Average Error: 35.8 → 32.0
Time: 8.3s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\]
\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \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{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}
double f(double g, double h, double a) {
        double r154232 = 1.0;
        double r154233 = 2.0;
        double r154234 = a;
        double r154235 = r154233 * r154234;
        double r154236 = r154232 / r154235;
        double r154237 = g;
        double r154238 = -r154237;
        double r154239 = r154237 * r154237;
        double r154240 = h;
        double r154241 = r154240 * r154240;
        double r154242 = r154239 - r154241;
        double r154243 = sqrt(r154242);
        double r154244 = r154238 + r154243;
        double r154245 = r154236 * r154244;
        double r154246 = cbrt(r154245);
        double r154247 = r154238 - r154243;
        double r154248 = r154236 * r154247;
        double r154249 = cbrt(r154248);
        double r154250 = r154246 + r154249;
        return r154250;
}
↓
double f(double g, double h, double a) {
        double r154251 = 1.0;
        double r154252 = g;
        double r154253 = -r154252;
        double r154254 = r154252 * r154252;
        double r154255 = h;
        double r154256 = r154255 * r154255;
        double r154257 = r154254 - r154256;
        double r154258 = sqrt(r154257);
        double r154259 = r154253 + r154258;
        double r154260 = r154251 * r154259;
        double r154261 = cbrt(r154260);
        double r154262 = 2.0;
        double r154263 = a;
        double r154264 = r154262 * r154263;
        double r154265 = cbrt(r154264);
        double r154266 = r154261 / r154265;
        double r154267 = r154251 / r154264;
        double r154268 = cbrt(r154267);
        double r154269 = r154253 - r154258;
        double r154270 = cbrt(r154269);
        double r154271 = r154268 * r154270;
        double r154272 = r154266 + r154271;
        return r154272;
}

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 35.8

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

  3. Applied associate-*l/35.8

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

  4. Applied cbrt-div34.0

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

  6. Applied cbrt-prod32.0

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

  7. Final simplification32.0

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))