Average Error: 36.3 → 32.5
Time: 14.7s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h}\\ \sqrt[3]{t_0 - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \frac{\sqrt[3]{\left(g + t_0\right) \cdot -0.5}}{\sqrt[3]{a}} \end{array} \]
(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 (sqrt (- (* g g) (* h h)))))
   (+
    (* (cbrt (- t_0 g)) (cbrt (/ 0.5 a)))
    (/ (cbrt (* (+ g t_0) -0.5)) (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) {
	double t_0 = sqrt(((g * g) - (h * h)));
	return (cbrt((t_0 - g)) * cbrt((0.5 / a))) + (cbrt(((g + t_0) * -0.5)) / 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) {
	double t_0 = Math.sqrt(((g * g) - (h * h)));
	return (Math.cbrt((t_0 - g)) * Math.cbrt((0.5 / a))) + (Math.cbrt(((g + t_0) * -0.5)) / 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)
	t_0 = sqrt(Float64(Float64(g * g) - Float64(h * h)))
	return Float64(Float64(cbrt(Float64(t_0 - g)) * cbrt(Float64(0.5 / a))) + Float64(cbrt(Float64(Float64(g + t_0) * -0.5)) / 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_] := Block[{t$95$0 = N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[N[(t$95$0 - g), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(g + t$95$0), $MachinePrecision] * -0.5), $MachinePrecision], 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)}
↓
\begin{array}{l}
t_0 := \sqrt{g \cdot g - h \cdot h}\\
\sqrt[3]{t_0 - g} \cdot \sqrt[3]{\frac{0.5}{a}} + \frac{\sqrt[3]{\left(g + t_0\right) \cdot -0.5}}{\sqrt[3]{a}}
\end{array}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 36.3

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

  2. Simplified36.3

    \[\leadsto \color{blue}{\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}} + \sqrt[3]{\frac{g + \sqrt{g \cdot g - h \cdot h}}{a} \cdot -0.5}} \]

  3. Applied associate-*l/_binary6436.3

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

  4. Applied cbrt-div_binary6434.2

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

  5. Applied div-inv_binary6434.2

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

  6. Applied cbrt-prod_binary6432.5

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

  7. Simplified32.5

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

  8. Final simplification32.5

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

Reproduce

herbie shell --seed 2022137 
(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))))))))