Average Error: 0.6 → 0.6
Time: 5.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(1 \cdot \mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(1 \cdot \mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)\right)
double f(double v) {
        double r227699 = 1.0;
        double r227700 = 5.0;
        double r227701 = v;
        double r227702 = r227701 * r227701;
        double r227703 = r227700 * r227702;
        double r227704 = r227699 - r227703;
        double r227705 = r227702 - r227699;
        double r227706 = r227704 / r227705;
        double r227707 = acos(r227706);
        return r227707;
}
↓
double f(double v) {
        double r227708 = 1.0;
        double r227709 = 1.0;
        double r227710 = 5.0;
        double r227711 = v;
        double r227712 = r227711 * r227711;
        double r227713 = r227710 * r227712;
        double r227714 = r227709 - r227713;
        double r227715 = r227712 - r227709;
        double r227716 = r227714 / r227715;
        double r227717 = acos(r227716);
        double r227718 = log1p(r227717);
        double r227719 = r227708 * r227718;
        double r227720 = expm1(r227719);
        return r227720;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 expm1-log1p-u0.6

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

  5. Applied *-un-lft-identity0.6

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

  6. Final simplification0.6

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))