Average Error: 35.8 → 2.6
Time: 21.6s
Precision: binary64
Cost: 20160
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + g\right)} + \frac{-\sqrt[3]{g}}{\sqrt[3]{a}} \]
(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 (* (/ 1.0 (* 2.0 a)) (+ (- g) g))) (/ (- (cbrt g)) (cbrt a))))
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(((1.0 / (2.0 * a)) * (-g + g))) + (-cbrt(g) / cbrt(a));
}

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

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(1.0 / Float64(2.0 * a)) * Float64(Float64(-g) + g))) + Float64(Float64(-cbrt(g)) / cbrt(a)))
end

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

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

Error

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 35.8

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

  2. Taylor expanded in g around inf 48.8

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

  3. Taylor expanded in g around inf 17.3

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

  4. Applied egg-rr2.6

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

  5. Applied egg-rr2.6

    \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + g\right)} + \frac{\color{blue}{\frac{0}{-1} - \sqrt[3]{g}}}{\sqrt[3]{a}} \]

  6. Simplified2.6

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

Alternatives

Alternative 1
Error17.3
Cost13824
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + g\right)} + \sqrt[3]{\frac{-1}{a} \cdot g} \]
Alternative 2
Error17.3
Cost13760
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + g\right)} + \left(-\sqrt[3]{\frac{g}{a}}\right) \]

Error

Reproduce

herbie shell --seed 2023010 
(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))))))))