Average Error: 0.6 → 0.6
Time: 44.7s
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 r7010286 = 1.0;
        double r7010287 = 5.0;
        double r7010288 = v;
        double r7010289 = r7010288 * r7010288;
        double r7010290 = r7010287 * r7010289;
        double r7010291 = r7010286 - r7010290;
        double r7010292 = r7010289 - r7010286;
        double r7010293 = r7010291 / r7010292;
        double r7010294 = acos(r7010293);
        return r7010294;
}
↓
double f(double v) {
        double r7010295 = 1.0;
        double r7010296 = v;
        double r7010297 = r7010296 * r7010296;
        double r7010298 = 5.0;
        double r7010299 = r7010297 * r7010298;
        double r7010300 = r7010295 - r7010299;
        double r7010301 = r7010297 - r7010295;
        double r7010302 = r7010300 / r7010301;
        double r7010303 = acos(r7010302);
        double r7010304 = log1p(r7010303);
        double r7010305 = expm1(r7010304);
        return r7010305;
}

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. Final simplification0.6

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))