Average Error: 0.6 → 0.6
Time: 6.0s
Precision: 64
\[\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
↓
\[\frac{\pi}{2} - \sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)\]
\cos^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
↓
\frac{\pi}{2} - \sin^{-1} \left(\frac{1}{{v}^{2} - 1} - \frac{5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)
double f(double v) {
        double r328412 = 1.0;
        double r328413 = 5.0;
        double r328414 = v;
        double r328415 = r328414 * r328414;
        double r328416 = r328413 * r328415;
        double r328417 = r328412 - r328416;
        double r328418 = r328415 - r328412;
        double r328419 = r328417 / r328418;
        double r328420 = acos(r328419);
        return r328420;
}
↓
double f(double v) {
        double r328421 = atan2(1.0, 0.0);
        double r328422 = 2.0;
        double r328423 = r328421 / r328422;
        double r328424 = 1.0;
        double r328425 = v;
        double r328426 = pow(r328425, r328422);
        double r328427 = r328426 - r328424;
        double r328428 = r328424 / r328427;
        double r328429 = 5.0;
        double r328430 = r328425 * r328425;
        double r328431 = r328429 * r328430;
        double r328432 = r328430 - r328424;
        double r328433 = r328431 / r328432;
        double r328434 = r328428 - r328433;
        double r328435 = asin(r328434);
        double r328436 = r328423 - r328435;
        return r328436;
}

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 acos-asin0.6

    \[\leadsto \color{blue}{\frac{\pi}{2} - \sin^{-1} \left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{v \cdot v - 1}\right)}\]
  4. Using strategy rm

  5. Applied div-sub0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020049 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))