Average Error: 0.6 → 0.6
Time: 24.1s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}
double f(double v) {
        double r151235 = 1.0;
        double r151236 = 5.0;
        double r151237 = v;
        double r151238 = r151237 * r151237;
        double r151239 = r151236 * r151238;
        double r151240 = r151235 - r151239;
        double r151241 = r151238 - r151235;
        double r151242 = r151240 / r151241;
        double r151243 = acos(r151242);
        return r151243;
}
↓
double f(double v) {
        double r151244 = 1.0;
        double r151245 = 5.0;
        double r151246 = v;
        double r151247 = 2.0;
        double r151248 = pow(r151246, r151247);
        double r151249 = r151245 * r151248;
        double r151250 = r151244 - r151249;
        double r151251 = 6.0;
        double r151252 = pow(r151246, r151251);
        double r151253 = 3.0;
        double r151254 = pow(r151244, r151253);
        double r151255 = r151252 - r151254;
        double r151256 = r151250 / r151255;
        double r151257 = r151246 * r151246;
        double r151258 = r151244 + r151257;
        double r151259 = r151244 * r151258;
        double r151260 = 4.0;
        double r151261 = pow(r151246, r151260);
        double r151262 = r151259 + r151261;
        double r151263 = r151256 * r151262;
        double r151264 = acos(r151263);
        double r151265 = log(r151264);
        double r151266 = exp(r151265);
        return r151266;
}

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 flip3--0.6

    \[\leadsto \cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{{\left(v \cdot v\right)}^{3} - {1}^{3}}{\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)}}}\right)\]

  4. Applied associate-/r/0.6

    \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{\left(v \cdot v\right)}^{3} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)}\]

  5. Simplified0.6

    \[\leadsto \cos^{-1} \left(\color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{6} - {1}^{3}}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\]
  6. Using strategy rm

  7. Applied add-exp-log0.6

    \[\leadsto \color{blue}{e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{{v}^{6} - {1}^{3}} \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + \left(1 \cdot 1 + \left(v \cdot v\right) \cdot 1\right)\right)\right)\right)}}\]

  8. Simplified0.6

    \[\leadsto e^{\color{blue}{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}}\]

  9. Final simplification0.6

    \[\leadsto e^{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{6} - {1}^{3}} \cdot \left(1 \cdot \left(1 + v \cdot v\right) + {v}^{4}\right)\right)\right)}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))