2-ancestry mixing, positive discriminant

Average Error: 35.7 → 3.0

Time: 16.1s

Precision: binary64

Cost: 33088

Math

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

↓

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

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 (* (/ 0.5 a) (- g g)))
  (* (* (cbrt g) (cbrt (/ 1.0 a))) (* (cbrt -0.5) (cbrt 2.0)))))

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) * cbrt((1.0 / a))) * (cbrt(-0.5) * cbrt(2.0)));
}

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) * Math.cbrt((1.0 / a))) * (Math.cbrt(-0.5) * Math.cbrt(2.0)));
}

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

Derivation

1. Initial program 35.7

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

2. Simplified 35.7

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

   [Start] 35.7	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   +-commutative [=>] 35.7	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   associate-/r* [=>] 35.7	\[ \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [=>] 35.7	\[ \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   unsub-neg [=>] 35.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   sub-neg [=>] 35.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   distribute-neg-out [=>] 35.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   neg-mul-1 [=>] 35.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   associate-*r* [=>] 35.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]
   *-commutative [<=] 35.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \left(\frac{1}{2 \cdot a} \cdot -1\right)}} \]
   associate-*l/ [=>] 35.7	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \color{blue}{\frac{1 \cdot -1}{2 \cdot a}}} \]

3. Taylor expanded in g around inf 49.4

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

4. Taylor expanded in h around 0 42.0

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

5. Applied egg-rr 3.0

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

6. Final simplification 3.0

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

Alternatives

Alternative 1
Error	3.1
Cost	32960

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

Alternative 2
Error	13.0
Cost	27472

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ t_1 := t_0 + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ t_2 := g + \sqrt{g \cdot g - h \cdot h}\\ \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;g \leq -1.12 \cdot 10^{-152}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot t_2} + \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \mathbf{elif}\;g \leq 4.4 \cdot 10^{-157}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(g + \frac{h}{\frac{g}{-h}} \cdot \left(0.5 + \frac{0}{g}\right)\right) - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right)\right)}\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.0
Cost	27472

\[\begin{array}{l} t_0 := g + \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ t_2 := t_1 + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;g \leq -8 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt[3]{-0.5 \cdot \left(g + g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-0.5}{a} \cdot t_0}\\ \mathbf{elif}\;g \leq 9.2 \cdot 10^{-157}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(g + \frac{h}{\frac{g}{-h}} \cdot \left(0.5 + \frac{0}{g}\right)\right) - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right)\right)}\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1 + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	12.8
Cost	27472

\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ t_1 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ t_2 := t_1 + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ \mathbf{if}\;g \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;g \leq -1.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(t_0 - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(0.5 \cdot \frac{h \cdot h}{g}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{elif}\;g \leq 7.8 \cdot 10^{-158}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \left(\left(g + \frac{h}{\frac{g}{-h}} \cdot \left(0.5 + \frac{0}{g}\right)\right) - g\right)} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + \mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right)\right)}\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1 + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	14.3
Cost	27208

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{0.5}{a} \cdot \left(g - g\right)}\\ \mathbf{if}\;g \leq 6.8 \cdot 10^{-151}:\\ \;\;\;\;\sqrt[3]{\frac{0.5}{a} \cdot \mathsf{fma}\left(0.5, \frac{-h}{\frac{g}{h}}, 0\right)} + \sqrt[3]{\frac{-g}{a}}\\ \mathbf{elif}\;g \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_0 + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \left(\sqrt[3]{-0.5} \cdot \sqrt[3]{2}\right) \cdot \frac{1}{\sqrt[3]{\frac{a}{g}}}\\ \end{array} \]

Alternative 6
Error	16.9
Cost	26688

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

Alternative 7
Error	15.9
Cost	20672

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

Alternative 8
Error	15.9
Cost	20288

\[\sqrt[3]{\frac{0.5}{a} \cdot \mathsf{fma}\left(0.5, \frac{-h}{\frac{g}{h}}, 0\right)} + \sqrt[3]{\frac{-g}{a}} \]

Alternative 9
Error	16.7
Cost	14592

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

Alternative 10
Error	16.9
Cost	13568

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

Alternative 11
Error	62.1
Cost	6848

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

Reproduce

herbie shell --seed 2023053 
(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))))))))