Average Error: 36.3 → 32.2
Time: 1.0m
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 r5382276 = 1.0;
        double r5382277 = 2.0;
        double r5382278 = a;
        double r5382279 = r5382277 * r5382278;
        double r5382280 = r5382276 / r5382279;
        double r5382281 = g;
        double r5382282 = -r5382281;
        double r5382283 = r5382281 * r5382281;
        double r5382284 = h;
        double r5382285 = r5382284 * r5382284;
        double r5382286 = r5382283 - r5382285;
        double r5382287 = sqrt(r5382286);
        double r5382288 = r5382282 + r5382287;
        double r5382289 = r5382280 * r5382288;
        double r5382290 = cbrt(r5382289);
        double r5382291 = r5382282 - r5382287;
        double r5382292 = r5382280 * r5382291;
        double r5382293 = cbrt(r5382292);
        double r5382294 = r5382290 + r5382293;
        return r5382294;
}
↓
double f(double g, double h, double a) {
        double r5382295 = g;
        double r5382296 = 8.889202414083867e-196;
        bool r5382297 = r5382295 <= r5382296;
        double r5382298 = 1.0;
        double r5382299 = a;
        double r5382300 = 2.0;
        double r5382301 = r5382299 * r5382300;
        double r5382302 = r5382298 / r5382301;
        double r5382303 = -r5382295;
        double r5382304 = r5382295 * r5382295;
        double r5382305 = h;
        double r5382306 = r5382305 * r5382305;
        double r5382307 = r5382304 - r5382306;
        double r5382308 = sqrt(r5382307);
        double r5382309 = r5382303 - r5382308;
        double r5382310 = r5382302 * r5382309;
        double r5382311 = cbrt(r5382310);
        double r5382312 = r5382295 + r5382295;
        double r5382313 = -r5382312;
        double r5382314 = cbrt(r5382313);
        double r5382315 = cbrt(r5382302);
        double r5382316 = r5382314 * r5382315;
        double r5382317 = r5382311 + r5382316;
        double r5382318 = r5382303 + r5382308;
        double r5382319 = r5382302 * r5382318;
        double r5382320 = cbrt(r5382319);
        double r5382321 = cbrt(r5382307);
        double r5382322 = sqrt(r5382321);
        double r5382323 = r5382321 * r5382321;
        double r5382324 = sqrt(r5382323);
        double r5382325 = r5382322 * r5382324;
        double r5382326 = r5382303 - r5382325;
        double r5382327 = r5382326 * r5382298;
        double r5382328 = cbrt(r5382327);
        double r5382329 = cbrt(r5382301);
        double r5382330 = r5382328 / r5382329;
        double r5382331 = r5382320 + r5382330;
        double r5382332 = r5382297 ? r5382317 : r5382331;
        return r5382332;
}

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))))))))