2-ancestry mixing, positive discriminant

Average Error: 35.3 → 31.7

Time: 1.4m

Precision: 64

Math

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

↓

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

C

double f(double g, double h, double a) {
        double r22531521 = 1.0;
        double r22531522 = 2.0;
        double r22531523 = a;
        double r22531524 = r22531522 * r22531523;
        double r22531525 = r22531521 / r22531524;
        double r22531526 = g;
        double r22531527 = -r22531526;
        double r22531528 = r22531526 * r22531526;
        double r22531529 = h;
        double r22531530 = r22531529 * r22531529;
        double r22531531 = r22531528 - r22531530;
        double r22531532 = sqrt(r22531531);
        double r22531533 = r22531527 + r22531532;
        double r22531534 = r22531525 * r22531533;
        double r22531535 = cbrt(r22531534);
        double r22531536 = r22531527 - r22531532;
        double r22531537 = r22531525 * r22531536;
        double r22531538 = cbrt(r22531537);
        double r22531539 = r22531535 + r22531538;
        return r22531539;
}

↓

double f(double g, double h, double a) {
        double r22531540 = 0.5;
        double r22531541 = sqrt(r22531540);
        double r22531542 = a;
        double r22531543 = cbrt(r22531542);
        double r22531544 = g;
        double r22531545 = r22531544 * r22531544;
        double r22531546 = h;
        double r22531547 = r22531546 * r22531546;
        double r22531548 = r22531545 - r22531547;
        double r22531549 = sqrt(r22531548);
        double r22531550 = cbrt(r22531549);
        double r22531551 = r22531550 * r22531550;
        double r22531552 = r22531551 * r22531550;
        double r22531553 = r22531552 - r22531544;
        double r22531554 = cbrt(r22531553);
        double r22531555 = r22531543 / r22531554;
        double r22531556 = r22531549 - r22531544;
        double r22531557 = cbrt(r22531556);
        double r22531558 = r22531543 / r22531557;
        double r22531559 = r22531555 * r22531558;
        double r22531560 = r22531541 / r22531559;
        double r22531561 = cbrt(r22531560);
        double r22531562 = cbrt(r22531548);
        double r22531563 = sqrt(r22531562);
        double r22531564 = fabs(r22531562);
        double r22531565 = r22531563 * r22531564;
        double r22531566 = r22531565 - r22531544;
        double r22531567 = cbrt(r22531566);
        double r22531568 = r22531543 / r22531541;
        double r22531569 = r22531567 / r22531568;
        double r22531570 = cbrt(r22531569);
        double r22531571 = r22531561 * r22531570;
        double r22531572 = -r22531544;
        double r22531573 = r22531572 - r22531549;
        double r22531574 = 2.0;
        double r22531575 = r22531573 / r22531574;
        double r22531576 = cbrt(r22531575);
        double r22531577 = 1.0;
        double r22531578 = r22531577 / r22531542;
        double r22531579 = cbrt(r22531578);
        double r22531580 = r22531576 * r22531579;
        double r22531581 = r22531571 + r22531580;
        return r22531581;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Initial program 35.3

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

2. Simplified 35.3

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

4. Applied div-inv 35.3

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

5. Applied *-un-lft-identity 35.3

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

6. Applied times-frac 35.3

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

7. Applied cbrt-prod 33.5

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

9. Applied add-sqr-sqrt 33.5

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

10. Applied add-cube-cbrt 33.5

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

11. Applied times-frac 33.5

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

12. Applied add-cube-cbrt 33.5

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

13. Applied times-frac 33.5

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

14. Applied cbrt-prod 31.6

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

15. Simplified 31.6

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

17. Applied add-cube-cbrt 31.6

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

18. Applied sqrt-prod 31.6

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

19. Simplified 31.6

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

21. Applied add-cube-cbrt 31.7

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

22. Final simplification 31.7

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

Reproduce

herbie shell --seed 2019124 
(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))))))))