2-ancestry mixing, positive discriminant

Average Accuracy: 43.0% → 96.0%

Time: 14.8s

Precision: binary64

Cost: 20672

Math

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

↓

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

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

Derivation

1. Initial program 43.0%

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

2. Simplified 43.0%

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

   [Start] 43.0	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   +-commutative [=>] 43.0	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   associate-/r* [=>] 43.0	\[ \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [=>] 43.0	\[ \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   unsub-neg [=>] 43.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   sub-neg [=>] 43.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   distribute-neg-out [=>] 43.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   neg-mul-1 [=>] 43.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   associate-*r* [=>] 43.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]
   *-commutative [<=] 43.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \left(\frac{1}{2 \cdot a} \cdot -1\right)}} \]
   associate-*l/ [=>] 43.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \color{blue}{\frac{1 \cdot -1}{2 \cdot a}}} \]

3. Applied egg-rr 46.2%

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

   [Start] 43.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   associate-*l/ [=>] 43.0	\[ \sqrt[3]{\color{blue}{\frac{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}{a}}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   cbrt-div [=>] 46.2	\[ \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

4. Taylor expanded in g around -inf 30.8%

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

5. Simplified 30.8%

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

   [Start] 30.8	\[ \frac{\sqrt[3]{0.5 \cdot \left(-1 \cdot g - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   mul-1-neg [=>] 30.8	\[ \frac{\sqrt[3]{0.5 \cdot \left(\color{blue}{\left(-g\right)} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

6. Taylor expanded in g around -inf 90.4%

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

7. Simplified 96.0%

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

   [Start] 90.4	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(0.5 \cdot \frac{{h}^{2}}{g} + -1 \cdot g\right)\right) \cdot \frac{-0.5}{a}} \]
   mul-1-neg [=>] 90.4	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(0.5 \cdot \frac{{h}^{2}}{g} + \color{blue}{\left(-g\right)}\right)\right) \cdot \frac{-0.5}{a}} \]
   unsub-neg [=>] 90.4	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \color{blue}{\left(0.5 \cdot \frac{{h}^{2}}{g} - g\right)}\right) \cdot \frac{-0.5}{a}} \]
   unpow2 [=>] 90.4	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(0.5 \cdot \frac{\color{blue}{h \cdot h}}{g} - g\right)\right) \cdot \frac{-0.5}{a}} \]
   associate-/l* [=>] 96.0	\[ \frac{\sqrt[3]{0.5 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \left(0.5 \cdot \color{blue}{\frac{h}{\frac{g}{h}}} - g\right)\right) \cdot \frac{-0.5}{a}} \]

8. Final simplification 96.0%

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

Alternatives

Alternative 1
Accuracy	95.7%
Cost	20160

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

Alternative 2
Accuracy	72.4%
Cost	13760

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

Alternative 3
Accuracy	72.5%
Cost	13568

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

Alternative 4
Accuracy	3.0%
Cost	6848

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

Reproduce

herbie shell --seed 2023137 
(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))))))))