Average Error: 0.5 → 0.6
Time: 4.7s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{1 \cdot 1 - \left(5 \cdot \left(v \cdot v\right)\right) \cdot \left(5 \cdot \left(v \cdot v\right)\right)}{\left(v \cdot v - 1\right) \cdot \left(1 + 5 \cdot \left(v \cdot v\right)\right)}\right)
double f(double v) {
        double r222421 = 1.0;
        double r222422 = 5.0;
        double r222423 = v;
        double r222424 = r222423 * r222423;
        double r222425 = r222422 * r222424;
        double r222426 = r222421 - r222425;
        double r222427 = r222424 - r222421;
        double r222428 = r222426 / r222427;
        double r222429 = acos(r222428);
        return r222429;
}
↓
double f(double v) {
        double r222430 = 1.0;
        double r222431 = r222430 * r222430;
        double r222432 = 5.0;
        double r222433 = v;
        double r222434 = r222433 * r222433;
        double r222435 = r222432 * r222434;
        double r222436 = r222435 * r222435;
        double r222437 = r222431 - r222436;
        double r222438 = r222434 - r222430;
        double r222439 = r222430 + r222435;
        double r222440 = r222438 * r222439;
        double r222441 = r222437 / r222440;
        double r222442 = acos(r222441);
        return r222442;
}

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 flip--0.6

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

  4. Applied associate-/l/0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))