2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 21.4s

Precision: 64

Math

\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

↓

\[2 \cdot \left(\log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right)\right)\]

C

double f(double g, double h) {
        double r5932719 = 2.0;
        double r5932720 = atan2(1.0, 0.0);
        double r5932721 = r5932719 * r5932720;
        double r5932722 = 3.0;
        double r5932723 = r5932721 / r5932722;
        double r5932724 = g;
        double r5932725 = -r5932724;
        double r5932726 = h;
        double r5932727 = r5932725 / r5932726;
        double r5932728 = acos(r5932727);
        double r5932729 = r5932728 / r5932722;
        double r5932730 = r5932723 + r5932729;
        double r5932731 = cos(r5932730);
        double r5932732 = r5932719 * r5932731;
        return r5932732;
}

↓

double f(double g, double h) {
        double r5932733 = 2.0;
        double r5932734 = 3.0;
        double r5932735 = r5932733 / r5932734;
        double r5932736 = atan2(1.0, 0.0);
        double r5932737 = g;
        double r5932738 = -r5932737;
        double r5932739 = h;
        double r5932740 = r5932738 / r5932739;
        double r5932741 = acos(r5932740);
        double r5932742 = r5932741 / r5932734;
        double r5932743 = fma(r5932735, r5932736, r5932742);
        double r5932744 = cos(r5932743);
        double r5932745 = exp(r5932744);
        double r5932746 = cbrt(r5932745);
        double r5932747 = r5932746 * r5932746;
        double r5932748 = log(r5932747);
        double r5932749 = log(r5932746);
        double r5932750 = r5932748 + r5932749;
        double r5932751 = r5932733 * r5932750;
        return r5932751;
}

Error

Bits error versus g

Bits error versus h

Derivation

1. Initial program 1.0

   \[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]

2. Simplified 1.0

   \[\leadsto \color{blue}{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right) \cdot 2}\]
3. Using strategy rm

4. Applied add-log-exp 1.0

   \[\leadsto \color{blue}{\log \left(e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}\right)} \cdot 2\]
5. Using strategy rm

6. Applied add-cube-cbrt 0.1

   \[\leadsto \log \color{blue}{\left(\left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right) \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right)} \cdot 2\]

7. Applied log-prod 0.1

   \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right)\right)} \cdot 2\]

8. Final simplification 0.1

   \[\leadsto 2 \cdot \left(\log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\frac{2}{3}, \pi, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right)\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2.0 (cos (+ (/ (* 2.0 PI) 3.0) (/ (acos (/ (- g) h)) 3.0)))))