Average Error: 35.7 → 32.1
Time: 11.4s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{t_0 - g} \cdot \sqrt[3]{\frac{-1}{a}}\right) + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - t_0} \end{array} \]
\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}\\
\sqrt[3]{-0.5} \cdot \left(\sqrt[3]{t_0 - g} \cdot \sqrt[3]{\frac{-1}{a}}\right) + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - t_0}
\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)))))
   (+
    (* (cbrt -0.5) (* (cbrt (- t_0 g)) (cbrt (/ -1.0 a))))
    (* (cbrt (/ 0.5 a)) (cbrt (- (- g) 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 = sqrt((g * g) - (h * h));
	return (cbrt(-0.5) * (cbrt(t_0 - g) * cbrt(-1.0 / a))) + (cbrt(0.5 / a) * cbrt(-g - t_0));
}

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 35.7

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

  3. Applied pow1/3_binary6443.4

    \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{{\left(\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)\right)}^{0.3333333333333333}} \]

  4. Applied unpow-prod-down_binary6456.8

    \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{{\left(\frac{1}{2 \cdot a}\right)}^{0.3333333333333333} \cdot {\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}^{0.3333333333333333}} \]

  5. Simplified50.0

    \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \color{blue}{\sqrt[3]{\frac{0.5}{a}}} \cdot {\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}^{0.3333333333333333} \]

  6. Simplified33.8

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

  7. Taylor expanded around -inf 48.8

    \[\leadsto \color{blue}{\sqrt[3]{-0.5} \cdot e^{0.3333333333333333 \cdot \left(\log \left(\sqrt{{g}^{2} - {h}^{2}} - g\right) + \log \left(\frac{-1}{a}\right)\right)}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \]

  8. Simplified32.1

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

  9. Final simplification32.1

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

Reproduce

herbie shell --seed 2021211 
(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))))))))