Average Error: 36.0 → 32.4
Time: 15.7s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;g \leq 5.72792437368683 \cdot 10^{-152}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{g} \cdot \sqrt[3]{g}\\ t_2 := \sqrt[3]{g} \cdot t_1\\ t_3 := \sqrt[3]{t_0}\\ \sqrt[3]{\mathsf{fma}\left(t_3 \cdot t_3, t_3, -t_2\right) + \mathsf{fma}\left(-\sqrt[3]{g}, t_1, t_2\right)} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{g + t_0}{a} \cdot -0.5} \end{array}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{t_0 - g}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g + h} \cdot \sqrt{g - h}\right)}}{\sqrt[3]{a}}\\ \end{array} \]
\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}\\
\mathbf{if}\;g \leq 5.72792437368683 \cdot 10^{-152}:\\
\;\;\;\;\begin{array}{l}
t_1 := \sqrt[3]{g} \cdot \sqrt[3]{g}\\
t_2 := \sqrt[3]{g} \cdot t_1\\
t_3 := \sqrt[3]{t_0}\\
\sqrt[3]{\mathsf{fma}\left(t_3 \cdot t_3, t_3, -t_2\right) + \mathsf{fma}\left(-\sqrt[3]{g}, t_1, t_2\right)} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{g + t_0}{a} \cdot -0.5}
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{t_0 - g}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g + h} \cdot \sqrt{g - h}\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)))))
   (if (<= g 5.72792437368683e-152)
     (let* ((t_1 (* (cbrt g) (cbrt g)))
            (t_2 (* (cbrt g) t_1))
            (t_3 (cbrt t_0)))
       (+
        (*
         (cbrt (+ (fma (* t_3 t_3) t_3 (- t_2)) (fma (- (cbrt g)) t_1 t_2)))
         (cbrt (/ 0.5 a)))
        (cbrt (* (/ (+ g t_0) a) -0.5))))
     (+
      (cbrt (/ (- t_0 g) (* a 2.0)))
      (/ (cbrt (* -0.5 (+ g (* (sqrt (+ g h)) (sqrt (- g h)))))) (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 tmp;
	if (g <= 5.72792437368683e-152) {
		double t_1_1 = cbrt(g) * cbrt(g);
		double t_2_2 = cbrt(g) * t_1_1;
		double t_3_3 = cbrt(t_0);
		tmp = (cbrt(fma((t_3_3 * t_3_3), t_3_3, -t_2_2) + fma(-cbrt(g), t_1_1, t_2_2)) * cbrt(0.5 / a)) + cbrt(((g + t_0) / a) * -0.5);
	} else {
		tmp = cbrt((t_0 - g) / (a * 2.0)) + (cbrt(-0.5 * (g + (sqrt(g + h) * sqrt(g - h)))) / cbrt(a));
	}
	return tmp;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < 5.72792437368683024e-152

    1. Initial program 37.0

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

    2. Simplified37.0

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

    3. Applied div-inv_binary6437.0

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

    4. Applied cbrt-prod_binary6433.5

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

    5. Simplified33.5

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

    6. Applied add-cube-cbrt_binary6433.6

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

    7. Applied add-cube-cbrt_binary6433.5

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

    8. Applied prod-diff_binary6433.5

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

    if 5.72792437368683024e-152 < g

    1. Initial program 35.0

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

    2. Simplified35.0

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

    3. Applied associate-*l/_binary6435.0

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

    4. Applied cbrt-div_binary6431.1

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

    5. Applied difference-of-squares_binary6431.1

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

    6. Applied sqrt-prod_binary6431.1

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

    7. Simplified31.1

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

  4. Final simplification32.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 5.72792437368683 \cdot 10^{-152}:\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}, \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}, -\sqrt[3]{g} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{g}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{g}, \sqrt[3]{g} \cdot \sqrt[3]{g}, \sqrt[3]{g} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{g}\right)\right)} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g + h} \cdot \sqrt{g - h}\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021275 
(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))))))))