Average Error: 36.4 → 32.2
Time: 26.4s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := g + t_0\\ t_2 := t_0 - g\\ \mathbf{if}\;\begin{array}{l} t_3 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_2} + \sqrt[3]{t_1 \cdot \frac{-1}{2 \cdot a}}\\ t_3 \leq -8.639119737926423 \cdot 10^{-107} \lor \neg \left(t_3 \leq 0\right) \end{array}:\\ \;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{t_2} \cdot \sqrt[3]{\frac{1}{a}}\right) + \frac{\sqrt[3]{t_1 \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{t_2}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \end{array} \]
\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}\\
t_1 := g + t_0\\
t_2 := t_0 - g\\
\mathbf{if}\;\begin{array}{l}
t_3 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_2} + \sqrt[3]{t_1 \cdot \frac{-1}{2 \cdot a}}\\
t_3 \leq -8.639119737926423 \cdot 10^{-107} \lor \neg \left(t_3 \leq 0\right)
\end{array}:\\
\;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{t_2} \cdot \sqrt[3]{\frac{1}{a}}\right) + \frac{\sqrt[3]{t_1 \cdot -0.5}}{\sqrt[3]{a}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{t_2}{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)))) (t_1 (+ g t_0)) (t_2 (- t_0 g)))
   (if (let* ((t_3
               (+
                (cbrt (* (/ 1.0 (* 2.0 a)) t_2))
                (cbrt (* t_1 (/ -1.0 (* 2.0 a)))))))
         (or (<= t_3 -8.639119737926423e-107) (not (<= t_3 0.0))))
     (+
      (* (cbrt 0.5) (* (cbrt t_2) (cbrt (/ 1.0 a))))
      (/ (cbrt (* t_1 -0.5)) (cbrt a)))
     (+ (cbrt (/ t_2 (* 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));
	double t_1 = g + t_0;
	double t_2 = t_0 - g;
	double t_3 = cbrt((1.0 / (2.0 * a)) * t_2) + cbrt(t_1 * (-1.0 / (2.0 * a)));
	double tmp;
	if ((t_3 <= -8.639119737926423e-107) || !(t_3 <= 0.0)) {
		tmp = (cbrt(0.5) * (cbrt(t_2) * cbrt(1.0 / a))) + (cbrt(t_1 * -0.5) / cbrt(a));
	} else {
		tmp = cbrt(t_2 / (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 (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < -8.6391197379264234e-107 or 0.0 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))

    1. Initial program 35.5

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

    2. Simplified35.5

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

    3. Applied associate-*l/_binary6435.5

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

    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 add-cube-cbrt_binary6441.2

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

    6. Applied add-cube-cbrt_binary6434.4

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

    7. Applied prod-diff_binary6434.4

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

    8. Applied *-un-lft-identity_binary6434.4

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

    9. Applied times-frac_binary6434.4

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

    10. Applied cbrt-prod_binary6434.4

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

    11. Applied div-inv_binary6434.4

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

    12. Applied cbrt-prod_binary6433.2

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

    13. Simplified33.1

      \[\leadsto \sqrt[3]{\frac{1}{2}} \cdot \left(\color{blue}{\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) + 0}} \cdot \sqrt[3]{\frac{1}{a}}\right) + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}} \]

    if -8.6391197379264234e-107 < (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) < 0.0

    1. Initial program 60.7

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

    2. Simplified60.3

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

    3. Applied associate-*l/_binary6460.3

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

    4. Applied cbrt-div_binary6440.5

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

    5. Taylor expanded in g around inf 7.8

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

  4. Final simplification32.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -8.639119737926423 \cdot 10^{-107} \lor \neg \left(\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0\right):\\ \;\;\;\;\sqrt[3]{0.5} \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{1}{a}}\right) + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022005 
(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))))))))