Average Error: 36.1 → 31.8
Time: 14.5s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := g + t_0\\ \mathbf{if}\;g \leq 3.464167688716374 \cdot 10^{-160}:\\ \;\;\;\;\sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{t_1}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{t_0 - g}{a \cdot 2}} + \frac{\sqrt[3]{t_1 \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array} \]
\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 := g + t_0\\
\mathbf{if}\;g \leq 3.464167688716374 \cdot 10^{-160}:\\
\;\;\;\;\sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{t_1}{a} \cdot -0.5}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{t_0 - g}{a \cdot 2}} + \frac{\sqrt[3]{t_1 \cdot -0.5}}{\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 (+ g t_0)))
   (if (<= g 3.464167688716374e-160)
     (+
      (* (cbrt (- (- (* 0.5 (/ (* h h) g)) g) g)) (cbrt (/ 0.5 a)))
      (cbrt (* (/ t_1 a) -0.5)))
     (+ (cbrt (/ (- t_0 g) (* a 2.0))) (/ (cbrt (* t_1 -0.5)) (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 = g + t_0;
	double tmp;
	if (g <= 3.464167688716374e-160) {
		tmp = (cbrt(((0.5 * ((h * h) / g)) - g) - g) * cbrt(0.5 / a)) + cbrt((t_1 / a) * -0.5);
	} else {
		tmp = cbrt((t_0 - g) / (a * 2.0)) + (cbrt(t_1 * -0.5) / 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 < 3.46416768871637378e-160

    1. Initial program 36.9

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

    2. Simplified36.8

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

    3. Applied div-inv_binary6436.9

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

    4. Applied cbrt-prod_binary6433.4

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

    5. Simplified33.3

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

    6. Taylor expanded in g around -inf 31.8

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

    7. Simplified31.8

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

    if 3.46416768871637378e-160 < g

    1. Initial program 35.4

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

    2. 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 associate-*l/_binary6435.3

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

    4. Applied cbrt-div_binary6431.7

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

  4. Final simplification31.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 3.464167688716374 \cdot 10^{-160}:\\ \;\;\;\;\sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g} - g\right) - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a \cdot 2}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022019 
(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))))))))