Average Error: 35.8 → 32.1
Time: 11.0s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\frac{1}{\sqrt{2}} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{\sqrt{2} \cdot a}}\]
\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)}
↓
\sqrt[3]{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{\frac{1}{\sqrt{2}} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{\sqrt{2} \cdot a}}
double code(double g, double h, double a) {
	return ((double) (((double) cbrt(((double) (((double) (1.0 / ((double) (2.0 * a)))) * ((double) (((double) -(g)) + ((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h)))))))))))) + ((double) cbrt(((double) (((double) (1.0 / ((double) (2.0 * a)))) * ((double) (((double) -(g)) - ((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h))))))))))))));
}
↓
double code(double g, double h, double a) {
	return ((double) (((double) (((double) cbrt(((double) (((double) cbrt(((double) (1.0 / ((double) (2.0 * a)))))) * ((double) cbrt(((double) (1.0 / ((double) (2.0 * a)))))))))) * ((double) cbrt(((double) (((double) cbrt(((double) (1.0 / ((double) (2.0 * a)))))) * ((double) (((double) -(g)) + ((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h)))))))))))))) + ((double) (((double) cbrt(((double) (((double) (1.0 / ((double) sqrt(2.0)))) * ((double) (((double) -(g)) - ((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h)))))))))))) / ((double) cbrt(((double) (((double) sqrt(2.0)) * a))))))));
}

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 add-sqr-sqrt35.8

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

  4. Applied associate-*l*35.8

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

  5. Applied associate-/r*35.8

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

  6. Applied associate-*l/35.8

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

  7. Applied cbrt-div33.8

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

  9. Applied add-cube-cbrt33.8

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

  10. Applied associate-*l*33.8

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

  11. Applied cbrt-prod32.1

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

  12. Final simplification32.1

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

Reproduce

herbie shell --seed 2020113 +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))))))))