Average Error: 36.2 → 33.1
Time: 9.6s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{2 \cdot a}\\ t_1 := \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{if}\;g \leq 4.246245010575633 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot -2}}{t_0} + t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{\sqrt[3]{-0.5 \cdot \frac{{h}^{2}}{g}}}{t_0}\\ \end{array} \]
\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}
t_0 := \sqrt[3]{2 \cdot a}\\
t_1 := \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\
\mathbf{if}\;g \leq 4.246245010575633 \cdot 10^{-182}:\\
\;\;\;\;\frac{\sqrt[3]{g \cdot -2}}{t_0} + t_1\\

\mathbf{else}:\\
\;\;\;\;t_1 + \frac{\sqrt[3]{-0.5 \cdot \frac{{h}^{2}}{g}}}{t_0}\\


\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
 (let* ((t_0 (cbrt (* 2.0 a)))
        (t_1 (cbrt (* (/ (+ g (sqrt (- (* g g) (* h h)))) a) -0.5))))
   (if (<= g 4.246245010575633e-182)
     (+ (/ (cbrt (* g -2.0)) t_0) t_1)
     (+ t_1 (/ (cbrt (* -0.5 (/ (pow h 2.0) g))) t_0)))))
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 t_0 = cbrt(2.0 * a);
	double t_1 = cbrt(((g + sqrt((g * g) - (h * h))) / a) * -0.5);
	double tmp;
	if (g <= 4.246245010575633e-182) {
		tmp = (cbrt(g * -2.0) / t_0) + t_1;
	} else {
		tmp = t_1 + (cbrt(-0.5 * (pow(h, 2.0) / g)) / t_0);
	}
	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 < 4.2462450105756327e-182

    1. Initial program 37.3

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

    2. Simplified37.3

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

    3. Applied cbrt-div_binary6433.7

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

    4. Taylor expanded in g around -inf 32.7

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

    if 4.2462450105756327e-182 < g

    1. Initial program 35.0

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

    2. Simplified35.0

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

    3. Applied cbrt-div_binary6434.9

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

    4. Taylor expanded in g around inf 33.5

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

  4. Final simplification33.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 4.246245010575633 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5} + \frac{\sqrt[3]{-0.5 \cdot \frac{{h}^{2}}{g}}}{\sqrt[3]{2 \cdot a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022088 
(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))))))))