2-ancestry mixing, positive discriminant

Average Error: 36.1 → 3.0

Time: 9.3s

Precision: binary64

Cost: 27200

Math

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

↓

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

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

Derivation

1. Initial program 36.1

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

2. Simplified 36.1

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

3. Taylor expanded in g around inf 47.2

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

4. Simplified 47.2

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

5. Taylor expanded in g around inf 18.7

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

6. Simplified 16.2

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

7. Applied egg-rr 3.0

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

8. Final simplification 3.0

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

Alternatives

Alternative 1
Error	3.3
Cost	26816

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

Alternative 2
Error	2.8
Cost	19968

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

Alternative 3
Error	16.1
Cost	14272

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

Alternative 4
Error	16.3
Cost	14016

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

Alternative 5
Error	16.3
Cost	13824

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

Alternative 6
Error	17.4
Cost	13760

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

Alternative 7
Error	17.3
Cost	13568

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

Reproduce

herbie shell --seed 2022221 
(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))))))))