v \cdot v, {v}^{4}\right)\right)\right)}\right)}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))