Average Error: 0.5 → 0.5
Time: 23.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)\right)\right)
double f(double v) {
        double r5817490 = 1.0;
        double r5817491 = 5.0;
        double r5817492 = v;
        double r5817493 = r5817492 * r5817492;
        double r5817494 = r5817491 * r5817493;
        double r5817495 = r5817490 - r5817494;
        double r5817496 = r5817493 - r5817490;
        double r5817497 = r5817495 / r5817496;
        double r5817498 = acos(r5817497);
        return r5817498;
}
↓
double f(double v) {
        double r5817499 = 1.0;
        double r5817500 = v;
        double r5817501 = r5817500 * r5817500;
        double r5817502 = 5.0;
        double r5817503 = r5817501 * r5817502;
        double r5817504 = r5817499 - r5817503;
        double r5817505 = r5817501 - r5817499;
        double r5817506 = r5817504 / r5817505;
        double r5817507 = acos(r5817506);
        double r5817508 = log1p(r5817507);
        double r5817509 = expm1(r5817508);
        return r5817509;
}

Error

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Final simplification0.5

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))