2-ancestry mixing, positive discriminant

Average Error: 42.9% → 98.1%

Time: 15.2s

Precision: binary64

Cost: 27328.00

Math

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

↓

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

FPCore

(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
 (+
  (/ (cbrt (* 0.5 (- (- (* h (* 0.5 (/ h g))) g) g))) (cbrt a))
  (/ (cbrt (/ (* h -0.5) (/ g (* 0.5 h)))) (cbrt a))))

C

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) {
	return (cbrt((0.5 * (((h * (0.5 * (h / g))) - g) - g))) / cbrt(a)) + (cbrt(((h * -0.5) / (g / (0.5 * h)))) / cbrt(a));
}

Java

public static double code(double g, double h, double a) {
	return Math.cbrt(((1.0 / (2.0 * a)) * (-g + Math.sqrt(((g * g) - (h * h)))))) + Math.cbrt(((1.0 / (2.0 * a)) * (-g - Math.sqrt(((g * g) - (h * h))))));
}

↓

public static double code(double g, double h, double a) {
	return (Math.cbrt((0.5 * (((h * (0.5 * (h / g))) - g) - g))) / Math.cbrt(a)) + (Math.cbrt(((h * -0.5) / (g / (0.5 * h)))) / Math.cbrt(a));
}

Julia

function code(g, h, a)
	return Float64(cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) + sqrt(Float64(Float64(g * g) - Float64(h * h)))))) + cbrt(Float64(Float64(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) - sqrt(Float64(Float64(g * g) - Float64(h * h)))))))
end

↓

function code(g, h, a)
	return Float64(Float64(cbrt(Float64(0.5 * Float64(Float64(Float64(h * Float64(0.5 * Float64(h / g))) - g) - g))) / cbrt(a)) + Float64(cbrt(Float64(Float64(h * -0.5) / Float64(g / Float64(0.5 * h)))) / cbrt(a)))
end

Wolfram

code[g_, h_, a_] := N[(N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[((-g) + N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(N[(1.0 / N[(2.0 * a), $MachinePrecision]), $MachinePrecision] * N[((-g) - N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

↓

code[g_, h_, a_] := N[(N[(N[Power[N[(0.5 * N[(N[(N[(h * N[(0.5 * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - g), $MachinePrecision] - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(h * -0.5), $MachinePrecision] / N[(g / N[(0.5 * h), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.9

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

2. Simplified 42.9

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

   [Start] 42.9	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   +-commutative [=>] 42.9	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   associate-/r* [=>] 42.9	\[ \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [=>] 42.9	\[ \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   unsub-neg [=>] 42.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   sub-neg [=>] 42.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   distribute-neg-out [=>] 42.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   neg-mul-1 [=>] 42.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   associate-*r* [=>] 42.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]
   *-commutative [<=] 42.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \left(\frac{1}{2 \cdot a} \cdot -1\right)}} \]
   associate-*l/ [=>] 42.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \color{blue}{\frac{1 \cdot -1}{2 \cdot a}}} \]

3. Taylor expanded in g around -inf 25.9

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

4. Simplified 25.9

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

   [Start] 25.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(0.5 \cdot \frac{{h}^{2}}{g} + -1 \cdot g\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   mul-1-neg [=>] 25.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(0.5 \cdot \frac{{h}^{2}}{g} + \color{blue}{\left(-g\right)}\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   unsub-neg [=>] 25.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{{h}^{2}}{g} - g\right)} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   unpow2 [=>] 25.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(0.5 \cdot \frac{\color{blue}{h \cdot h}}{g} - g\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   associate-/l* [=>] 25.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(0.5 \cdot \color{blue}{\frac{h}{\frac{g}{h}}} - g\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   associate-*r/ [=>] 25.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\color{blue}{\frac{0.5 \cdot h}{\frac{g}{h}}} - g\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

5. Taylor expanded in g around -inf 69.6

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

6. Simplified 72.9

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

   [Start] 69.6	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\left(g + \left(0.5 \cdot \frac{{h}^{2}}{g} + -1 \cdot g\right)\right) \cdot \frac{-0.5}{a}} \]
   mul-1-neg [=>] 69.6	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\left(g + \left(0.5 \cdot \frac{{h}^{2}}{g} + \color{blue}{\left(-g\right)}\right)\right) \cdot \frac{-0.5}{a}} \]
   unsub-neg [=>] 69.6	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\left(g + \color{blue}{\left(0.5 \cdot \frac{{h}^{2}}{g} - g\right)}\right) \cdot \frac{-0.5}{a}} \]
   unpow2 [=>] 69.6	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\left(g + \left(0.5 \cdot \frac{\color{blue}{h \cdot h}}{g} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   associate-/l* [=>] 72.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\left(g + \left(0.5 \cdot \color{blue}{\frac{h}{\frac{g}{h}}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   associate-*r/ [=>] 72.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\left(g + \left(\color{blue}{\frac{0.5 \cdot h}{\frac{g}{h}}} - g\right)\right) \cdot \frac{-0.5}{a}} \]

