Average Error: 0.6 → 0.7
Time: 22.8s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\mathsf{fma}\left(4, \mathsf{fma}\left(v \cdot v, v \cdot v, v \cdot v\right), -1\right)\right)\right)\right)\]
\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 \cdot v, v \cdot v, v \cdot v\right), -1\right)\right)\right)\right)
double f(double v) {
        double r5235074 = 1.0;
        double r5235075 = 5.0;
        double r5235076 = v;
        double r5235077 = r5235076 * r5235076;
        double r5235078 = r5235075 * r5235077;
        double r5235079 = r5235074 - r5235078;
        double r5235080 = r5235077 - r5235074;
        double r5235081 = r5235079 / r5235080;
        double r5235082 = acos(r5235081);
        return r5235082;
}
↓
double f(double v) {
        double r5235083 = 4.0;
        double r5235084 = v;
        double r5235085 = r5235084 * r5235084;
        double r5235086 = fma(r5235085, r5235085, r5235085);
        double r5235087 = -1.0;
        double r5235088 = fma(r5235083, r5235086, r5235087);
        double r5235089 = acos(r5235088);
        double r5235090 = log1p(r5235089);
        double r5235091 = expm1(r5235090);
        return r5235091;
}

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

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

  3. Taylor expanded around 0 0.7

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

  4. Simplified0.7

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

  6. Applied expm1-log1p-u0.7

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

  7. Final simplification0.7

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

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))