Average Error: 36.0 → 32.3
Time: 21.7s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{g + t_0}\\ \frac{\sqrt[3]{t_0 - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(t_1 \cdot \left(t_1 \cdot t_1\right)\right) \cdot -0.5}}{\sqrt[3]{a}} \end{array} \]
\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 := \sqrt[3]{g + t_0}\\
\frac{\sqrt[3]{t_0 - g}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{\left(t_1 \cdot \left(t_1 \cdot t_1\right)\right) \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 (cbrt (+ g t_0))))
   (+
    (/ (cbrt (- t_0 g)) (cbrt (* 2.0 a)))
    (/ (cbrt (* (* t_1 (* t_1 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 = cbrt(g + t_0);
	return (cbrt(t_0 - g) / cbrt(2.0 * a)) + (cbrt((t_1 * (t_1 * t_1)) * -0.5) / cbrt(a));
}

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. Initial program 36.0

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

  2. Simplified36.0

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

  3. Applied cbrt-div_binary6434.0

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

  4. Applied add-cube-cbrt_binary6434.1

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

  5. Applied associate-*l/_binary6434.1

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

  6. Applied cbrt-div_binary6432.3

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

  7. Final simplification32.3

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

Reproduce

herbie shell --seed 2021291 
(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))))))))