2-ancestry mixing, positive discriminant

Average Error: 55.74% → 4.17%

Time: 14.0s

Precision: binary64

Cost: 20928

Math

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

↓

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

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

Derivation

1. Initial program 55.74

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

2. Simplified 55.73

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

   [Start] 55.74	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   +-commutative [=>] 55.74	\[ \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   associate-/r* [=>] 55.73	\[ \sqrt[3]{\color{blue}{\frac{\frac{1}{2}}{a}} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   metadata-eval [=>] 55.73	\[ \sqrt[3]{\frac{\color{blue}{0.5}}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   unsub-neg [=>] 55.73	\[ \sqrt[3]{\frac{0.5}{a} \cdot \color{blue}{\left(\sqrt{g \cdot g - h \cdot h} - g\right)}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
   sub-neg [=>] 55.73	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(\left(-g\right) + \left(-\sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   distribute-neg-out [=>] 55.73	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-\left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   neg-mul-1 [=>] 55.73	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \color{blue}{\left(-1 \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)\right)}} \]
   associate-*r* [=>] 55.73	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(\frac{1}{2 \cdot a} \cdot -1\right) \cdot \left(g + \sqrt{g \cdot g - h \cdot h}\right)}} \]
   *-commutative [<=] 55.73	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\color{blue}{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \left(\frac{1}{2 \cdot a} \cdot -1\right)}} \]
   associate-*l/ [=>] 55.73	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \color{blue}{\frac{1 \cdot -1}{2 \cdot a}}} \]

3. Taylor expanded in g around -inf 73.35

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

4. Simplified 73.35

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

   [Start] 73.35	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(0.5 \cdot \frac{{h}^{2}}{g} + -1 \cdot g\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   mul-1-neg [=>] 73.35	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(0.5 \cdot \frac{{h}^{2}}{g} + \color{blue}{\left(-g\right)}\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   unsub-neg [=>] 73.35	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{{h}^{2}}{g} - g\right)} - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   unpow2 [=>] 73.35	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(0.5 \cdot \frac{\color{blue}{h \cdot h}}{g} - g\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   associate-/l* [=>] 73.35	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(0.5 \cdot \color{blue}{\frac{h}{\frac{g}{h}}} - g\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   associate-*r/ [=>] 73.35	\[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\left(\color{blue}{\frac{0.5 \cdot h}{\frac{g}{h}}} - g\right) - g\right)} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

5. Applied egg-rr 68.42

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

6. Simplified 68.42

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

   [Start] 68.42	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\mathsf{fma}\left(0.5 \cdot h, \frac{h}{g}, -g\right) - g} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   fma-neg [<=] 68.42	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\color{blue}{\left(\left(0.5 \cdot h\right) \cdot \frac{h}{g} - g\right)} - g} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   *-commutative [=>] 68.42	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(\color{blue}{\left(h \cdot 0.5\right)} \cdot \frac{h}{g} - g\right) - g} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]
   associate-*l* [=>] 68.42	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(\color{blue}{h \cdot \left(0.5 \cdot \frac{h}{g}\right)} - g\right) - g} + \sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{-0.5}{a}} \]

7. Taylor expanded in g around -inf 8.96

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

8. Simplified 4.17

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

   [Start] 8.96	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g} + \sqrt[3]{\left(0.5 \cdot \frac{{h}^{2}}{g}\right) \cdot \frac{-0.5}{a}} \]
   unpow2 [=>] 8.96	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g} + \sqrt[3]{\left(0.5 \cdot \frac{\color{blue}{h \cdot h}}{g}\right) \cdot \frac{-0.5}{a}} \]
   associate-/l* [=>] 4.17	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g} + \sqrt[3]{\left(0.5 \cdot \color{blue}{\frac{h}{\frac{g}{h}}}\right) \cdot \frac{-0.5}{a}} \]
   associate-/r/ [=>] 4.17	\[ \sqrt[3]{\frac{0.5}{a}} \cdot \sqrt[3]{\left(h \cdot \left(0.5 \cdot \frac{h}{g}\right) - g\right) - g} + \sqrt[3]{\left(0.5 \cdot \color{blue}{\left(\frac{h}{g} \cdot h\right)}\right) \cdot \frac{-0.5}{a}} \]

9. Final simplification 4.17

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

Alternatives

Alternative 1
Error	4.3%
Cost	20160

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

Alternative 2
Error	25.9%
Cost	14528

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

Alternative 3
Error	27.28%
Cost	14080

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

Alternative 4
Error	27.66%
Cost	13568

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

Alternative 5
Error	97.05%
Cost	6848

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

Reproduce

herbie shell --seed 2023093 
(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))))))))