Average Error: 1.0 → 0.0
Time: 23.3s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[2 \cdot \left(\cos \left(\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}} \cdot \left(\sqrt{\frac{1}{\sqrt{3}}} \cdot \sqrt{\frac{1}{\sqrt{3}}}\right)\right)\right)\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
2 \cdot \left(\cos \left(\frac{1}{\sqrt{3}} \cdot \frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}\right) \cdot \cos \left(\frac{2}{3} \cdot \pi\right) - \sin \left(\frac{2}{3} \cdot \pi\right) \cdot \sin \left(\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}} \cdot \left(\sqrt{\frac{1}{\sqrt{3}}} \cdot \sqrt{\frac{1}{\sqrt{3}}}\right)\right)\right)
double f(double g, double h) {
        double r4423459 = 2.0;
        double r4423460 = atan2(1.0, 0.0);
        double r4423461 = r4423459 * r4423460;
        double r4423462 = 3.0;
        double r4423463 = r4423461 / r4423462;
        double r4423464 = g;
        double r4423465 = -r4423464;
        double r4423466 = h;
        double r4423467 = r4423465 / r4423466;
        double r4423468 = acos(r4423467);
        double r4423469 = r4423468 / r4423462;
        double r4423470 = r4423463 + r4423469;
        double r4423471 = cos(r4423470);
        double r4423472 = r4423459 * r4423471;
        return r4423472;
}
↓
double f(double g, double h) {
        double r4423473 = 2.0;
        double r4423474 = 1.0;
        double r4423475 = 3.0;
        double r4423476 = sqrt(r4423475);
        double r4423477 = r4423474 / r4423476;
        double r4423478 = g;
        double r4423479 = h;
        double r4423480 = r4423478 / r4423479;
        double r4423481 = -r4423480;
        double r4423482 = acos(r4423481);
        double r4423483 = r4423482 / r4423476;
        double r4423484 = r4423477 * r4423483;
        double r4423485 = cos(r4423484);
        double r4423486 = 0.6666666666666666;
        double r4423487 = atan2(1.0, 0.0);
        double r4423488 = r4423486 * r4423487;
        double r4423489 = cos(r4423488);
        double r4423490 = r4423485 * r4423489;
        double r4423491 = sin(r4423488);
        double r4423492 = sqrt(r4423477);
        double r4423493 = r4423492 * r4423492;
        double r4423494 = r4423483 * r4423493;
        double r4423495 = sin(r4423494);
        double r4423496 = r4423491 * r4423495;
        double r4423497 = r4423490 - r4423496;
        double r4423498 = r4423473 * r4423497;
        return r4423498;
}

Error

Bits error versus g

Bits error versus h

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 fma-udef1.0

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

  9. Applied cos-sum1.0

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

  11. Applied add-sqr-sqrt0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))