Average Error: 35.5 → 30.9
Time: 14.8s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \leq 3.1238534360467275 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g - g}{2 \cdot a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array}\]
\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 3.1238534360467275 \cdot 10^{-176}:\\
\;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{g - g}{2 \cdot a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{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 3.1238534360467275e-176)
   (+
    (/ (cbrt (- (- g) g)) (cbrt (* 2.0 a)))
    (cbrt (* (/ (+ g (sqrt (- (* g g) (* h h)))) a) -0.5)))
   (+
    (cbrt (/ (- g g) (* 2.0 a)))
    (/ (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) -0.5)) (cbrt a)))))
double code(double g, double h, double a) {
	return cbrt((1.0 / (2.0 * a)) * (-g + sqrt((g * g) - (h * h)))) + cbrt((1.0 / (2.0 * a)) * (-g - sqrt((g * g) - (h * h))));
}
↓
double code(double g, double h, double a) {
	double tmp;
	if (g <= 3.1238534360467275e-176) {
		tmp = (cbrt(-g - g) / cbrt(2.0 * a)) + cbrt(((g + sqrt((g * g) - (h * h))) / a) * -0.5);
	} else {
		tmp = cbrt((g - g) / (2.0 * a)) + (cbrt((g + sqrt((g * g) - (h * h))) * -0.5) / cbrt(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 < 3.12385343604672747e-176

    1. Initial program 36.4

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

    2. Simplified36.4

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}}\]
    3. Using strategy rm

    4. Applied cbrt-div_binary64_352032.5

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\]

    5. Taylor expanded around -inf 31.5

      \[\leadsto \frac{\sqrt[3]{\color{blue}{-1 \cdot g} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\]

    if 3.12385343604672747e-176 < g

    1. Initial program 34.5

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

    2. Simplified34.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}}\]
    3. Using strategy rm

    4. Applied associate-*l/_binary64_343134.5

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\color{blue}{\frac{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}{a}}}\]

    5. Applied cbrt-div_binary64_352030.1

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \color{blue}{\frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}}\]

    6. Simplified30.1

      \[\leadsto \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\color{blue}{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}}}{\sqrt[3]{a}}\]

    7. Taylor expanded around inf 30.3

      \[\leadsto \sqrt[3]{\frac{\color{blue}{g} - g}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{a}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification30.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 3.1238534360467275 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g - g}{2 \cdot a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021022 
(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))))))))