Average Error: 35.6 → 30.9
Time: 7.1s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \leq -7.035285471056203 \cdot 10^{-159}:\\ \;\;\;\;\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{elif}\;g \leq 1.037351012381759 \cdot 10^{-131}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{2}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array}\]
\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}
\mathbf{if}\;g \leq -7.035285471056203 \cdot 10^{-159}:\\
\;\;\;\;\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\

\mathbf{elif}\;g \leq 1.037351012381759 \cdot 10^{-131}:\\
\;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{2}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\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
 (if (<= g -7.035285471056203e-159)
   (+
    (* (cbrt (- (sqrt (- (* g g) (* h h))) g)) (cbrt (/ 0.5 a)))
    (cbrt (* (/ (+ g (sqrt (- (* g g) (* h h)))) a) -0.5)))
   (if (<= g 1.037351012381759e-131)
     (+
      (cbrt (/ (- (sqrt (- (* g g) (* h h))) g) (* a 2.0)))
      (/ (cbrt (* -0.5 (+ g g))) (cbrt a)))
     (+
      (*
       (cbrt
        (/
         (*
          (cbrt (- (sqrt (- (* g g) (* h h))) g))
          (cbrt (- (sqrt (- (* g g) (* h h))) g)))
         2.0))
       (cbrt (/ (cbrt (- (sqrt (- (* g g) (* h h))) g)) a)))
      (/ (cbrt (* (+ g (sqrt (- (* g g) (* h h)))) -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 tmp;
	if (g <= -7.035285471056203e-159) {
		tmp = (cbrt(sqrt((g * g) - (h * h)) - g) * cbrt(0.5 / a)) + cbrt(((g + sqrt((g * g) - (h * h))) / a) * -0.5);
	} else if (g <= 1.037351012381759e-131) {
		tmp = cbrt((sqrt((g * g) - (h * h)) - g) / (a * 2.0)) + (cbrt(-0.5 * (g + g)) / cbrt(a));
	} else {
		tmp = (cbrt((cbrt(sqrt((g * g) - (h * h)) - g) * cbrt(sqrt((g * g) - (h * h)) - g)) / 2.0) * cbrt(cbrt(sqrt((g * g) - (h * h)) - g) / a)) + (cbrt((g + sqrt((g * g) - (h * h))) * -0.5) / cbrt(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 g < -7.0352854710562029e-159

    1. Initial program 33.9

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

    2. Simplified33.9

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

    4. Applied div-inv_binary64_246233.9

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

    5. Applied cbrt-prod_binary64_249630.3

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

    6. Simplified30.3

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

    if -7.0352854710562029e-159 < g < 1.03735101238175904e-131

    1. Initial program 48.3

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

    2. Simplified48.3

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

    4. Applied associate-*l/_binary64_240848.3

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

    5. Applied cbrt-div_binary64_249742.9

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

    6. Taylor expanded around inf 32.1

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

    if 1.03735101238175904e-131 < g

    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. Simplified35.2

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

    4. Applied associate-*l/_binary64_240835.2

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

    5. Applied cbrt-div_binary64_249731.4

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

    7. Applied add-cube-cbrt_binary64_250031.4

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

    8. Applied times-frac_binary64_247131.4

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

    9. Applied cbrt-prod_binary64_249631.4

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

  4. Final simplification30.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -7.035285471056203 \cdot 10^{-159}:\\ \;\;\;\;\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}\\ \mathbf{elif}\;g \leq 1.037351012381759 \cdot 10^{-131}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a \cdot 2}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{2}} \cdot \sqrt[3]{\frac{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}{a}} + \frac{\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot -0.5}}{\sqrt[3]{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020343 
(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))))))))