Average Error: 36.1 → 32.1
Time: 6.5s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \leq -1.1576464449940226 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 -1.1576464449940226 \cdot 10^{-189}:\\
\;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}} - g\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)}\\

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

\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 -1.1576464449940226e-189)
   (+
    (/
     (cbrt
      (*
       1.0
       (-
        (* (fabs (cbrt (- (* g g) (* h h)))) (sqrt (cbrt (- (* g g) (* h h)))))
        g)))
     (cbrt (* 2.0 a)))
    (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h)))))))
   (+
    (cbrt (* (/ 1.0 (* 2.0 a)) (- (sqrt (- (* g g) (* h h))) g)))
    (/ (cbrt (* 1.0 (- (- g) g))) (cbrt (* 2.0 a))))))
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 <= -1.1576464449940226e-189)) {
		tmp = ((double) ((((double) cbrt(((double) (1.0 * ((double) (((double) (((double) fabs(((double) cbrt(((double) (((double) (g * g)) - ((double) (h * h)))))))) * ((double) sqrt(((double) cbrt(((double) (((double) (g * g)) - ((double) (h * h)))))))))) - g)))))) / ((double) cbrt(((double) (2.0 * a))))) + ((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) (((double) sqrt(((double) (((double) (g * g)) - ((double) (h * h)))))) - g)))))) + (((double) cbrt(((double) (1.0 * ((double) (((double) -(g)) - g)))))) / ((double) cbrt(((double) (2.0 * a)))))));
	}
	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 < -1.1576464449940226e-189

    1. Initial program 35.3

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

    2. Simplified35.3

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

    4. Applied associate-*l/_binary6435.3

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

    5. Applied cbrt-div_binary6431.8

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

    7. Applied add-cube-cbrt_binary6431.8

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

    8. Applied sqrt-prod_binary6431.8

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

    9. Simplified31.8

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

    if -1.1576464449940226e-189 < g

    1. Initial program 36.9

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

    2. Simplified36.9

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

    4. Applied associate-*l/_binary6436.9

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

    5. Applied cbrt-div_binary6433.6

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

    6. Taylor expanded around inf 32.3

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

  4. Final simplification32.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.1576464449940226 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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))))))))