Average Error: 35.4 → 31.3
Time: 17.1s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{\left(g + t_0\right) \cdot \frac{-1}{a \cdot 2}}\\ t_2 := \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{if}\;g \leq 8.577312417414274 \cdot 10^{-162}:\\ \;\;\;\;t_2 \cdot \sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g} + t_1\\ \mathbf{elif}\;g \leq 7.483790593490912 \cdot 10^{+24}:\\ \;\;\;\;t_1 + \frac{\sqrt[3]{-0.5 \cdot \frac{{h}^{2}}{g}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(t_0 - g\right)} + t_2 \cdot \sqrt[3]{\left(-g\right) - t_0}\\ \end{array} \]
\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{g \cdot g - h \cdot h}\\
t_1 := \sqrt[3]{\left(g + t_0\right) \cdot \frac{-1}{a \cdot 2}}\\
t_2 := \sqrt[3]{\frac{0.5}{a}}\\
\mathbf{if}\;g \leq 8.577312417414274 \cdot 10^{-162}:\\
\;\;\;\;t_2 \cdot \sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g} + t_1\\

\mathbf{elif}\;g \leq 7.483790593490912 \cdot 10^{+24}:\\
\;\;\;\;t_1 + \frac{\sqrt[3]{-0.5 \cdot \frac{{h}^{2}}{g}}}{\sqrt[3]{a \cdot 2}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(t_0 - g\right)} + t_2 \cdot \sqrt[3]{\left(-g\right) - 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 (sqrt (- (* g g) (* h h))))
        (t_1 (cbrt (* (+ g t_0) (/ -1.0 (* a 2.0)))))
        (t_2 (cbrt (/ 0.5 a))))
   (if (<= g 8.577312417414274e-162)
     (+ (* t_2 (cbrt (- (- (* 0.5 (/ (* h h) g)) g) g))) t_1)
     (if (<= g 7.483790593490912e+24)
       (+ t_1 (/ (cbrt (* -0.5 (/ (pow h 2.0) g))) (cbrt (* a 2.0))))
       (+
        (cbrt (* (/ 1.0 (* a 2.0)) (- t_0 g)))
        (* t_2 (cbrt (- (- 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 = sqrt((g * g) - (h * h));
	double t_1 = cbrt((g + t_0) * (-1.0 / (a * 2.0)));
	double t_2 = cbrt(0.5 / a);
	double tmp;
	if (g <= 8.577312417414274e-162) {
		tmp = (t_2 * cbrt(((0.5 * ((h * h) / g)) - g) - g)) + t_1;
	} else if (g <= 7.483790593490912e+24) {
		tmp = t_1 + (cbrt(-0.5 * (pow(h, 2.0) / g)) / cbrt(a * 2.0));
	} else {
		tmp = cbrt((1.0 / (a * 2.0)) * (t_0 - g)) + (t_2 * cbrt(-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 3 regimes

  2. if g < 8.57731241741427389e-162

    1. Initial program 36.8

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

    2. Applied cbrt-prod_binary6433.2

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

    3. Simplified33.1

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

    4. Simplified33.1

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

    5. Taylor expanded in g around -inf 31.9

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

    6. Simplified31.9

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

    if 8.57731241741427389e-162 < g < 7.483790593490912e24

    1. Initial program 6.5

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

    2. Applied associate-*l/_binary646.5

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

    3. Applied cbrt-div_binary646.5

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

    4. Simplified6.5

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

    5. Taylor expanded in g around inf 5.4

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

    if 7.483790593490912e24 < g

    1. Initial program 43.7

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

    2. Applied cbrt-prod_binary6439.7

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

    3. Simplified39.7

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\sqrt[3]{\frac{0.5}{a}}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification31.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 8.577312417414274 \cdot 10^{-162}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}}\\ \mathbf{elif}\;g \leq 7.483790593490912 \cdot 10^{+24}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \frac{{h}^{2}}{g}}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022055 
(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))))))))