Average Error: 36.2 → 31.4
Time: 15.1s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := g + \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{2 \cdot a}\\ \mathbf{if}\;g \leq 1.854914823509544 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot -2}}{t_1} + \sqrt[3]{\frac{t_0}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \sqrt{t_0}\\ \frac{\sqrt[3]{-0.5 \cdot \frac{h \cdot h}{g}}}{t_1} + \frac{\sqrt[3]{-0.5 \cdot \left(t_2 \cdot t_2\right)}}{\sqrt[3]{a}} \end{array}\\ \end{array} \]
\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 := g + \sqrt{g \cdot g - h \cdot h}\\
t_1 := \sqrt[3]{2 \cdot a}\\
\mathbf{if}\;g \leq 1.854914823509544 \cdot 10^{-163}:\\
\;\;\;\;\frac{\sqrt[3]{g \cdot -2}}{t_1} + \sqrt[3]{\frac{t_0}{a} \cdot -0.5}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_2 := \sqrt{t_0}\\
\frac{\sqrt[3]{-0.5 \cdot \frac{h \cdot h}{g}}}{t_1} + \frac{\sqrt[3]{-0.5 \cdot \left(t_2 \cdot t_2\right)}}{\sqrt[3]{a}}
\end{array}\\


\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 (+ g (sqrt (- (* g g) (* h h))))) (t_1 (cbrt (* 2.0 a))))
   (if (<= g 1.854914823509544e-163)
     (+ (/ (cbrt (* g -2.0)) t_1) (cbrt (* (/ t_0 a) -0.5)))
     (let* ((t_2 (sqrt t_0)))
       (+
        (/ (cbrt (* -0.5 (/ (* h h) g))) t_1)
        (/ (cbrt (* -0.5 (* t_2 t_2))) (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 t_0 = g + sqrt((g * g) - (h * h));
	double t_1 = cbrt(2.0 * a);
	double tmp;
	if (g <= 1.854914823509544e-163) {
		tmp = (cbrt(g * -2.0) / t_1) + cbrt((t_0 / a) * -0.5);
	} else {
		double t_2 = sqrt(t_0);
		tmp = (cbrt(-0.5 * ((h * h) / g)) / t_1) + (cbrt(-0.5 * (t_2 * t_2)) / 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 < 1.854914823509544e-163

    1. Initial program 37.1

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

    2. Simplified37.1

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

    3. Applied cbrt-div_binary6433.2

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

    4. Taylor expanded in g around -inf 32.1

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

    if 1.854914823509544e-163 < g

    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

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

    3. Applied cbrt-div_binary6435.2

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

    4. Applied associate-*l/_binary6435.2

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

    5. Applied cbrt-div_binary6431.2

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

    6. Simplified31.2

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

    7. Applied add-sqr-sqrt_binary6431.2

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

    8. Taylor expanded in g around inf 30.7

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

    9. Simplified30.7

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

  4. Final simplification31.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 1.854914823509544 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{-0.5 \cdot \frac{h \cdot h}{g}}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(\sqrt{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{g + \sqrt{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022104 
(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))))))))