?

Average Error: 36.4 → 18.0
Time: 22.7s
Precision: binary64
Cost: 14016

?

\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \frac{1}{a + a}\\ \sqrt[3]{t_0 \cdot \left(g - g\right)} + \sqrt[3]{t_0 \cdot \left(g \cdot -2\right)} \end{array} \]
(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
 (let* ((t_0 (/ 1.0 (+ a a))))
   (+ (cbrt (* t_0 (- g g))) (cbrt (* t_0 (* g -2.0))))))
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) {
	double t_0 = 1.0 / (a + a);
	return cbrt((t_0 * (g - g))) + cbrt((t_0 * (g * -2.0)));
}

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) {
	double t_0 = 1.0 / (a + a);
	return Math.cbrt((t_0 * (g - g))) + Math.cbrt((t_0 * (g * -2.0)));
}

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)
	t_0 = Float64(1.0 / Float64(a + a))
	return Float64(cbrt(Float64(t_0 * Float64(g - g))) + cbrt(Float64(t_0 * Float64(g * -2.0))))
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_] := Block[{t$95$0 = N[(1.0 / N[(a + a), $MachinePrecision]), $MachinePrecision]}, N[(N[Power[N[(t$95$0 * N[(g - g), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[Power[N[(t$95$0 * N[(g * -2.0), $MachinePrecision]), $MachinePrecision], 1/3], $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)}
↓
\begin{array}{l}
t_0 := \frac{1}{a + a}\\
\sqrt[3]{t_0 \cdot \left(g - g\right)} + \sqrt[3]{t_0 \cdot \left(g \cdot -2\right)}
\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 36.4

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

  2. Simplified36.4

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

    [Start]36.4

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

  3. Taylor expanded in g around inf 49.7

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

  4. Taylor expanded in g around inf 18.0

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

  5. Simplified18.0

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

    [Start]18.0

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

    rational_best_oopsla_all_46_json_45_simplify-74 [=>]18.0

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

  6. Final simplification18.0

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

Alternatives

Alternative 1
Error18.0
Cost13696
\[\sqrt[3]{\frac{1}{a + a} \cdot \left(g - g\right)} + \sqrt[3]{-\frac{g}{a}} \]

Error

Reproduce?

herbie shell --seed 2023090 
(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))))))))