Average Error: 36.1 → 31.4
Time: 16.4s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{a \cdot 2}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := t_1 - g\\ t_3 := \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{t_2}\\ \mathbf{if}\;g \leq -1.2893397083911 \cdot 10^{-188}:\\ \;\;\;\;t_3 + \frac{\sqrt[3]{\frac{h \cdot h}{g} \cdot -0.5}}{t_0}\\ \mathbf{elif}\;g \leq 4.467477958425206 \cdot 10^{-133}:\\ \;\;\;\;\sqrt[3]{t_2 \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{g \cdot -2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := \sqrt{t_1}\\ t_3 + \frac{\sqrt[3]{\left(-g\right) - t_4 \cdot t_4}}{t_0} \end{array}\\ \end{array} \]
\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[3]{a \cdot 2}\\
t_1 := \sqrt{g \cdot g - h \cdot h}\\
t_2 := t_1 - g\\
t_3 := \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{t_2}\\
\mathbf{if}\;g \leq -1.2893397083911 \cdot 10^{-188}:\\
\;\;\;\;t_3 + \frac{\sqrt[3]{\frac{h \cdot h}{g} \cdot -0.5}}{t_0}\\

\mathbf{elif}\;g \leq 4.467477958425206 \cdot 10^{-133}:\\
\;\;\;\;\sqrt[3]{t_2 \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{g \cdot -2}}{t_0}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_4 := \sqrt{t_1}\\
t_3 + \frac{\sqrt[3]{\left(-g\right) - t_4 \cdot t_4}}{t_0}
\end{array}\\


\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 (cbrt (* a 2.0)))
        (t_1 (sqrt (- (* g g) (* h h))))
        (t_2 (- t_1 g))
        (t_3 (* (cbrt (/ 0.5 a)) (cbrt t_2))))
   (if (<= g -1.2893397083911e-188)
     (+ t_3 (/ (cbrt (* (/ (* h h) g) -0.5)) t_0))
     (if (<= g 4.467477958425206e-133)
       (+ (cbrt (* t_2 (/ 1.0 (* a 2.0)))) (/ (cbrt (* g -2.0)) t_0))
       (let* ((t_4 (sqrt t_1)))
         (+ t_3 (/ (cbrt (- (- g) (* t_4 t_4))) t_0)))))))
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 = cbrt(a * 2.0);
	double t_1 = sqrt((g * g) - (h * h));
	double t_2 = t_1 - g;
	double t_3 = cbrt(0.5 / a) * cbrt(t_2);
	double tmp;
	if (g <= -1.2893397083911e-188) {
		tmp = t_3 + (cbrt(((h * h) / g) * -0.5) / t_0);
	} else if (g <= 4.467477958425206e-133) {
		tmp = cbrt(t_2 * (1.0 / (a * 2.0))) + (cbrt(g * -2.0) / t_0);
	} else {
		double t_4 = sqrt(t_1);
		tmp = t_3 + (cbrt(-g - (t_4 * t_4)) / t_0);
	}
	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 g < -1.2893397083910999e-188

    1. Initial program 35.2

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

    2. Applied associate-*l/_binary6435.2

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

    3. Applied cbrt-div_binary6435.2

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

    4. Simplified35.2

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

    5. Applied cbrt-prod_binary6431.3

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

    6. Simplified31.3

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

    7. Simplified31.3

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

    8. Taylor expanded in g around -inf 30.7

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

    9. Simplified30.7

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

    if -1.2893397083910999e-188 < g < 4.46747795842520638e-133

    1. Initial program 49.4

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

    2. Applied associate-*l/_binary6449.4

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

    3. Applied cbrt-div_binary6444.1

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

    4. Simplified44.1

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

    5. Taylor expanded in g around inf 33.4

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

    if 4.46747795842520638e-133 < g

    1. Initial program 35.1

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

    2. Applied associate-*l/_binary6435.1

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

    3. Applied cbrt-div_binary6431.9

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

    4. Simplified31.9

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

    5. Applied cbrt-prod_binary6431.8

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

    6. Simplified31.8

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

    7. Simplified31.8

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

    8. Applied add-sqr-sqrt_binary6431.8

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

  4. Final simplification31.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.2893397083911 \cdot 10^{-188}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\sqrt[3]{\frac{h \cdot h}{g} \cdot -0.5}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{elif}\;g \leq 4.467477958425206 \cdot 10^{-133}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{1}{a \cdot 2}} + \frac{\sqrt[3]{g \cdot -2}}{\sqrt[3]{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}}}}{\sqrt[3]{a \cdot 2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(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))))))))