Average Error: 0.6 → 0.6
Time: 42.6s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\cos^{-1} \left(\sqrt[3]{\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1} \cdot \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\cos^{-1} \left(\sqrt[3]{\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1} \cdot \left(\frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1} \cdot \frac{1 - \left(v \cdot v\right) \cdot 5}{v \cdot v - 1}\right)}\right)
double f(double v) {
        double r5920357 = 1.0;
        double r5920358 = 5.0;
        double r5920359 = v;
        double r5920360 = r5920359 * r5920359;
        double r5920361 = r5920358 * r5920360;
        double r5920362 = r5920357 - r5920361;
        double r5920363 = r5920360 - r5920357;
        double r5920364 = r5920362 / r5920363;
        double r5920365 = acos(r5920364);
        return r5920365;
}
↓
double f(double v) {
        double r5920366 = 1.0;
        double r5920367 = v;
        double r5920368 = r5920367 * r5920367;
        double r5920369 = 5.0;
        double r5920370 = r5920368 * r5920369;
        double r5920371 = r5920366 - r5920370;
        double r5920372 = r5920368 - r5920366;
        double r5920373 = r5920371 / r5920372;
        double r5920374 = r5920373 * r5920373;
        double r5920375 = r5920373 * r5920374;
        double r5920376 = cbrt(r5920375);
        double r5920377 = acos(r5920376);
        return r5920377;
}

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 add-cbrt-cube0.6

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

  4. Applied add-cbrt-cube0.6

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

  5. Applied cbrt-undiv0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))