2-ancestry mixing, negative discriminant

Average Error: 1.0 → 0.1

Time: 38.8s

Precision: 64

Math

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

↓

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

C

double f(double g, double h) {
        double r5992403 = 2.0;
        double r5992404 = atan2(1.0, 0.0);
        double r5992405 = r5992403 * r5992404;
        double r5992406 = 3.0;
        double r5992407 = r5992405 / r5992406;
        double r5992408 = g;
        double r5992409 = -r5992408;
        double r5992410 = h;
        double r5992411 = r5992409 / r5992410;
        double r5992412 = acos(r5992411);
        double r5992413 = r5992412 / r5992406;
        double r5992414 = r5992407 + r5992413;
        double r5992415 = cos(r5992414);
        double r5992416 = r5992403 * r5992415;
        return r5992416;
}

↓

double f(double g, double h) {
        double r5992417 = 2.0;
        double r5992418 = 3.0;
        double r5992419 = r5992417 / r5992418;
        double r5992420 = atan2(1.0, 0.0);
        double r5992421 = g;
        double r5992422 = -r5992421;
        double r5992423 = h;
        double r5992424 = r5992422 / r5992423;
        double r5992425 = acos(r5992424);
        double r5992426 = r5992425 / r5992418;
        double r5992427 = cbrt(r5992426);
        double r5992428 = r5992427 * r5992427;
        double r5992429 = r5992427 * r5992428;
        double r5992430 = fma(r5992419, r5992420, r5992429);
        double r5992431 = cos(r5992430);
        double r5992432 = exp(r5992431);
        double r5992433 = cbrt(r5992432);
        double r5992434 = log(r5992433);
        double r5992435 = fma(r5992419, r5992420, r5992426);
        double r5992436 = cos(r5992435);
        double r5992437 = exp(r5992436);
        double r5992438 = cbrt(r5992437);
        double r5992439 = r5992438 * r5992438;
        double r5992440 = log(r5992439);
        double r5992441 = r5992434 + r5992440;
        double r5992442 = r5992417 * r5992441;
        return r5992442;
}

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. Using strategy rm

9. Applied add-cube-cbrt 0.1

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

10. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019168 +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)))))