Average Error: 1.0 → 0.1
Time: 23.7s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(2 \cdot \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\cos \left(\mathsf{fma}\left(\pi, \frac{2}{3}, \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\right)}}\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(2 \cdot \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) + \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)
double f(double g, double h) {
        double r79383 = 2.0;
        double r79384 = atan2(1.0, 0.0);
        double r79385 = r79383 * r79384;
        double r79386 = 3.0;
        double r79387 = r79385 / r79386;
        double r79388 = g;
        double r79389 = -r79388;
        double r79390 = h;
        double r79391 = r79389 / r79390;
        double r79392 = acos(r79391);
        double r79393 = r79392 / r79386;
        double r79394 = r79387 + r79393;
        double r79395 = cos(r79394);
        double r79396 = r79383 * r79395;
        return r79396;
}
↓
double f(double g, double h) {
        double r79397 = 2.0;
        double r79398 = 2.0;
        double r79399 = atan2(1.0, 0.0);
        double r79400 = 3.0;
        double r79401 = r79397 / r79400;
        double r79402 = g;
        double r79403 = -r79402;
        double r79404 = h;
        double r79405 = r79403 / r79404;
        double r79406 = acos(r79405);
        double r79407 = r79406 / r79400;
        double r79408 = fma(r79399, r79401, r79407);
        double r79409 = cos(r79408);
        double r79410 = exp(r79409);
        double r79411 = cbrt(r79410);
        double r79412 = log(r79411);
        double r79413 = r79398 * r79412;
        double r79414 = r79413 + r79412;
        double r79415 = r79397 * r79414;
        return r79415;
}

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

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

  4. Applied add-sqr-sqrt1.0

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

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

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

  6. Applied times-frac1.0

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

  8. Applied add-log-exp1.0

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

  9. Simplified1.0

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

  11. Applied add-cube-cbrt0.1

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

  12. Applied log-prod0.1

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

  13. Simplified0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  :precision binary64
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))