2-ancestry mixing, positive discriminant

Average Error: 36.1 → 2.9

Time: 20.4s

Precision: binary64

Cost: 19776

Math

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

↓

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

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 a)) (+ (cbrt (* g -2.0)) (cbrt 0.0))))

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)) * (cbrt((g * -2.0)) + cbrt(0.0));
}

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)) * (Math.cbrt((g * -2.0)) + Math.cbrt(0.0));
}

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(0.5 / a)) * Float64(cbrt(Float64(g * -2.0)) + cbrt(0.0)))
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[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision] * N[(N[Power[N[(g * -2.0), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[0.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.1

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

2. Simplified 32.3

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

   [Start] 36.1	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   exponential-simplify-18 [=>] 34.1	\[ \color{blue}{\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best-simplify-2 [=>] 34.1	\[ \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   exponential-simplify-18 [=>] 32.3	\[ \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} + \color{blue}{\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}} \]
   rational_best-simplify-2 [=>] 32.3	\[ \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}} + \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}} \]
   rational_best-simplify-47 [=>] 32.3	\[ \color{blue}{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \left(\sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}} + \sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}\right)} \]

3. Taylor expanded in g around inf 47.6

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

4. Taylor expanded in g around inf 2.9

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

5. Simplified 2.9

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

   [Start] 2.9	\[ \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \left(\sqrt[3]{-2 \cdot g} + \sqrt[3]{\left(-g\right) + g}\right) \]
   rational_best-simplify-2 [<=] 2.9	\[ \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \left(\sqrt[3]{\color{blue}{g \cdot -2}} + \sqrt[3]{\left(-g\right) + g}\right) \]

6. Applied egg-rr 2.9

   \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a + a}} \cdot \left(\sqrt[3]{g \cdot -2} + \sqrt[3]{0}\right) + 0} \]

7. Simplified 2.9

   \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{a + a}} \cdot \left(\sqrt[3]{g \cdot -2} + \sqrt[3]{0}\right)} \]
   Proof

   [Start] 2.9	\[ \sqrt[3]{\frac{1}{a + a}} \cdot \left(\sqrt[3]{g \cdot -2} + \sqrt[3]{0}\right) + 0 \]
   rational_best-simplify-4 [=>] 2.9	\[ \color{blue}{\sqrt[3]{\frac{1}{a + a}} \cdot \left(\sqrt[3]{g \cdot -2} + \sqrt[3]{0}\right)} \]

8. Taylor expanded in a around 0 2.9

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

9. Final simplification 2.9

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

Alternatives

Alternative 1
Error	17.9
Cost	13632

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

Reproduce

herbie shell --seed 2023092 
(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))))))))