Average Error: 36.3 → 32.2
Time: 59.9s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 8.889202414083867012167858399548234734645 \cdot 10^{-196}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(\left(-g\right) - \sqrt{\sqrt[3]{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h} \cdot \sqrt[3]{g \cdot g - h \cdot h}}\right) \cdot 1}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]
\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 8.889202414083867012167858399548234734645 \cdot 10^{-196}:\\
\;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{1}{a \cdot 2}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r5997416 = 1.0;
        double r5997417 = 2.0;
        double r5997418 = a;
        double r5997419 = r5997417 * r5997418;
        double r5997420 = r5997416 / r5997419;
        double r5997421 = g;
        double r5997422 = -r5997421;
        double r5997423 = r5997421 * r5997421;
        double r5997424 = h;
        double r5997425 = r5997424 * r5997424;
        double r5997426 = r5997423 - r5997425;
        double r5997427 = sqrt(r5997426);
        double r5997428 = r5997422 + r5997427;
        double r5997429 = r5997420 * r5997428;
        double r5997430 = cbrt(r5997429);
        double r5997431 = r5997422 - r5997427;
        double r5997432 = r5997420 * r5997431;
        double r5997433 = cbrt(r5997432);
        double r5997434 = r5997430 + r5997433;
        return r5997434;
}
↓
double f(double g, double h, double a) {
        double r5997435 = g;
        double r5997436 = 8.889202414083867e-196;
        bool r5997437 = r5997435 <= r5997436;
        double r5997438 = 1.0;
        double r5997439 = a;
        double r5997440 = 2.0;
        double r5997441 = r5997439 * r5997440;
        double r5997442 = r5997438 / r5997441;
        double r5997443 = -r5997435;
        double r5997444 = r5997435 * r5997435;
        double r5997445 = h;
        double r5997446 = r5997445 * r5997445;
        double r5997447 = r5997444 - r5997446;
        double r5997448 = sqrt(r5997447);
        double r5997449 = r5997443 - r5997448;
        double r5997450 = r5997442 * r5997449;
        double r5997451 = cbrt(r5997450);
        double r5997452 = r5997435 + r5997435;
        double r5997453 = -r5997452;
        double r5997454 = cbrt(r5997453);
        double r5997455 = cbrt(r5997442);
        double r5997456 = r5997454 * r5997455;
        double r5997457 = r5997451 + r5997456;
        double r5997458 = r5997443 + r5997448;
        double r5997459 = r5997442 * r5997458;
        double r5997460 = cbrt(r5997459);
        double r5997461 = cbrt(r5997447);
        double r5997462 = sqrt(r5997461);
        double r5997463 = r5997461 * r5997461;
        double r5997464 = sqrt(r5997463);
        double r5997465 = r5997462 * r5997464;
        double r5997466 = r5997443 - r5997465;
        double r5997467 = r5997466 * r5997438;
        double r5997468 = cbrt(r5997467);
        double r5997469 = cbrt(r5997441);
        double r5997470 = r5997468 / r5997469;
        double r5997471 = r5997460 + r5997470;
        double r5997472 = r5997437 ? r5997457 : r5997471;
        return r5997472;
}

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 < 8.889202414083867e-196

    1. Initial program 37.4

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

    3. Applied cbrt-prod34.0

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

    4. Taylor expanded around -inf 33.0

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

    5. Simplified33.0

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

    if 8.889202414083867e-196 < g

    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. Using strategy rm

    3. Applied associate-*l/35.1

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

    4. Applied cbrt-div31.2

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

    6. Applied add-cube-cbrt31.2

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

    7. Applied sqrt-prod31.2

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

  4. Final simplification32.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 8.889202414083867012167858399548234734645 \cdot 10^{-196}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\left(\left(-g\right) - \sqrt{\sqrt[3]{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h} \cdot \sqrt[3]{g \cdot g - h \cdot h}}\right) \cdot 1}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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))))))))