2-ancestry mixing, positive discriminant

Average Error: 35.0 → 1.8

Time: 15.5s

Precision: binary64

Cost: 39616

Math

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

↓

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

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 (* (* -0.25 h) (/ h g))) (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(((-0.25 * h) * (h / g))) / 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(((-0.25 * h) * (h / g))) / 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(-0.25 * h) * Float64(h / g))) / 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[(-0.25 * h), $MachinePrecision] * N[(h / g), $MachinePrecision]), $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 35.0

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

2. Simplified 35.0

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

   [Start] 35.0	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   +-commutative [=>] 35.0	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   associate-/r* [=>] 35.0	\[ \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [=>] 35.0	\[ \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   unsub-neg [=>] 35.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   sub-neg [=>] 35.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   distribute-neg-out [=>] 35.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   neg-mul-1 [=>] 35.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   associate-*r* [=>] 35.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]
   *-commutative [<=] 35.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \left(\frac{1}{2 \cdot a} \cdot -1\right)}} \]
   associate-*l/ [=>] 35.0	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \color{blue}{\frac{1 \cdot -1}{2 \cdot a}}} \]

3. Taylor expanded in h around 0 55.7

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

4. Simplified 46.7

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

   [Start] 55.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + {\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333} \cdot \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \]
   *-commutative [=>] 55.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \color{blue}{\left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot {\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333}} \]
   associate-*l* [=>] 55.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \color{blue}{\sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot {\left(\frac{1 \cdot g}{a}\right)}^{0.3333333333333333}\right)} \]
   unpow1/3 [=>] 46.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \color{blue}{\sqrt[3]{\frac{1 \cdot g}{a}}}\right) \]
   *-lft-identity [=>] 46.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{\color{blue}{g}}{a}}\right) \]

5. Applied egg-rr 43.6

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

6. Taylor expanded in g around inf 6.2

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

7. Simplified 3.1

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

   [Start] 6.2	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{{h}^{2}}{g}\right)} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   unpow2 [=>] 6.2	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{\color{blue}{h \cdot h}}{g}\right)} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   associate-/l* [=>] 3.1	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \color{blue}{\frac{h}{\frac{g}{h}}}\right)} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]

8. Applied egg-rr 1.8

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

9. Simplified 1.8

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

   [Start] 1.8	\[ \frac{\sqrt[3]{-0.25 \cdot \left(h \cdot \frac{h}{g}\right)}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]
   associate-*r* [=>] 1.8	\[ \frac{\sqrt[3]{\color{blue}{\left(-0.25 \cdot h\right) \cdot \frac{h}{g}}}}{\sqrt[3]{a}} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{a}}\right)\right) \]

10. Final simplification 1.8

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

Alternatives

Alternative 1
Error	13.5
Cost	137808

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{a}{g}}\\ t_1 := \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)}\\ t_2 := \sqrt{g \cdot g - h \cdot h}\\ t_3 := g + t_2\\ t_4 := t_2 - g\\ t_5 := \sqrt[3]{\frac{1}{a \cdot 2} \cdot t_4} + \sqrt[3]{t_3 \cdot \frac{-1}{a \cdot 2}}\\ \mathbf{if}\;t_5 \leq -1 \cdot 10^{-98}:\\ \;\;\;\;t_1 + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \frac{1}{t_0}\right)\\ \mathbf{elif}\;t_5 \leq 0:\\ \;\;\;\;\frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}} + \sqrt[3]{t_3 \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;t_5 \leq 10^{+97}:\\ \;\;\;\;t_1 + \sqrt[3]{-0.5} \cdot \left({2}^{0.3333333333333333} \cdot \sqrt[3]{\frac{g}{a}}\right)\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_4}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(0.5 \cdot \frac{h \cdot h}{g}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right) - g\right)} + \frac{\sqrt[3]{-1}}{t_0}\\ \end{array} \]

Alternative 2
Error	2.8
Cost	33408

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

Alternative 3
Error	3.3
Cost	33344

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

Alternative 4
Error	3.1
Cost	33344

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

Alternative 5
Error	2.5
Cost	33216

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

Alternative 6
Error	2.5
Cost	33216

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

Alternative 7
Error	2.5
Cost	33216

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

Alternative 8
Error	3.1
Cost	33088

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

Alternative 9
Error	14.9
Cost	27208

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{a}{g}}\\ \mathbf{if}\;g \leq -1.55 \cdot 10^{+139}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \left(h \cdot \frac{h}{g}\right)\right)} + \sqrt[3]{-0.5} \cdot \left(\sqrt[3]{2} \cdot \frac{1}{t_0}\right)\\ \mathbf{elif}\;g \leq -6.4 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right) - g\right)} + \frac{\sqrt[3]{-1}}{t_0}\\ \end{array} \]

Alternative 10
Error	16.1
Cost	26944

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

Alternative 11
Error	16.1
Cost	26816

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

Alternative 12
Error	16.5
Cost	26688

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

Alternative 13
Error	16.6
Cost	20352

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

Alternative 14
Error	16.9
Cost	13568

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

Alternative 15
Error	62.1
Cost	6848

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

Reproduce

herbie shell --seed 2023066 
(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))))))))