Average Error: 35.6 → 32.0
Time: 17.6s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \leq 3.1405143144072282 \cdot 10^{-183}:\\ \;\;\;\;\left(\sqrt[3]{g \cdot -2} \cdot \sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{-0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}}\\ \mathbf{elif}\;g \leq 7.721957821369504 \cdot 10^{+69}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} + \frac{\sqrt[3]{0.5 \cdot \frac{h \cdot h}{a}}}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]
\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}
\mathbf{if}\;g \leq 3.1405143144072282 \cdot 10^{-183}:\\
\;\;\;\;\left(\sqrt[3]{g \cdot -2} \cdot \sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{-0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}}\\

\mathbf{elif}\;g \leq 7.721957821369504 \cdot 10^{+69}:\\
\;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} + \frac{\sqrt[3]{0.5 \cdot \frac{h \cdot h}{a}}}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}\\

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

\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
 (if (<= g 3.1405143144072282e-183)
   (+
    (* (* (cbrt (* g -2.0)) (cbrt (/ -1.0 a))) (cbrt -0.5))
    (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -1.0 (* a 2.0)))))
   (if (<= g 7.721957821369504e+69)
     (+
      (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) (/ -1.0 (* a 2.0))))
      (/
       (cbrt (* 0.5 (/ (* h h) a)))
       (cbrt (- (- g) (sqrt (- (* g g) (* h h)))))))
     (+
      (cbrt (* (/ 1.0 (* a 2.0)) (- (sqrt (- (* g g) (* h h))) g)))
      (/ (cbrt (- (- g) (sqrt (- (* g g) (* h h))))) (cbrt (* a 2.0)))))))
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 tmp;
	if (g <= 3.1405143144072282e-183) {
		tmp = ((cbrt(g * -2.0) * cbrt(-1.0 / a)) * cbrt(-0.5)) + cbrt((g + sqrt((g * g) - (h * h))) * (-1.0 / (a * 2.0)));
	} else if (g <= 7.721957821369504e+69) {
		tmp = cbrt((g + sqrt((g * g) - (h * h))) * (-1.0 / (a * 2.0))) + (cbrt(0.5 * ((h * h) / a)) / cbrt(-g - sqrt((g * g) - (h * h))));
	} else {
		tmp = cbrt((1.0 / (a * 2.0)) * (sqrt((g * g) - (h * h)) - g)) + (cbrt(-g - sqrt((g * g) - (h * h))) / cbrt(a * 2.0));
	}
	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 < 3.1405143144072282e-183

    1. Initial program 36.3

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

    2. Taylor expanded around -inf 49.6

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

    3. Simplified32.8

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

    4. Taylor expanded around -inf 31.7

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

    if 3.1405143144072282e-183 < g < 7.7219578213695045e69

    1. Initial program 9.3

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

    3. Applied flip-+_binary64_55089.0

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

    4. Applied associate-*r/_binary64_54769.1

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

    5. Applied cbrt-div_binary64_55669.1

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

    6. Simplified8.0

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

    if 7.7219578213695045e69 < g

    1. Initial program 48.2

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

    3. Applied associate-*l/_binary64_547748.2

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

    4. Applied cbrt-div_binary64_556644.9

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

    5. Simplified44.9

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \frac{\color{blue}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}}{\sqrt[3]{2 \cdot a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification32.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq 3.1405143144072282 \cdot 10^{-183}:\\ \;\;\;\;\left(\sqrt[3]{g \cdot -2} \cdot \sqrt[3]{\frac{-1}{a}}\right) \cdot \sqrt[3]{-0.5} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}}\\ \mathbf{elif}\;g \leq 7.721957821369504 \cdot 10^{+69}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-1}{a \cdot 2}} + \frac{\sqrt[3]{0.5 \cdot \frac{h \cdot h}{a}}}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a \cdot 2} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}}{\sqrt[3]{a \cdot 2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(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))))))))