Average Error: 1.0 → 0.0
Time: 36.1s
Precision: 64
\[2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)\]
↓
\[\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) - \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \frac{2}{3}\right)\right) \cdot \sin \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right)\right) \cdot 2\]
2 \cdot \cos \left(\frac{2 \cdot \pi}{3} + \frac{\cos^{-1} \left(\frac{-g}{h}\right)}{3}\right)
↓
\left(\cos \left(\frac{2}{3} \cdot \pi\right) \cdot \cos \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right) - \sin \left(\sqrt{\pi} \cdot \left(\sqrt{\pi} \cdot \frac{2}{3}\right)\right) \cdot \sin \left(\frac{\frac{\cos^{-1} \left(-\frac{g}{h}\right)}{\sqrt{3}}}{\sqrt{3}}\right)\right) \cdot 2
double f(double g, double h) {
        double r33889452 = 2.0;
        double r33889453 = atan2(1.0, 0.0);
        double r33889454 = r33889452 * r33889453;
        double r33889455 = 3.0;
        double r33889456 = r33889454 / r33889455;
        double r33889457 = g;
        double r33889458 = -r33889457;
        double r33889459 = h;
        double r33889460 = r33889458 / r33889459;
        double r33889461 = acos(r33889460);
        double r33889462 = r33889461 / r33889455;
        double r33889463 = r33889456 + r33889462;
        double r33889464 = cos(r33889463);
        double r33889465 = r33889452 * r33889464;
        return r33889465;
}
↓
double f(double g, double h) {
        double r33889466 = 0.6666666666666666;
        double r33889467 = atan2(1.0, 0.0);
        double r33889468 = r33889466 * r33889467;
        double r33889469 = cos(r33889468);
        double r33889470 = g;
        double r33889471 = h;
        double r33889472 = r33889470 / r33889471;
        double r33889473 = -r33889472;
        double r33889474 = acos(r33889473);
        double r33889475 = 3.0;
        double r33889476 = sqrt(r33889475);
        double r33889477 = r33889474 / r33889476;
        double r33889478 = r33889477 / r33889476;
        double r33889479 = cos(r33889478);
        double r33889480 = r33889469 * r33889479;
        double r33889481 = sqrt(r33889467);
        double r33889482 = r33889481 * r33889466;
        double r33889483 = r33889481 * r33889482;
        double r33889484 = sin(r33889483);
        double r33889485 = sin(r33889478);
        double r33889486 = r33889484 * r33889485;
        double r33889487 = r33889480 - r33889486;
        double r33889488 = 2.0;
        double r33889489 = r33889487 * r33889488;
        return r33889489;
}

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

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

  4. Applied add-sqr-sqrt1.0

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

  5. Applied associate-/r*1.0

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

  7. Applied fma-udef1.0

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

  8. Applied cos-sum1.0

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

  10. Applied add-sqr-sqrt0.0

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

  11. Applied associate-*r*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (g h)
  :name "2-ancestry mixing, negative discriminant"
  (* 2 (cos (+ (/ (* 2 PI) 3) (/ (acos (/ (- g) h)) 3)))))