2-ancestry mixing, positive discriminant

Average Error: 35.8 → 1.8

Time: 16.4s

Precision: binary64

Cost: 39744

Math

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

↓

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

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 (/ h g))) 0.5)) (cbrt a))
  (* (cbrt -0.5) (* (cbrt 2.0) (* (cbrt g) (cbrt (/ 1.0 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 * (h / g))) * 0.5)) / cbrt(a)) + (cbrt(-0.5) * (cbrt(2.0) * (cbrt(g) * cbrt((1.0 / 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 * (h / g))) * 0.5)) / Math.cbrt(a)) + (Math.cbrt(-0.5) * (Math.cbrt(2.0) * (Math.cbrt(g) * Math.cbrt((1.0 / 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(Float64(-0.5 * Float64(h * Float64(h / g))) * 0.5)) / cbrt(a)) + Float64(cbrt(-0.5) * Float64(cbrt(2.0) * Float64(cbrt(g) * cbrt(Float64(1.0 / 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[(N[(-0.5 * N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[(N[Power[2.0, 1/3], $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.8

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

2. Simplified 35.8

   \[\leadsto \color{blue}{\sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}} \]
   Proof
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 (Rewrite<= metadata-eval (/.f64 1 2)) a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 2 a))) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 14 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (Rewrite<= unsub-neg_binary64 (+.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) (neg.f64 g))))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 14 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (Rewrite<= +-commutative_binary64 (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 -1/2 a)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 -1)) a)))): 14 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (/.f64 (*.f64 (Rewrite<= metadata-eval (/.f64 1 2)) -1) a)))): 14 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (/.f64 1 2) a) -1))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))) (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 1 (*.f64 2 a))) -1)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (Rewrite=> *-commutative_binary64 (*.f64 (*.f64 (/.f64 1 (*.f64 2 a)) -1) (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (Rewrite<= associate-*r*_binary64 (*.f64 (/.f64 1 (*.f64 2 a)) (*.f64 -1 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (+.f64 g (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (Rewrite<= distribute-neg-out_binary64 (+.f64 (neg.f64 g) (neg.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))))))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (+.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))) (cbrt.f64 (*.f64 (/.f64 1 (*.f64 2 a)) (Rewrite<= sub-neg_binary64 (-.f64 (neg.f64 g) (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h)))))))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in h around 0 55.8

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

4. Simplified 47.3

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \color{blue}{\sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right)} \]
   Proof
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (cbrt.f64 (/.f64 g a))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (cbrt.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 g)) a))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (Rewrite<= unpow1/3_binary64 (pow.f64 (/.f64 (*.f64 1 g) a) 1/3))))): 0 points increase in error, 5 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cbrt.f64 -1/2) (cbrt.f64 2)) (pow.f64 (/.f64 (*.f64 1 g) a) 1/3)))): 3 points increase in error, 2 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (/.f64 (*.f64 1 g) a) 1/3) (*.f64 (cbrt.f64 -1/2) (cbrt.f64 2))))): 2 points increase in error, 0 points decrease in error

5. Taylor expanded in g around inf 19.9

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

6. Simplified 17.7

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right)} - g\right)} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   Proof
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (cbrt.f64 (/.f64 g a))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (cbrt.f64 (/.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 g)) a))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (Rewrite<= unpow1/3_binary64 (pow.f64 (/.f64 (*.f64 1 g) a) 1/3))))): 0 points increase in error, 5 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (cbrt.f64 -1/2) (cbrt.f64 2)) (pow.f64 (/.f64 (*.f64 1 g) a) 1/3)))): 3 points increase in error, 2 points decrease in error
   (+.f64 (cbrt.f64 (*.f64 (/.f64 1/2 a) (-.f64 (sqrt.f64 (-.f64 (*.f64 g g) (*.f64 h h))) g))) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (/.f64 (*.f64 1 g) a) 1/3) (*.f64 (cbrt.f64 -1/2) (cbrt.f64 2))))): 2 points increase in error, 0 points decrease in error

