Average Error: 36.2 → 32.0
Time: 10.2s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h} - g\\ \mathbf{if}\;g \leq -2.893269831035563 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt[3]{t_0}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \left(0.5 \cdot \frac{h \cdot h}{g} - g\right)}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{t_0}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \end{array} \]
\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} - g\\
\mathbf{if}\;g \leq -2.893269831035563 \cdot 10^{-182}:\\
\;\;\;\;\frac{\sqrt[3]{t_0}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \left(0.5 \cdot \frac{h \cdot h}{g} - g\right)}{a} \cdot -0.5}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{t_0}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\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))) g)))
   (if (<= g -2.893269831035563e-182)
     (+
      (/ (cbrt t_0) (cbrt (* 2.0 a)))
      (cbrt (* (/ (+ g (- (* 0.5 (/ (* h h) g)) g)) a) -0.5)))
     (+ (cbrt (/ t_0 (* 2.0 a))) (/ (cbrt (* -0.5 (+ g g))) (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)) - g;
	double tmp;
	if (g <= -2.893269831035563e-182) {
		tmp = (cbrt(t_0) / cbrt(2.0 * a)) + cbrt(((g + ((0.5 * ((h * h) / g)) - g)) / a) * -0.5);
	} else {
		tmp = cbrt(t_0 / (2.0 * a)) + (cbrt(-0.5 * (g + g)) / 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 < -2.8932698310355632e-182

    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. Simplified34.9

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

    3. Applied cbrt-div_binary6431.3

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

    4. Taylor expanded in g around -inf 31.3

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

    5. Simplified31.3

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

    if -2.8932698310355632e-182 < g

    1. Initial program 37.4

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

    2. Simplified37.4

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

    3. Applied associate-*l/_binary6437.4

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

    4. Applied cbrt-div_binary6433.6

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

    5. Taylor expanded in g around inf 32.6

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

  4. Final simplification32.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -2.893269831035563 \cdot 10^{-182}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{g + \left(0.5 \cdot \frac{h \cdot h}{g} - g\right)}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021224 
(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))))))))