Average Error: 35.4 → 31.5
Time: 9.2s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 3.0797478610904441 \cdot 10^{-234}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g + h} \cdot \sqrt{g - h}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 \le 3.0797478610904441 \cdot 10^{-234}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\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)}\\

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

\end{array}
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 temp;
	if ((g <= 3.079747861090444e-234)) {
		temp = ((cbrt((1.0 / (2.0 * a))) * cbrt((-g + (sqrt(sqrt(((g * g) - (h * h)))) * sqrt(sqrt(((g * g) - (h * h)))))))) + cbrt(((1.0 / (2.0 * a)) * (-g - sqrt(((g * g) - (h * h)))))));
	} else {
		temp = (cbrt(((1.0 / (2.0 * a)) * (-g + sqrt(((g * g) - (h * h)))))) + (cbrt((1.0 * (-g - (sqrt((g + h)) * sqrt((g - h)))))) / cbrt((2.0 * a))));
	}
	return temp;
}

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 < 3.079747861090444e-234

    1. Initial program 35.9

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

    3. Applied cbrt-prod32.3

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

    5. Applied add-sqr-sqrt32.3

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

    6. Applied sqrt-prod32.3

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

    if 3.079747861090444e-234 < g

    1. Initial program 34.9

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

    3. Applied associate-*l/34.9

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

    4. Applied cbrt-div31.4

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

    6. Applied difference-of-squares31.4

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

    7. Applied sqrt-prod30.8

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

  4. Final simplification31.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 3.0797478610904441 \cdot 10^{-234}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g + h} \cdot \sqrt{g - h}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))