Average Error: 0.5 → 0.5
Time: 4.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(-1, 1, {v}^{4}\right)} \cdot \left(v \cdot v + 1\right)\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(-1, 1, {v}^{4}\right)} \cdot \left(v \cdot v + 1\right)\right)\right)\right)
double f(double v) {
        double r180197 = 1.0;
        double r180198 = 5.0;
        double r180199 = v;
        double r180200 = r180199 * r180199;
        double r180201 = r180198 * r180200;
        double r180202 = r180197 - r180201;
        double r180203 = r180200 - r180197;
        double r180204 = r180202 / r180203;
        double r180205 = acos(r180204);
        return r180205;
}
↓
double f(double v) {
        double r180206 = 1.0;
        double r180207 = 5.0;
        double r180208 = v;
        double r180209 = r180208 * r180208;
        double r180210 = r180207 * r180209;
        double r180211 = r180206 - r180210;
        double r180212 = -r180206;
        double r180213 = 4.0;
        double r180214 = pow(r180208, r180213);
        double r180215 = fma(r180212, r180206, r180214);
        double r180216 = r180211 / r180215;
        double r180217 = r180209 + r180206;
        double r180218 = r180216 * r180217;
        double r180219 = acos(r180218);
        double r180220 = log1p(r180219);
        double r180221 = expm1(r180220);
        return r180221;
}

Error

Bits error versus v

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 flip--0.5

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

  4. Applied associate-/r/0.5

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

  5. Simplified0.5

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

  7. Applied expm1-log1p-u0.5

    \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(-1, 1, {v}^{4}\right)} \cdot \left(v \cdot v + 1\right)\right)\right)\right)}\]

  8. Final simplification0.5

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\mathsf{fma}\left(-1, 1, {v}^{4}\right)} \cdot \left(v \cdot v + 1\right)\right)\right)\right)\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))