2-ancestry mixing, positive discriminant

Average Accuracy: 43.3% → 95.7%

Time: 22.3s

Precision: binary64

Cost: 19840

Math

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

↓

\[\sqrt[3]{0 \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}} \]

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

Derivation

1. Initial program 47.5%

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

2. Simplified 47.8%

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

   [Start] 47.5	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

3. Taylor expanded in g around inf 23.5%

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

4. Simplified 23.5%

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

   [Start] 23.5	\[ \sqrt[3]{\left(-0.5 \cdot \left(h + -1 \cdot h\right)\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]
   distribute-rgt1-in [=>] 23.5	\[ \sqrt[3]{\left(-0.5 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]
   metadata-eval [=>] 23.5	\[ \sqrt[3]{\left(-0.5 \cdot \left(\color{blue}{0} \cdot h\right)\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]
   mul0-lft [=>] 23.5	\[ \sqrt[3]{\left(-0.5 \cdot \color{blue}{0}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]
   metadata-eval [=>] 23.5	\[ \sqrt[3]{\color{blue}{0} \cdot \frac{-0.5}{a}} + \sqrt[3]{\left(g + \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} \]

5. Taylor expanded in g around inf 77.6%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{-1 \cdot \frac{g}{a}}} \]

6. Simplified 77.6%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-g}{a}}} \]
   Proof

   [Start] 77.6	\[ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{-1 \cdot \frac{g}{a}} \]
   associate-*r/ [=>] 77.6	\[ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-1 \cdot g}{a}}} \]
   mul-1-neg [=>] 77.6	\[ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{\color{blue}{-g}}{a}} \]

7. Applied egg-rr 96.3%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\frac{\sqrt[3]{g}}{\sqrt[3]{-a}}} \]
   Proof

   [Start] 77.6	\[ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}} \]
   frac-2neg [=>] 77.6	\[ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\color{blue}{\frac{-\left(-g\right)}{-a}}} \]
   cbrt-div [=>] 96.3	\[ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \color{blue}{\frac{\sqrt[3]{-\left(-g\right)}}{\sqrt[3]{-a}}} \]
   remove-double-neg [=>] 96.3	\[ \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{\color{blue}{g}}}{\sqrt[3]{-a}} \]

8. Final simplification 96.3%

   \[\leadsto \sqrt[3]{0 \cdot \frac{-0.5}{a}} + \frac{\sqrt[3]{g}}{\sqrt[3]{-a}} \]

Alternatives

Alternative 1
Accuracy	71.7%
Cost	14144

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

Alternative 2
Accuracy	7.0%
Cost	13508

\[\begin{array}{l} t_0 := \sqrt[3]{0 \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -2 \cdot 10^{-309}:\\ \;\;\;\;t_0 + \sqrt[3]{g + g}\\ \mathbf{else}:\\ \;\;\;\;t_0 + 2 \cdot \left(-0.5 \cdot g\right)\\ \end{array} \]

Alternative 3
Accuracy	7.9%
Cost	13508

\[\begin{array}{l} t_0 := \sqrt[3]{0 \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -2 \cdot 10^{-309}:\\ \;\;\;\;t_0 + \sqrt[3]{g + g}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \sqrt[3]{g} \cdot -2\\ \end{array} \]

Alternative 4
Accuracy	71.7%
Cost	13440

\[\sqrt[3]{0 \cdot \frac{-0.5}{a}} + \sqrt[3]{\frac{-g}{a}} \]

Alternative 5
Accuracy	6.2%
Cost	7236

\[\begin{array}{l} t_0 := \sqrt[3]{0 \cdot \frac{-0.5}{a}}\\ \mathbf{if}\;a \leq -2 \cdot 10^{-309}:\\ \;\;\;\;t_0 + g \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_0 + 2 \cdot \left(-0.5 \cdot g\right)\\ \end{array} \]

Alternative 6
Accuracy	5.1%
Cost	7232

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

Alternative 7
Accuracy	4.4%
Cost	6848

\[\sqrt[3]{0 \cdot \frac{-0.5}{a}} + -2 \]

Reproduce

herbie shell --seed 2023157 
(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))))))))