Average Error: 35.6 → 31.9
Time: 8.1s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \leq -1.590876763403097 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h\right)}}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]
\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 \leq -1.590876763403097 \cdot 10^{-162}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\

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

\end{array}
(FPCore (g h a)
 :precision binary64
 (+
  (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h))))))
  (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))
↓
(FPCore (g h a)
 :precision binary64
 (if (<= g -1.590876763403097e-162)
   (+
    (* (cbrt (/ 1.0 (* 2.0 a))) (cbrt (- (sqrt (- (* g g) (* h h))) g)))
    (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h)))))))
   (+
    (/
     (cbrt (* (/ 1.0 (* 2.0 a)) (* h h)))
     (cbrt (- (- g) (sqrt (- (* g g) (* h h))))))
    (* (cbrt (/ 1.0 (* 2.0 a))) (cbrt (- (- g) (sqrt (- (* g g) (* h h)))))))))
double code(double g, double h, double a) {
	return ((double) (((double) cbrt(((double) ((1.0 / ((double) (2.0 * a))) * ((double) (((double) -(g)) + ((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h)))))))))))) + ((double) cbrt(((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) {
	double tmp;
	if ((g <= -1.590876763403097e-162)) {
		tmp = ((double) (((double) (((double) cbrt((1.0 / ((double) (2.0 * a))))) * ((double) cbrt(((double) (((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h)))))) - g)))))) + ((double) cbrt(((double) ((1.0 / ((double) (2.0 * a))) * ((double) (((double) -(g)) - ((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h))))))))))))));
	} else {
		tmp = ((double) ((((double) cbrt(((double) ((1.0 / ((double) (2.0 * a))) * ((double) (h * h)))))) / ((double) cbrt(((double) (((double) -(g)) - ((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h))))))))))) + ((double) (((double) cbrt((1.0 / ((double) (2.0 * a))))) * ((double) cbrt(((double) (((double) -(g)) - ((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h))))))))))))));
	}
	return tmp;
}

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 < -1.59087676340309706e-162

    1. Initial program Error: 34.9 bits

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

    3. Applied cbrt-prodError: 30.8 bits

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

    4. SimplifiedError: 30.8 bits

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

    if -1.59087676340309706e-162 < g

    1. Initial program Error: 36.2 bits

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

    3. Applied cbrt-prodError: 33.0 bits

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

    5. Applied flip-+Error: 32.9 bits

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

    6. Applied associate-*r/Error: 32.9 bits

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

    7. Applied cbrt-divError: 32.9 bits

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

    8. SimplifiedError: 32.9 bits

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

  4. Final simplificationError: 31.9 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.590876763403097 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h\right)}}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))