7. Applied egg-rr 2.9

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

8. Applied egg-rr 2.8

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

9. Simplified 1.8

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot 0.5}}{\sqrt[3]{a}}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   Proof
   (+.f64 (/.f64 (cbrt.f64 (*.f64 (*.f64 -1/2 (*.f64 h (/.f64 h g))) 1/2)) (cbrt.f64 a)) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (*.f64 (cbrt.f64 g) (cbrt.f64 (/.f64 1 a)))))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (cbrt.f64 (*.f64 (Rewrite<= +-lft-identity_binary64 (+.f64 0 (*.f64 -1/2 (*.f64 h (/.f64 h g))))) 1/2)) (cbrt.f64 a)) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (*.f64 (cbrt.f64 g) (cbrt.f64 (/.f64 1 a)))))): 6 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (cbrt.f64 (*.f64 (+.f64 (Rewrite<= +-inverses_binary64 (-.f64 g g)) (*.f64 -1/2 (*.f64 h (/.f64 h g)))) 1/2)) (cbrt.f64 a)) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (*.f64 (cbrt.f64 g) (cbrt.f64 (/.f64 1 a)))))): 6 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (cbrt.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 (*.f64 -1/2 (*.f64 h (/.f64 h g))) (-.f64 g g))) 1/2)) (cbrt.f64 a)) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (*.f64 (cbrt.f64 g) (cbrt.f64 (/.f64 1 a)))))): 0 points increase in error, 6 points decrease in error
   (+.f64 (/.f64 (cbrt.f64 (*.f64 (Rewrite=> associate-+r-_binary64 (-.f64 (+.f64 (*.f64 -1/2 (*.f64 h (/.f64 h g))) g) g)) 1/2)) (cbrt.f64 a)) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (*.f64 (cbrt.f64 g) (cbrt.f64 (/.f64 1 a)))))): 6 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (cbrt.f64 (*.f64 (-.f64 (Rewrite<= fma-udef_binary64 (fma.f64 -1/2 (*.f64 h (/.f64 h g)) g)) g) 1/2)) (cbrt.f64 a)) (*.f64 (cbrt.f64 -1/2) (*.f64 (cbrt.f64 2) (*.f64 (cbrt.f64 g) (cbrt.f64 (/.f64 1 a)))))): 6 points increase in error, 0 points decrease in error

10. Final simplification 1.8

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

Alternatives

Alternative 1
Error	2.4
Cost	34121

\[\begin{array}{l} t_0 := \frac{1}{a \cdot 2}\\ t_1 := \sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+101} \lor \neg \left(t_0 \leq 1000000\right):\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + t_1 \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot t_1\right) + \sqrt[3]{\left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot \frac{0.5}{a}}\\ \end{array} \]

Alternative 2
Error	2.9
Cost	33600

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

Alternative 3
Error	2.9
Cost	33088

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

Alternative 4
Error	3.1
Cost	32960

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

Alternative 5
Error	5.9
Cost	27844

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a}}\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{\frac{h \cdot h}{g - t_1} \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;g \leq 9.5 \cdot 10^{-155}:\\ \;\;\;\;t_0 \cdot \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}} - g\right) - g} + t_2\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + t_1\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_2 + t_0 \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array} \]

Alternative 6
Error	6.0
Cost	27529

\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;g \leq 4 \cdot 10^{-155} \lor \neg \left(g \leq 1.35 \cdot 10^{+154}\right):\\ \;\;\;\;\sqrt[3]{\frac{h \cdot h}{g - t_0} \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)} + \frac{\sqrt[3]{-0.5 \cdot \left(g + t_0\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Alternative 7
Error	13.9
Cost	27472

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ t_1 := g + \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{\left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot \frac{0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right)\\ \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;g \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - g} + \sqrt[3]{\frac{-0.5}{a} \cdot t_1}\\ \mathbf{elif}\;g \leq 4.5 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;g \leq 2.9 \cdot 10^{+148}:\\ \;\;\;\;t_0 + \frac{\sqrt[3]{-0.5 \cdot t_1}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ \end{array} \]

Alternative 8
Error	13.6
Cost	27472

\[\begin{array}{l} t_0 := \sqrt[3]{\left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot \frac{0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right)\\ t_1 := \sqrt{g \cdot g - h \cdot h}\\ t_2 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;g \leq -3.3 \cdot 10^{-156}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{t_1 - g} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(0.5 \cdot \frac{h \cdot h}{g}\right)}\\ \mathbf{elif}\;g \leq 3 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;g \leq 2.9 \cdot 10^{+148}:\\ \;\;\;\;t_2 + \frac{\sqrt[3]{-0.5 \cdot \left(g + t_1\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_2 + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ \end{array} \]

Alternative 9
Error	15.3
Cost	27208

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{if}\;g \leq 4.5 \cdot 10^{-139}:\\ \;\;\;\;\sqrt[3]{\left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right) \cdot \frac{0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right)\\ \mathbf{elif}\;g \leq 2.9 \cdot 10^{+148}:\\ \;\;\;\;t_0 + \frac{\sqrt[3]{-0.5 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ \end{array} \]

Alternative 10
Error	17.0
Cost	26816

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

Alternative 11
Error	17.4
Cost	13568

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

Alternative 12
Error	62.1
Cost	6848

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

Reproduce

herbie shell --seed 2022343 
(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))))))))