7. Applied egg-rr 96.3

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

   [Start] 72.9	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   associate-*l/ [=>] 72.9	\[ \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)}{a}}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   cbrt-div [=>] 96.3	\[ \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right) - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   div-inv [=>] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(\color{blue}{\left(0.5 \cdot h\right) \cdot \frac{1}{\frac{g}{h}}} - g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   fma-neg [=>] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\color{blue}{\mathsf{fma}\left(0.5 \cdot h, \frac{1}{\frac{g}{h}}, -g\right)} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   clear-num [<=] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\mathsf{fma}\left(0.5 \cdot h, \color{blue}{\frac{h}{g}}, -g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]

8. Simplified 96.3

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

   [Start] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\mathsf{fma}\left(0.5 \cdot h, \frac{h}{g}, -g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   fma-neg [<=] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\color{blue}{\left(\left(0.5 \cdot h\right) \cdot \frac{h}{g} - g\right)} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   *-commutative [=>] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(\color{blue}{\left(h \cdot 0.5\right)} \cdot \frac{h}{g} - g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   associate-*l* [=>] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(\color{blue}{h \cdot \left(0.5 \cdot \frac{h}{g}\right)} - g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]

9. Applied egg-rr 96.6

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

   [Start] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   associate-*r/ [=>] 96.3	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\color{blue}{\frac{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot -0.5}{a}}} \]
   cbrt-div [=>] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \color{blue}{\frac{\sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot -0.5}}{\sqrt[3]{a}}} \]
   associate-+r- [=>] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\color{blue}{\left(\left(g + \frac{0.5 \cdot h}{\frac{g}{h}}\right) - g\right)} \cdot -0.5}}{\sqrt[3]{a}} \]
   +-commutative [=>] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(\color{blue}{\left(\frac{0.5 \cdot h}{\frac{g}{h}} + g\right)} - g\right) \cdot -0.5}}{\sqrt[3]{a}} \]
   associate-/r/ [=>] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(\left(\color{blue}{\frac{0.5 \cdot h}{g} \cdot h} + g\right) - g\right) \cdot -0.5}}{\sqrt[3]{a}} \]
   fma-def [=>] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(\color{blue}{\mathsf{fma}\left(\frac{0.5 \cdot h}{g}, h, g\right)} - g\right) \cdot -0.5}}{\sqrt[3]{a}} \]
   associate-/l* [=>] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(\mathsf{fma}\left(\color{blue}{\frac{0.5}{\frac{g}{h}}}, h, g\right) - g\right) \cdot -0.5}}{\sqrt[3]{a}} \]

10. Simplified 98.1

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

    [Start] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\left(\mathsf{fma}\left(\frac{0.5}{\frac{g}{h}}, h, g\right) - g\right) \cdot -0.5}}{\sqrt[3]{a}} \]
    *-commutative [=>] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\color{blue}{-0.5 \cdot \left(\mathsf{fma}\left(\frac{0.5}{\frac{g}{h}}, h, g\right) - g\right)}}}{\sqrt[3]{a}} \]
    fma-udef [=>] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(\color{blue}{\left(\frac{0.5}{\frac{g}{h}} \cdot h + g\right)} - g\right)}}{\sqrt[3]{a}} \]
    *-commutative [<=] 96.6	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(\left(\color{blue}{h \cdot \frac{0.5}{\frac{g}{h}}} + g\right) - g\right)}}{\sqrt[3]{a}} \]
    associate--l+ [=>] 98.1	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \color{blue}{\left(h \cdot \frac{0.5}{\frac{g}{h}} + \left(g - g\right)\right)}}}{\sqrt[3]{a}} \]
    +-inverses [=>] 98.1	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(h \cdot \frac{0.5}{\frac{g}{h}} + \color{blue}{0}\right)}}{\sqrt[3]{a}} \]
    +-rgt-identity [=>] 98.1	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \color{blue}{\left(h \cdot \frac{0.5}{\frac{g}{h}}\right)}}}{\sqrt[3]{a}} \]
    associate-*r/ [=>] 98.1	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \color{blue}{\frac{h \cdot 0.5}{\frac{g}{h}}}}}{\sqrt[3]{a}} \]
    associate-/l* [=>] 98.1	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{-0.5 \cdot \color{blue}{\frac{h}{\frac{\frac{g}{h}}{0.5}}}}}{\sqrt[3]{a}} \]
    associate-*r/ [=>] 98.1	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\color{blue}{\frac{-0.5 \cdot h}{\frac{\frac{g}{h}}{0.5}}}}}{\sqrt[3]{a}} \]
    associate-/l/ [=>] 98.1	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g\right)}}{\sqrt[3]{a}} + \frac{\sqrt[3]{\frac{-0.5 \cdot h}{\color{blue}{\frac{g}{0.5 \cdot h}}}}}{\sqrt[3]{a}} \]

11. Final simplification 98.1

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

Alternatives

Alternative 1
Error	96.3%
Cost	21184.00

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

Alternative 2
Error	96.3%
Cost	20928.00

\[\sqrt[3]{\left(g + \left(\frac{0.5 \cdot h}{\frac{g}{h}} - g\right)\right) \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{\left(h \cdot \frac{h}{g}\right) \cdot 0.25 - g}}{\sqrt[3]{a}} \]

Alternative 3
Error	95.9%
Cost	13760.00

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

Alternative 4
Error	72.6%
Cost	13568.00

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

Alternative 5
Error	2.9%
Cost	6848.00

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

Reproduce

herbie shell --seed 2023121 
(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))))))))