Average Error: 0.5 → 0.6
Time: 33.3s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
e^{\sqrt{\log \left(\cos^{-1} \left(\frac{1 - \log \left(e^{5 \cdot \left(v \cdot v\right)}\right)}{v \cdot v - 1}\right)\right)} \cdot \sqrt{\log \left(\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\right)}}
double f(double v) {
        double r139259 = 1.0;
        double r139260 = 5.0;
        double r139261 = v;
        double r139262 = r139261 * r139261;
        double r139263 = r139260 * r139262;
        double r139264 = r139259 - r139263;
        double r139265 = r139262 - r139259;
        double r139266 = r139264 / r139265;
        double r139267 = acos(r139266);
        return r139267;
}
↓
double f(double v) {
        double r139268 = 1.0;
        double r139269 = 5.0;
        double r139270 = v;
        double r139271 = r139270 * r139270;
        double r139272 = r139269 * r139271;
        double r139273 = exp(r139272);
        double r139274 = log(r139273);
        double r139275 = r139268 - r139274;
        double r139276 = r139271 - r139268;
        double r139277 = r139275 / r139276;
        double r139278 = acos(r139277);
        double r139279 = log(r139278);
        double r139280 = sqrt(r139279);
        double r139281 = r139268 - r139272;
        double r139282 = r139281 / r139276;
        double r139283 = acos(r139282);
        double r139284 = log(r139283);
        double r139285 = sqrt(r139284);
        double r139286 = r139280 * r139285;
        double r139287 = exp(r139286);
        return r139287;
}

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 add-exp-log0.5

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

  5. Applied add-sqr-sqrt0.6

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

  7. Applied add-log-exp0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))