Average Error: 0.5 → 0.6
Time: 15.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}\right)
double f(double v) {
        double r159308 = 1.0;
        double r159309 = 5.0;
        double r159310 = v;
        double r159311 = r159310 * r159310;
        double r159312 = r159309 * r159311;
        double r159313 = r159308 - r159312;
        double r159314 = r159311 - r159308;
        double r159315 = r159313 / r159314;
        double r159316 = acos(r159315);
        return r159316;
}
↓
double f(double v) {
        double r159317 = 1.0;
        double r159318 = 5.0;
        double r159319 = v;
        double r159320 = 2.0;
        double r159321 = pow(r159319, r159320);
        double r159322 = r159318 * r159321;
        double r159323 = r159317 - r159322;
        double r159324 = r159321 - r159317;
        double r159325 = r159323 / r159324;
        double r159326 = 3.0;
        double r159327 = pow(r159325, r159326);
        double r159328 = cbrt(r159327);
        double r159329 = acos(r159328);
        return r159329;
}

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-log-exp0.6

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

  5. Applied add-cbrt-cube0.6

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

  6. Applied add-cbrt-cube0.6

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

  7. Applied cbrt-undiv0.6

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

  8. Simplified0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{\color{blue}{{\left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}}\right)\]

  9. Final simplification0.6

    \[\leadsto \cos^{-1} \left(\sqrt[3]{{\left(\frac{1 - 5 \cdot {v}^{2}}{{v}^{2} - 1}\right)}^{3}}\right)\]

Reproduce

herbie shell --seed 2019303 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))