Average Error: 0.5 → 0.5
Time: 20.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)\]
\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{\mathsf{fma}\left(-5 \cdot v, v, 1\right)}{\mathsf{fma}\left(v, v, -1\right)}\right)\right)\right)
double f(double v) {
        double r3716787 = 1.0;
        double r3716788 = 5.0;
        double r3716789 = v;
        double r3716790 = r3716789 * r3716789;
        double r3716791 = r3716788 * r3716790;
        double r3716792 = r3716787 - r3716791;
        double r3716793 = r3716790 - r3716787;
        double r3716794 = r3716792 / r3716793;
        double r3716795 = acos(r3716794);
        return r3716795;
}
↓
double f(double v) {
        double r3716796 = -5.0;
        double r3716797 = v;
        double r3716798 = r3716796 * r3716797;
        double r3716799 = 1.0;
        double r3716800 = fma(r3716798, r3716797, r3716799);
        double r3716801 = -1.0;
        double r3716802 = fma(r3716797, r3716797, r3716801);
        double r3716803 = r3716800 / r3716802;
        double r3716804 = acos(r3716803);
        double r3716805 = log1p(r3716804);
        double r3716806 = expm1(r3716805);
        return r3716806;
}

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. Simplified0.5

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

  4. Applied expm1-log1p-u0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))