2-ancestry mixing, positive discriminant

Average Accuracy: 44.2% → 97.1%

Time: 17.2s

Precision: binary64

Cost: 39616

Math

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

↓

\[\frac{\sqrt[3]{\left(h \cdot \frac{h}{g}\right) \cdot -0.25}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]

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 (* (* h (/ h g)) -0.25)) (cbrt a))
  (* (cbrt -0.5) (* (cbrt 2.0) (* (cbrt g) (cbrt (/ 1.0 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(((h * (h / g)) * -0.25)) / cbrt(a)) + (cbrt(-0.5) * (cbrt(2.0) * (cbrt(g) * cbrt((1.0 / 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(((h * (h / g)) * -0.25)) / Math.cbrt(a)) + (Math.cbrt(-0.5) * (Math.cbrt(2.0) * (Math.cbrt(g) * Math.cbrt((1.0 / 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(Float64(h * Float64(h / g)) * -0.25)) / cbrt(a)) + Float64(cbrt(-0.5) * Float64(cbrt(2.0) * Float64(cbrt(g) * cbrt(Float64(1.0 / 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[(N[(h * N[(h / g), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[-0.5, 1/3], $MachinePrecision] * N[(N[Power[2.0, 1/3], $MachinePrecision] * N[(N[Power[g, 1/3], $MachinePrecision] * N[Power[N[(1.0 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.3%

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

2. Simplified 47.3%

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

   [Start] 47.3	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]

3. Taylor expanded in h around 0 12.7%

   \[\leadsto \sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \color{blue}{{\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right)} \]

4. Simplified 30.8%

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

   [Start] 12.7	\[ \sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + {\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]
   *-commutative [=>] 12.7	\[ \sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \color{blue}{\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot {\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333}} \]
   associate-*l* [=>] 12.7	\[ \sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \color{blue}{\sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot {\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333}\right)} \]
   unpow1/3 [=>] 30.8	\[ \sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \color{blue}{\sqrt[3]{\frac{1 \cdot g}{a}}}\right) \]
   *-lft-identity [=>] 30.8	\[ \sqrt[3]{\left(g - \sqrt{\left(g + h\right) \cdot \left(g - h\right)}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{\color{blue}{g}}{a}}\right) \]

5. Taylor expanded in g around -inf 0.0%

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

6. Simplified 80.1%

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

   [Start] 0.0	\[ \sqrt[3]{\left(-0.5 \cdot \left(-1 \cdot h + h\right) + \left(0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g} + \left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g\right)\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   fma-def [=>] 0.0	\[ \sqrt[3]{\color{blue}{\mathsf{fma}\left(-0.5, -1 \cdot h + h, 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g} + \left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g\right)} \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   distribute-lft1-in [=>] 0.0	\[ \sqrt[3]{\mathsf{fma}\left(-0.5, \color{blue}{\left(-1 + 1\right) \cdot h}, 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g} + \left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   metadata-eval [=>] 0.0	\[ \sqrt[3]{\mathsf{fma}\left(-0.5, \color{blue}{0} \cdot h, 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g} + \left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   mul0-lft [=>] 0.0	\[ \sqrt[3]{\mathsf{fma}\left(-0.5, \color{blue}{0}, 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g} + \left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   fma-udef [=>] 0.0	\[ \sqrt[3]{\color{blue}{\left(-0.5 \cdot 0 + \left(0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g} + \left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g\right)\right)} \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   metadata-eval [=>] 0.0	\[ \sqrt[3]{\left(\color{blue}{0} + \left(0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g} + \left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g\right)\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   +-lft-identity [=>] 0.0	\[ \sqrt[3]{\color{blue}{\left(0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g} + \left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g\right)} \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   +-commutative [=>] 0.0	\[ \sqrt[3]{\color{blue}{\left(\left(1 + {\left(\sqrt{-1}\right)}^{2}\right) \cdot g + 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g}\right)} \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   fma-def [=>] 0.0	\[ \sqrt[3]{\color{blue}{\mathsf{fma}\left(1 + {\left(\sqrt{-1}\right)}^{2}, g, 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g}\right)} \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   unpow2 [=>] 0.0	\[ \sqrt[3]{\mathsf{fma}\left(1 + \color{blue}{\sqrt{-1} \cdot \sqrt{-1}}, g, 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   rem-square-sqrt [=>] 0.0	\[ \sqrt[3]{\mathsf{fma}\left(1 + \color{blue}{-1}, g, 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   metadata-eval [=>] 0.0	\[ \sqrt[3]{\mathsf{fma}\left(\color{blue}{0}, g, 0.5 \cdot \frac{{h}^{2} - {\left(0.5 \cdot \frac{h + -1 \cdot h}{\sqrt{-1}}\right)}^{2}}{g}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]

7. Applied egg-rr 94.0%

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

   [Start] 80.1	\[ \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{g}{a}}\right) \]
   div-inv [=>] 80.1	\[ \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\color{blue}{g \cdot \frac{1}{a}}}\right) \]
   cbrt-prod [=>] 94.0	\[ \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \color{blue}{\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)}\right) \]

8. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(h \cdot \frac{h}{g}\right) \cdot -0.25}}{\sqrt[3]{a}}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   Step-by-step derivation

   [Start] 94.0	\[ \sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot \frac{-0.5}{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   associate-*r/ [=>] 94.0	\[ \sqrt[3]{\color{blue}{\frac{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot -0.5}{a}}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   cbrt-div [=>] 96.6	\[ \color{blue}{\frac{\sqrt[3]{\left(0.5 \cdot \frac{h}{\frac{g}{h}}\right) \cdot -0.5}}{\sqrt[3]{a}}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   *-commutative [=>] 96.6	\[ \frac{\sqrt[3]{\color{blue}{\left(\frac{h}{\frac{g}{h}} \cdot 0.5\right)} \cdot -0.5}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   associate-*l* [=>] 96.6	\[ \frac{\sqrt[3]{\color{blue}{\frac{h}{\frac{g}{h}} \cdot \left(0.5 \cdot -0.5\right)}}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   associate-/r/ [=>] 96.6	\[ \frac{\sqrt[3]{\color{blue}{\left(\frac{h}{g} \cdot h\right)} \cdot \left(0.5 \cdot -0.5\right)}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   *-commutative [<=] 96.6	\[ \frac{\sqrt[3]{\color{blue}{\left(h \cdot \frac{h}{g}\right)} \cdot \left(0.5 \cdot -0.5\right)}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   metadata-eval [=>] 96.6	\[ \frac{\sqrt[3]{\left(h \cdot \frac{h}{g}\right) \cdot \color{blue}{-0.25}}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]

9. Final simplification 96.6%

   \[\leadsto \frac{\sqrt[3]{\left(h \cdot \frac{h}{g}\right) \cdot -0.25}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]

Alternatives

Alternative 1
Accuracy	95.9%
Cost	20032

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

Alternative 2
Accuracy	95.9%
Cost	19968

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

Alternative 3
Accuracy	73.7%
Cost	13568

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

Alternative 4
Accuracy	1.4%
Cost	13504

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

Reproduce

herbie shell --seed 2023158 
(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))))))))