2-ancestry mixing, positive discriminant

Average Error: 36.1 → 17.9

Time: 11.6s

Precision: binary64

Cost: 13568

Math

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

↓

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

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 a) (- g g))) (cbrt (- (/ g 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 / a) * (g - g))) + cbrt(-(g / 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 / a) * (g - g))) + Math.cbrt(-(g / 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(cbrt(Float64(Float64(0.5 / a) * Float64(g - g))) + cbrt(Float64(-Float64(g / 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[Power[N[(N[(0.5 / a), $MachinePrecision] * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[(-N[(g / a), $MachinePrecision]), 1/3], $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 36.1

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

   [Start] 36.1	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-43 [=>] 36.1	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-63 [=>] 36.1	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \color{blue}{\left(0 - g\right)}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-108 [=>] 36.1	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(0 + \sqrt{g \cdot g - h \cdot h}\right) - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-43 [=>] 36.1	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + 0\right)} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-11 [=>] 36.1	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\color{blue}{\sqrt{g \cdot g - h \cdot h}} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-67 [=>] 36.1	\[ \sqrt[3]{\frac{1}{\color{blue}{a \cdot 2}} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [<=] 36.1	\[ \sqrt[3]{\frac{1}{a \cdot \color{blue}{\left(1 + 1\right)}} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-51 [<=] 36.1	\[ \sqrt[3]{\frac{1}{\color{blue}{1 \cdot a + a \cdot 1}} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-67 [<=] 36.1	\[ \sqrt[3]{\frac{1}{\color{blue}{a \cdot 1} + a \cdot 1} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-37 [=>] 36.1	\[ \sqrt[3]{\frac{1}{\color{blue}{a} + a \cdot 1} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   rational_best_45_simplify-37 [=>] 36.1	\[ \sqrt[3]{\frac{1}{a + \color{blue}{a}} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

3. Taylor expanded in g around inf 49.6

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

4. Taylor expanded in g around inf 17.9

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

5. Simplified 17.9

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

   [Start] 17.9	\[ \sqrt[3]{\frac{1}{a + a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{1}{a + a} \cdot \left(-2 \cdot g\right)} \]
   rational_best_45_simplify-67 [=>] 17.9	\[ \sqrt[3]{\frac{1}{a + a} \cdot \left(g - g\right)} + \sqrt[3]{\frac{1}{a + a} \cdot \color{blue}{\left(g \cdot -2\right)}} \]

6. Taylor expanded in a around 0 17.9

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

7. Simplified 17.9

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

   [Start] 17.9	\[ \sqrt[3]{\frac{1}{a + a} \cdot \left(g - g\right)} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]
   rational_best_45_simplify-67 [=>] 17.9	\[ \sqrt[3]{\frac{1}{a + a} \cdot \left(g - g\right)} + \sqrt[3]{\color{blue}{\frac{g}{a} \cdot -1}} \]
   rational_best_45_simplify-53 [=>] 17.9	\[ \sqrt[3]{\frac{1}{a + a} \cdot \left(g - g\right)} + \sqrt[3]{\color{blue}{-\frac{g}{a}}} \]

8. Taylor expanded in a around 0 17.9

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

9. Final simplification 17.9

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

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))))))))