Average Error: 35.5 → 30.5
Time: 14.9s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := t_0 - g\\ t_2 := \sqrt[3]{\frac{0.5}{a}}\\ \mathbf{if}\;g \leq -5.625096900594615 \cdot 10^{-182}:\\ \;\;\;\;\sqrt[3]{t_1} \cdot t_2 + \frac{\sqrt[3]{\left(0.5 \cdot \frac{{h}^{2}}{g}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;g \leq 8.469717689810226 \cdot 10^{-144}:\\ \;\;\;\;\sqrt[3]{\frac{t_1}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sqrt[3]{-0.5 \cdot \frac{h \cdot h}{g}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + t_0\right)}}{\sqrt[3]{a}}\\ \end{array} \]
\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}
t_0 := \sqrt{g \cdot g - h \cdot h}\\
t_1 := t_0 - g\\
t_2 := \sqrt[3]{\frac{0.5}{a}}\\
\mathbf{if}\;g \leq -5.625096900594615 \cdot 10^{-182}:\\
\;\;\;\;\sqrt[3]{t_1} \cdot t_2 + \frac{\sqrt[3]{\left(0.5 \cdot \frac{{h}^{2}}{g}\right) \cdot -0.5}}{\sqrt[3]{a}}\\

\mathbf{elif}\;g \leq 8.469717689810226 \cdot 10^{-144}:\\
\;\;\;\;\sqrt[3]{\frac{t_1}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\

\mathbf{else}:\\
\;\;\;\;t_2 \cdot \sqrt[3]{-0.5 \cdot \frac{h \cdot h}{g}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + t_0\right)}}{\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
 (let* ((t_0 (sqrt (- (* g g) (* h h))))
        (t_1 (- t_0 g))
        (t_2 (cbrt (/ 0.5 a))))
   (if (<= g -5.625096900594615e-182)
     (+
      (* (cbrt t_1) t_2)
      (/ (cbrt (* (* 0.5 (/ (pow h 2.0) g)) -0.5)) (cbrt a)))
     (if (<= g 8.469717689810226e-144)
       (+ (cbrt (/ t_1 (* a 2.0))) (/ (cbrt (* -0.5 (+ g g))) (cbrt a)))
       (+
        (* t_2 (cbrt (* -0.5 (/ (* h h) g))))
        (/ (cbrt (* -0.5 (+ g t_0))) (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 = sqrt((g * g) - (h * h));
	double t_1 = t_0 - g;
	double t_2 = cbrt(0.5 / a);
	double tmp;
	if (g <= -5.625096900594615e-182) {
		tmp = (cbrt(t_1) * t_2) + (cbrt((0.5 * (pow(h, 2.0) / g)) * -0.5) / cbrt(a));
	} else if (g <= 8.469717689810226e-144) {
		tmp = cbrt(t_1 / (a * 2.0)) + (cbrt(-0.5 * (g + g)) / cbrt(a));
	} else {
		tmp = (t_2 * cbrt(-0.5 * ((h * h) / g))) + (cbrt(-0.5 * (g + t_0)) / 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 3 regimes

  2. if g < -5.62509690059461486e-182

    1. Initial program 34.3

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

    2. Simplified34.3

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

    3. Applied associate-*l/_binary6434.3

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

    4. Applied cbrt-div_binary6434.3

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

    5. Applied div-inv_binary6434.3

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

    6. Applied cbrt-prod_binary6430.6

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

    7. Simplified30.6

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

    8. Taylor expanded in g around -inf 30.3

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

    if -5.62509690059461486e-182 < g < 8.46971768981022565e-144

    1. Initial program 50.6

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

    2. Simplified50.6

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

    3. Applied associate-*l/_binary6450.6

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

    4. Applied cbrt-div_binary6446.1

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

    5. Taylor expanded in g around inf 33.4

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

    if 8.46971768981022565e-144 < g

    1. Initial program 34.8

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

    2. Simplified34.8

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

    3. Applied associate-*l/_binary6434.8

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

    4. Applied cbrt-div_binary6430.9

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

    5. Applied div-inv_binary6430.9

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

    6. Applied cbrt-prod_binary6430.8

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

    7. Simplified30.8

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

    8. Taylor expanded in g around inf 30.4

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

    9. Simplified30.4

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

  4. Final simplification30.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -5.625096900594615 \cdot 10^{-182}:\\ \;\;\;\;\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \frac{\sqrt[3]{\left(0.5 \cdot \frac{{h}^{2}}{g}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;g \leq 8.469717689810226 \cdot 10^{-144}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{-0.5 \cdot \frac{h \cdot h}{g}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(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))))))))