Average Error: 0.5 → 0.8
Time: 25.9s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\right)\right)\]
\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(\mathsf{fma}\left(4, \mathsf{fma}\left(v, v, {v}^{4}\right), -1\right)\right)\right)\right)
double f(double v) {
        double r185208 = 1.0;
        double r185209 = 5.0;
        double r185210 = v;
        double r185211 = r185210 * r185210;
        double r185212 = r185209 * r185211;
        double r185213 = r185208 - r185212;
        double r185214 = r185211 - r185208;
        double r185215 = r185213 / r185214;
        double r185216 = acos(r185215);
        return r185216;
}
↓
double f(double v) {
        double r185217 = 4.0;
        double r185218 = v;
        double r185219 = 4.0;
        double r185220 = pow(r185218, r185219);
        double r185221 = fma(r185218, r185218, r185220);
        double r185222 = 1.0;
        double r185223 = -r185222;
        double r185224 = fma(r185217, r185221, r185223);
        double r185225 = acos(r185224);
        double r185226 = log1p(r185225);
        double r185227 = expm1(r185226);
        return r185227;
}

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 expm1-log1p-u0.5

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

  4. Taylor expanded around 0 0.8

    \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\left(4 \cdot {v}^{2} + 4 \cdot {v}^{4}\right) - 1\right)}\right)\right)\]

  5. Simplified0.8

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

  6. Final simplification0.8

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))