2-ancestry mixing, positive discriminant

Average Error: 36.6 → 2.8

Time: 14.5s

Precision: binary64

Cost: 20224

Math

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

↓

\[\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\frac{0.25}{a}} \cdot \sqrt[3]{g \cdot -4} \]

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 a) (+ (- g) g))) (* (cbrt (/ 0.25 a)) (cbrt (* g -4.0)))))

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 / a) * (-g + g))) + (cbrt((0.25 / a)) * cbrt((g * -4.0)));
}

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 / a) * (-g + g))) + (Math.cbrt((0.25 / a)) * Math.cbrt((g * -4.0)));
}

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(cbrt(Float64(Float64(0.5 / a) * Float64(Float64(-g) + g))) + Float64(cbrt(Float64(0.25 / a)) * cbrt(Float64(g * -4.0))))
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[Power[N[(N[(0.5 / a), $MachinePrecision] * N[((-g) + g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(0.25 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(g * -4.0), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.6

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

2. Simplified 36.6

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

   [Start] 36.6	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational.json-simplify-46 [=>] 36.6	\[ \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [=>] 36.6	\[ \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational.json-simplify-46 [=>] 36.6	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [=>] 36.6	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

3. Taylor expanded in g around inf 49.6

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

4. Taylor expanded in g around inf 17.4

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

5. Simplified 17.4

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

   [Start] 17.4	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-2 \cdot g\right)} \]
   rational.json-simplify-2 [=>] 17.4	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(g \cdot -2\right)}} \]

6. Applied egg-rr 2.8

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + g\right)} + \color{blue}{\sqrt[3]{\frac{0.25}{a}} \cdot \sqrt[3]{g \cdot -4}} \]

7. Final simplification 2.8

   \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\frac{0.25}{a}} \cdot \sqrt[3]{g \cdot -4} \]

Alternatives

Alternative 1
Error	2.7
Cost	20096

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

Alternative 2
Error	2.8
Cost	19776

\[\sqrt[3]{\frac{0.5}{a}} \cdot \left(\sqrt[3]{g \cdot -2} + \sqrt[3]{0}\right) \]

Alternative 3
Error	17.4
Cost	13632

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

Reproduce

herbie shell --seed 2023067 
(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))))))))