Average Error: 35.6 → 32.2
Time: 25.7s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{\left(g + \sqrt{\left(g - h\right) \cdot \left(g + h\right)}\right) \cdot \frac{-1}{2}}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}} + \frac{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right) \cdot \frac{1}{2}}}{\sqrt[3]{a}}\]
\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)}
↓
\frac{\sqrt[3]{\left(g + \sqrt{\left(g - h\right) \cdot \left(g + h\right)}\right) \cdot \frac{-1}{2}}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt[3]{a}} \cdot \sqrt[3]{\sqrt[3]{a}}} + \frac{\sqrt[3]{\left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right) \cdot \frac{1}{2}}}{\sqrt[3]{a}}
double f(double g, double h, double a) {
        double r3928325 = 1.0;
        double r3928326 = 2.0;
        double r3928327 = a;
        double r3928328 = r3928326 * r3928327;
        double r3928329 = r3928325 / r3928328;
        double r3928330 = g;
        double r3928331 = -r3928330;
        double r3928332 = r3928330 * r3928330;
        double r3928333 = h;
        double r3928334 = r3928333 * r3928333;
        double r3928335 = r3928332 - r3928334;
        double r3928336 = sqrt(r3928335);
        double r3928337 = r3928331 + r3928336;
        double r3928338 = r3928329 * r3928337;
        double r3928339 = cbrt(r3928338);
        double r3928340 = r3928331 - r3928336;
        double r3928341 = r3928329 * r3928340;
        double r3928342 = cbrt(r3928341);
        double r3928343 = r3928339 + r3928342;
        return r3928343;
}
↓
double f(double g, double h, double a) {
        double r3928344 = g;
        double r3928345 = h;
        double r3928346 = r3928344 - r3928345;
        double r3928347 = r3928344 + r3928345;
        double r3928348 = r3928346 * r3928347;
        double r3928349 = sqrt(r3928348);
        double r3928350 = r3928344 + r3928349;
        double r3928351 = -0.5;
        double r3928352 = r3928350 * r3928351;
        double r3928353 = cbrt(r3928352);
        double r3928354 = a;
        double r3928355 = cbrt(r3928354);
        double r3928356 = r3928355 * r3928355;
        double r3928357 = cbrt(r3928356);
        double r3928358 = cbrt(r3928355);
        double r3928359 = r3928357 * r3928358;
        double r3928360 = r3928353 / r3928359;
        double r3928361 = r3928349 - r3928344;
        double r3928362 = 0.5;
        double r3928363 = r3928361 * r3928362;
        double r3928364 = cbrt(r3928363);
        double r3928365 = r3928364 / r3928355;
        double r3928366 = r3928360 + r3928365;
        return r3928366;
}

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. Initial program 35.6

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

  2. Simplified35.6

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

  4. Applied associate-*l/35.6

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

  5. Applied cbrt-div33.9

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

  7. Applied associate-*l/33.9

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

  8. Applied cbrt-div32.1

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

  10. Applied add-cube-cbrt32.2

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

  11. Applied cbrt-prod32.2

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

  12. Final simplification32.2

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

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(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))))))))