2-ancestry mixing, positive discriminant

Average Error: 35.2 → 32.4

Time: 1.3m

Precision: 64

Math

\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

↓

\[\sqrt[3]{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{1}{2}}}}} + \sqrt[3]{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\sqrt[3]{\sqrt[3]{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt[3]{g \cdot g - h \cdot h} \cdot \sqrt[3]{g \cdot g - h \cdot h}}} \cdot \left|\sqrt[3]{g \cdot g - h \cdot h}\right|}}{\frac{a}{\frac{1}{2}}}}\]

C

double f(double g, double h, double a) {
        double r18940378 = 1.0;
        double r18940379 = 2.0;
        double r18940380 = a;
        double r18940381 = r18940379 * r18940380;
        double r18940382 = r18940378 / r18940381;
        double r18940383 = g;
        double r18940384 = -r18940383;
        double r18940385 = r18940383 * r18940383;
        double r18940386 = h;
        double r18940387 = r18940386 * r18940386;
        double r18940388 = r18940385 - r18940387;
        double r18940389 = sqrt(r18940388);
        double r18940390 = r18940384 + r18940389;
        double r18940391 = r18940382 * r18940390;
        double r18940392 = cbrt(r18940391);
        double r18940393 = r18940384 - r18940389;
        double r18940394 = r18940382 * r18940393;
        double r18940395 = cbrt(r18940394);
        double r18940396 = r18940392 + r18940395;
        return r18940396;
}

↓

double f(double g, double h, double a) {
        double r18940397 = 0.5;
        double r18940398 = cbrt(r18940397);
        double r18940399 = a;
        double r18940400 = cbrt(r18940399);
        double r18940401 = r18940398 / r18940400;
        double r18940402 = r18940401 * r18940401;
        double r18940403 = cbrt(r18940402);
        double r18940404 = g;
        double r18940405 = r18940404 * r18940404;
        double r18940406 = h;
        double r18940407 = r18940406 * r18940406;
        double r18940408 = r18940405 - r18940407;
        double r18940409 = sqrt(r18940408);
        double r18940410 = r18940409 - r18940404;
        double r18940411 = r18940400 / r18940398;
        double r18940412 = r18940410 / r18940411;
        double r18940413 = cbrt(r18940412);
        double r18940414 = r18940403 * r18940413;
        double r18940415 = -r18940404;
        double r18940416 = r18940415 - r18940409;
        double r18940417 = cbrt(r18940416);
        double r18940418 = r18940417 * r18940417;
        double r18940419 = cbrt(r18940418);
        double r18940420 = cbrt(r18940408);
        double r18940421 = cbrt(r18940420);
        double r18940422 = r18940420 * r18940420;
        double r18940423 = cbrt(r18940422);
        double r18940424 = r18940421 * r18940423;
        double r18940425 = sqrt(r18940424);
        double r18940426 = fabs(r18940420);
        double r18940427 = r18940425 * r18940426;
        double r18940428 = r18940415 - r18940427;
        double r18940429 = cbrt(r18940428);
        double r18940430 = r18940399 / r18940397;
        double r18940431 = r18940429 / r18940430;
        double r18940432 = cbrt(r18940431);
        double r18940433 = r18940419 * r18940432;
        double r18940434 = r18940414 + r18940433;
        return r18940434;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Initial program 35.2

   \[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]

2. Simplified 35.2

   \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{a}{\frac{1}{2}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\frac{a}{\frac{1}{2}}}}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 35.2

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

5. Applied add-cube-cbrt 35.3

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

6. Applied times-frac 35.3

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

7. Applied *-un-lft-identity 35.3

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

8. Applied times-frac 35.3

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

9. Applied cbrt-prod 33.4

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

10. Simplified 33.3

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}}} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{1}{2}}}}} + \sqrt[3]{\frac{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}{\frac{a}{\frac{1}{2}}}}\]
11. Using strategy rm

12. Applied *-un-lft-identity 33.3

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

13. Applied add-cube-cbrt 33.4

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

14. Applied times-frac 33.4

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

15. Applied cbrt-prod 32.2

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

16. Simplified 32.2

    \[\leadsto \sqrt[3]{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{1}{2}}}}} + \color{blue}{\sqrt[3]{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\frac{a}{\frac{1}{2}}}}\]
17. Using strategy rm

18. Applied add-cube-cbrt 32.5

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

19. Applied sqrt-prod 32.5

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

20. Simplified 32.5

    \[\leadsto \sqrt[3]{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{1}{2}}}}} + \sqrt[3]{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \color{blue}{\left|\sqrt[3]{g \cdot g - h \cdot h}\right|} \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}}}}{\frac{a}{\frac{1}{2}}}}\]
21. Using strategy rm

22. Applied add-cube-cbrt 32.5

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

23. Applied cbrt-prod 32.4

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

24. Final simplification 32.4

    \[\leadsto \sqrt[3]{\frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}} \cdot \frac{\sqrt[3]{\frac{1}{2}}}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{\frac{\sqrt[3]{a}}{\sqrt[3]{\frac{1}{2}}}}} + \sqrt[3]{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} \cdot \sqrt[3]{\frac{\sqrt[3]{\left(-g\right) - \sqrt{\sqrt[3]{\sqrt[3]{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt[3]{g \cdot g - h \cdot h} \cdot \sqrt[3]{g \cdot g - h \cdot h}}} \cdot \left|\sqrt[3]{g \cdot g - h \cdot h}\right|}}{\frac{a}{\frac{1}{2}}}}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))