Average Error: 0.6 → 0.6
Time: 14.4s
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 r132375 = 1.0;
        double r132376 = 5.0;
        double r132377 = v;
        double r132378 = r132377 * r132377;
        double r132379 = r132376 * r132378;
        double r132380 = r132375 - r132379;
        double r132381 = r132378 - r132375;
        double r132382 = r132380 / r132381;
        double r132383 = acos(r132382);
        return r132383;
}
↓
double f(double v) {
        double r132384 = 1.0;
        double r132385 = r132384 * r132384;
        double r132386 = 5.0;
        double r132387 = v;
        double r132388 = r132387 * r132387;
        double r132389 = r132386 * r132388;
        double r132390 = r132389 * r132389;
        double r132391 = r132385 - r132390;
        double r132392 = r132388 - r132384;
        double r132393 = r132384 + r132389;
        double r132394 = r132392 * r132393;
        double r132395 = r132391 / r132394;
        double r132396 = acos(r132395);
        return r132396;
}

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 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 2019235 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))