Falkner and Boettcher, Appendix B, 1

Average Error: 0.6 → 0.6

Time: 25.9s

Precision: 64

Math

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

↓

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

C

double f(double v) {
        double r11610604 = 1.0;
        double r11610605 = 5.0;
        double r11610606 = v;
        double r11610607 = r11610606 * r11610606;
        double r11610608 = r11610605 * r11610607;
        double r11610609 = r11610604 - r11610608;
        double r11610610 = r11610607 - r11610604;
        double r11610611 = r11610609 / r11610610;
        double r11610612 = acos(r11610611);
        return r11610612;
}

↓

double f(double v) {
        double r11610613 = 1.0;
        double r11610614 = v;
        double r11610615 = r11610614 * r11610614;
        double r11610616 = 5.0;
        double r11610617 = r11610615 * r11610616;
        double r11610618 = r11610613 - r11610617;
        double r11610619 = r11610615 - r11610613;
        double r11610620 = r11610618 / r11610619;
        double r11610621 = acos(r11610620);
        double r11610622 = r11610621 * r11610621;
        double r11610623 = cbrt(r11610621);
        double r11610624 = r11610623 * r11610623;
        double r11610625 = r11610623 * r11610624;
        double r11610626 = r11610622 * r11610625;
        double r11610627 = cbrt(r11610626);
        return r11610627;
}

Error

Bits error versus v

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-cube 1.5

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

5. Applied add-cube-cbrt 0.6

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

6. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019170 
(FPCore (v)
  :name "Falkner and Boettcher, Appendix B, 1"
  (acos (/ (- 1.0 (* 5.0 (* v v))) (- (* v v) 1.0))))