Average Error: 1.0 → 0.1
Time: 24.4s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)}}\right)\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)}} \cdot \sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{3}\right)\right)}}\right)\right) \cdot 2
double f(double g, double h) {
        double r4087551 = 2.0;
        double r4087552 = atan2(1.0, 0.0);
        double r4087553 = r4087551 * r4087552;
        double r4087554 = 3.0;
        double r4087555 = r4087553 / r4087554;
        double r4087556 = g;
        double r4087557 = -r4087556;
        double r4087558 = h;
        double r4087559 = r4087557 / r4087558;
        double r4087560 = acos(r4087559);
        double r4087561 = r4087560 / r4087554;
        double r4087562 = r4087555 + r4087561;
        double r4087563 = cos(r4087562);
        double r4087564 = r4087551 * r4087563;
        return r4087564;
}
↓
double f(double g, double h) {
        double r4087565 = atan2(1.0, 0.0);
        double r4087566 = 2.0;
        double r4087567 = 3.0;
        double r4087568 = r4087566 / r4087567;
        double r4087569 = g;
        double r4087570 = h;
        double r4087571 = r4087569 / r4087570;
        double r4087572 = -r4087571;
        double r4087573 = acos(r4087572);
        double r4087574 = r4087573 / r4087567;
        double r4087575 = fma(r4087565, r4087568, r4087574);
        double r4087576 = cos(r4087575);
        double r4087577 = exp(r4087576);
        double r4087578 = cbrt(r4087577);
        double r4087579 = r4087578 * r4087578;
        double r4087580 = log(r4087579);
        double r4087581 = log(r4087578);
        double r4087582 = r4087580 + r4087581;
        double r4087583 = r4087582 * r4087566;
        return r4087583;
}

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. Simplified1.0

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied *-un-lft-identity1.0

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

  6. Applied times-frac1.0

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

  8. Applied add-log-exp1.0

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

  9. Simplified1.0

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

  11. Applied add-cube-cbrt0.1

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

  12. Applied log-prod0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))