Average Error: 0.5 → 0.5
Time: 6.7s
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 r185550 = 1.0;
        double r185551 = 5.0;
        double r185552 = v;
        double r185553 = r185552 * r185552;
        double r185554 = r185551 * r185553;
        double r185555 = r185550 - r185554;
        double r185556 = r185553 - r185550;
        double r185557 = r185555 / r185556;
        double r185558 = acos(r185557);
        return r185558;
}
↓
double f(double v) {
        double r185559 = atan2(1.0, 0.0);
        double r185560 = 2.0;
        double r185561 = r185559 / r185560;
        double r185562 = 1.0;
        double r185563 = v;
        double r185564 = pow(r185563, r185560);
        double r185565 = r185564 - r185562;
        double r185566 = r185562 / r185565;
        double r185567 = 5.0;
        double r185568 = r185563 * r185563;
        double r185569 = r185567 * r185568;
        double r185570 = r185568 - r185562;
        double r185571 = r185569 / r185570;
        double r185572 = r185566 - r185571;
        double r185573 = asin(r185572);
        double r185574 = r185561 - r185573;
        return r185574;
}

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 div-sub0.5

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

  4. Simplified0.5

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

  6. Applied acos-asin0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020036 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  :precision binary64
  (acos (/ (- 1 (* 5 (* v v))) (- (* v v) 1))))