Average Error: 36.4 → 32.4
Time: 11.2s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := g + t_0\\ t_2 := t_0 - g\\ t_3 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_2} + \sqrt[3]{t_1 \cdot \frac{-1}{2 \cdot a}}\\ t_4 := \sqrt[3]{t_2}\\ \mathbf{if}\;t_3 \leq -1.9430711755519203 \cdot 10^{-100}:\\ \;\;\;\;\frac{t_4}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{t_1 \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;t_3 \leq 0:\\ \;\;\;\;\sqrt[3]{\frac{t_2}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{-0.5 \cdot \frac{g + \left|t_0\right|}{a}}\\ \end{array} \]
\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\\
t_3 := \sqrt[3]{\frac{1}{2 \cdot a} \cdot t_2} + \sqrt[3]{t_1 \cdot \frac{-1}{2 \cdot a}}\\
t_4 := \sqrt[3]{t_2}\\
\mathbf{if}\;t_3 \leq -1.9430711755519203 \cdot 10^{-100}:\\
\;\;\;\;\frac{t_4}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{t_1 \cdot -0.5}}{\sqrt[3]{a}}\\

\mathbf{elif}\;t_3 \leq 0:\\
\;\;\;\;\sqrt[3]{\frac{t_2}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\

\mathbf{else}:\\
\;\;\;\;t_4 \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{-0.5 \cdot \frac{g + \left|t_0\right|}{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))
        (t_3
         (+
          (cbrt (* (/ 1.0 (* 2.0 a)) t_2))
          (cbrt (* t_1 (/ -1.0 (* 2.0 a))))))
        (t_4 (cbrt t_2)))
   (if (<= t_3 -1.9430711755519203e-100)
     (+ (/ t_4 (cbrt (* 2.0 a))) (/ (cbrt (* t_1 -0.5)) (cbrt a)))
     (if (<= t_3 0.0)
       (+ (cbrt (/ t_2 (* 2.0 a))) (/ (cbrt (* -0.5 (+ g g))) (cbrt a)))
       (+ (* t_4 (cbrt (/ 0.5 a))) (cbrt (* -0.5 (/ (+ g (fabs t_0)) 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 t_4 = cbrt(t_2);
	double tmp;
	if (t_3 <= -1.9430711755519203e-100) {
		tmp = (t_4 / cbrt(2.0 * a)) + (cbrt(t_1 * -0.5) / cbrt(a));
	} else if (t_3 <= 0.0) {
		tmp = cbrt(t_2 / (2.0 * a)) + (cbrt(-0.5 * (g + g)) / cbrt(a));
	} else {
		tmp = (t_4 * cbrt(0.5 / a)) + cbrt(-0.5 * ((g + fabs(t_0)) / 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 (+.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))))))) < -1.94307117555192034e-100

    1. Initial program 10.6

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

    2. Simplified10.5

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

    3. Applied associate-*l/_binary6410.5

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

    4. Applied cbrt-div_binary648.5

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

    5. Applied cbrt-div_binary646.2

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

    if -1.94307117555192034e-100 < (+.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 57.4

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

    2. Simplified57.3

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

    3. Applied associate-*l/_binary6457.3

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

    4. Applied cbrt-div_binary6440.2

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

    5. Taylor expanded in g around inf 9.1

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

    if -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 44.2

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

    2. Simplified44.2

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

    3. Applied div-inv_binary6444.2

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

    4. Applied cbrt-prod_binary6443.2

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

    5. Simplified43.2

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

    6. Applied add-sqr-sqrt_binary6443.2

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

    7. Applied rem-sqrt-square_binary6443.2

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

  4. Final simplification32.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq -1.9430711755519203 \cdot 10^{-100}:\\ \;\;\;\;\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \mathbf{elif}\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{2 \cdot a}} \leq 0:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{-0.5 \cdot \frac{g + \left|\sqrt{g \cdot g - h \cdot h}\right|}{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021215 
(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))))))))