Average Error: 0.6 → 0.6
Time: 17.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{v \cdot v - 1} - \frac{5}{1 - \frac{1}{v \cdot v}}\right)\right)} - 1\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\mathsf{log1p}\left(\cos^{-1} \left(\frac{1}{v \cdot v - 1} - \frac{5}{1 - \frac{1}{v \cdot v}}\right)\right)} - 1
double f(double v) {
        double r252543 = 1.0;
        double r252544 = 5.0;
        double r252545 = v;
        double r252546 = r252545 * r252545;
        double r252547 = r252544 * r252546;
        double r252548 = r252543 - r252547;
        double r252549 = r252546 - r252543;
        double r252550 = r252548 / r252549;
        double r252551 = acos(r252550);
        return r252551;
}
↓
double f(double v) {
        double r252552 = 1.0;
        double r252553 = v;
        double r252554 = r252553 * r252553;
        double r252555 = r252554 - r252552;
        double r252556 = r252552 / r252555;
        double r252557 = 5.0;
        double r252558 = 1.0;
        double r252559 = r252552 / r252554;
        double r252560 = r252558 - r252559;
        double r252561 = r252557 / r252560;
        double r252562 = r252556 - r252561;
        double r252563 = acos(r252562);
        double r252564 = log1p(r252563);
        double r252565 = exp(r252564);
        double r252566 = r252565 - r252558;
        return r252566;
}

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 div-sub0.6

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

  4. Simplified0.6

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

  6. Applied expm1-log1p-u0.6

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

  8. Applied expm1-udef0.6

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

  9. Final simplification0.6

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

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))