Average Error: 0.5 → 0.5
Time: 15.2s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\frac{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(1 - 5 \cdot {v}^{2}\right)}{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\frac{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(1 - 5 \cdot {v}^{2}\right)}{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)
double f(double v) {
        double r135219 = 1.0;
        double r135220 = 5.0;
        double r135221 = v;
        double r135222 = r135221 * r135221;
        double r135223 = r135220 * r135222;
        double r135224 = r135219 - r135223;
        double r135225 = r135222 - r135219;
        double r135226 = r135224 / r135225;
        double r135227 = acos(r135226);
        return r135227;
}
↓
double f(double v) {
        double r135228 = v;
        double r135229 = 4.0;
        double r135230 = pow(r135228, r135229);
        double r135231 = 1.0;
        double r135232 = r135231 * r135231;
        double r135233 = r135230 - r135232;
        double r135234 = 5.0;
        double r135235 = 2.0;
        double r135236 = pow(r135228, r135235);
        double r135237 = r135234 * r135236;
        double r135238 = r135231 - r135237;
        double r135239 = r135233 * r135238;
        double r135240 = r135228 * r135228;
        double r135241 = r135240 - r135231;
        double r135242 = r135233 * r135241;
        double r135243 = r135239 / r135242;
        double r135244 = acos(r135243);
        return r135244;
}

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

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

  4. Applied associate-/r/0.6

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

  5. Simplified0.6

    \[\leadsto \cos^{-1} \left(\color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{4} - 1 \cdot 1}} \cdot \left(v \cdot v + 1\right)\right)\]
  6. Using strategy rm

  7. Applied flip-+0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{4} - 1 \cdot 1} \cdot \color{blue}{\frac{\left(v \cdot v\right) \cdot \left(v \cdot v\right) - 1 \cdot 1}{v \cdot v - 1}}\right)\]

  8. Applied frac-times0.5

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{\left(1 - 5 \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) - 1 \cdot 1\right)}{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)}\]

  9. Simplified0.5

    \[\leadsto \cos^{-1} \left(\frac{\color{blue}{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(1 - 5 \cdot {v}^{2}\right)}}{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)\]

  10. Final simplification0.5

    \[\leadsto \cos^{-1} \left(\frac{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(1 - 5 \cdot {v}^{2}\right)}{\left({v}^{4} - 1 \cdot 1\right) \cdot \left(v \cdot v - 1\right)}\right)\]

Reproduce

herbie shell --seed 2019195 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))