Average Error: 35.3 → 31.0
Time: 30.2s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 9.191731187329127127438182160139194041057 \cdot 10^{-156}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\left(-g\right) - g} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-1\right) \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g\right)}}{\sqrt[3]{a \cdot 2}} + \sqrt{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right) \cdot \frac{\frac{1}{a}}{2}}} \cdot \sqrt{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right) \cdot \frac{\frac{1}{a}}{2}}}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;g \le 9.191731187329127127438182160139194041057 \cdot 10^{-156}:\\
\;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\left(-g\right) - g} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r133448 = 1.0;
        double r133449 = 2.0;
        double r133450 = a;
        double r133451 = r133449 * r133450;
        double r133452 = r133448 / r133451;
        double r133453 = g;
        double r133454 = -r133453;
        double r133455 = r133453 * r133453;
        double r133456 = h;
        double r133457 = r133456 * r133456;
        double r133458 = r133455 - r133457;
        double r133459 = sqrt(r133458);
        double r133460 = r133454 + r133459;
        double r133461 = r133452 * r133460;
        double r133462 = cbrt(r133461);
        double r133463 = r133454 - r133459;
        double r133464 = r133452 * r133463;
        double r133465 = cbrt(r133464);
        double r133466 = r133462 + r133465;
        return r133466;
}
↓
double f(double g, double h, double a) {
        double r133467 = g;
        double r133468 = 9.191731187329127e-156;
        bool r133469 = r133467 <= r133468;
        double r133470 = -r133467;
        double r133471 = r133467 * r133467;
        double r133472 = h;
        double r133473 = r133472 * r133472;
        double r133474 = r133471 - r133473;
        double r133475 = sqrt(r133474);
        double r133476 = r133470 - r133475;
        double r133477 = 1.0;
        double r133478 = a;
        double r133479 = 2.0;
        double r133480 = r133478 * r133479;
        double r133481 = r133477 / r133480;
        double r133482 = r133476 * r133481;
        double r133483 = cbrt(r133482);
        double r133484 = r133470 - r133467;
        double r133485 = cbrt(r133484);
        double r133486 = r133477 / r133479;
        double r133487 = r133486 / r133478;
        double r133488 = cbrt(r133487);
        double r133489 = r133485 * r133488;
        double r133490 = r133483 + r133489;
        double r133491 = -r133477;
        double r133492 = r133467 - r133472;
        double r133493 = r133472 + r133467;
        double r133494 = r133492 * r133493;
        double r133495 = sqrt(r133494);
        double r133496 = r133495 + r133467;
        double r133497 = r133491 * r133496;
        double r133498 = cbrt(r133497);
        double r133499 = cbrt(r133480);
        double r133500 = r133498 / r133499;
        double r133501 = r133495 - r133467;
        double r133502 = r133477 / r133478;
        double r133503 = r133502 / r133479;
        double r133504 = r133501 * r133503;
        double r133505 = cbrt(r133504);
        double r133506 = sqrt(r133505);
        double r133507 = r133506 * r133506;
        double r133508 = r133500 + r133507;
        double r133509 = r133469 ? r133490 : r133508;
        return r133509;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if g < 9.191731187329127e-156

    1. Initial program 36.3

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

    3. Applied cbrt-prod32.5

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

    4. Simplified32.4

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

    5. Simplified32.4

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

    6. Taylor expanded around -inf 31.2

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

    7. Simplified31.2

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

    if 9.191731187329127e-156 < g

    1. Initial program 34.3

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

    3. Applied associate-*l/34.3

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

    4. Applied cbrt-div30.5

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

    5. Simplified30.5

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

    7. Applied add-sqr-sqrt30.8

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

    8. Simplified30.8

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

    9. Simplified30.8

      \[\leadsto \sqrt{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right) \cdot \frac{\frac{1}{a}}{2}}} \cdot \color{blue}{\sqrt{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right) \cdot \frac{\frac{1}{a}}{2}}}} + \frac{\sqrt[3]{-1 \cdot \left(g + \sqrt{\left(g - h\right) \cdot \left(g + h\right)}\right)}}{\sqrt[3]{2 \cdot a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification31.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 9.191731187329127127438182160139194041057 \cdot 10^{-156}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{a \cdot 2}} + \sqrt[3]{\left(-g\right) - g} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-1\right) \cdot \left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g\right)}}{\sqrt[3]{a \cdot 2}} + \sqrt{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right) \cdot \frac{\frac{1}{a}}{2}}} \cdot \sqrt{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(h + g\right)} - g\right) \cdot \frac{\frac{1}{a}}{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))