Average Error: 36.0 → 31.6
Time: 9.7s
Precision: binary64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{g \cdot g - h \cdot h} - g\\ \mathbf{if}\;g \leq -1.6 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_0}}{\sqrt[3]{a}} + \sqrt[3]{\left(\left(h \cdot h\right) \cdot \frac{0.5}{g}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{t_0 \cdot \frac{0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right)\right)}}{\sqrt[3]{a}}\\ \end{array} \]
(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))) g)))
   (if (<= g -1.6e-180)
     (+
      (/ (cbrt (* 0.5 t_0)) (cbrt a))
      (cbrt (* (* (* h h) (/ 0.5 g)) (/ -0.5 a))))
     (+
      (cbrt (* t_0 (/ 0.5 a)))
      (/ (cbrt (* -0.5 (+ g (fma -0.5 (/ h (/ g h)) 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) {
	double t_0 = sqrt(((g * g) - (h * h))) - g;
	double tmp;
	if (g <= -1.6e-180) {
		tmp = (cbrt((0.5 * t_0)) / cbrt(a)) + cbrt((((h * h) * (0.5 / g)) * (-0.5 / a)));
	} else {
		tmp = cbrt((t_0 * (0.5 / a))) + (cbrt((-0.5 * (g + fma(-0.5, (h / (g / h)), g)))) / cbrt(a));
	}
	return tmp;
}

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(sqrt(Float64(Float64(g * g) - Float64(h * h))) - g)
	tmp = 0.0
	if (g <= -1.6e-180)
		tmp = Float64(Float64(cbrt(Float64(0.5 * t_0)) / cbrt(a)) + cbrt(Float64(Float64(Float64(h * h) * Float64(0.5 / g)) * Float64(-0.5 / a))));
	else
		tmp = Float64(cbrt(Float64(t_0 * Float64(0.5 / a))) + Float64(cbrt(Float64(-0.5 * Float64(g + fma(-0.5, Float64(h / Float64(g / h)), g)))) / cbrt(a)));
	end
	return tmp
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[(N[Sqrt[N[(N[(g * g), $MachinePrecision] - N[(h * h), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - g), $MachinePrecision]}, If[LessEqual[g, -1.6e-180], N[(N[(N[Power[N[(0.5 * t$95$0), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[(h * h), $MachinePrecision] * N[(0.5 / g), $MachinePrecision]), $MachinePrecision] * N[(-0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t$95$0 * N[(0.5 / a), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] + N[(N[Power[N[(-0.5 * N[(g + N[(-0.5 * N[(h / N[(g / h), $MachinePrecision]), $MachinePrecision] + g), $MachinePrecision]), $MachinePrecision]), $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} - g\\
\mathbf{if}\;g \leq -1.6 \cdot 10^{-180}:\\
\;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_0}}{\sqrt[3]{a}} + \sqrt[3]{\left(\left(h \cdot h\right) \cdot \frac{0.5}{g}\right) \cdot \frac{-0.5}{a}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{t_0 \cdot \frac{0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right)\right)}}{\sqrt[3]{a}}\\


\end{array}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

  1. Split input into 2 regimes

  2. if g < -1.60000000000000008e-180

    1. Initial program 35.4

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

    2. Simplified35.3

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

    3. Applied egg-rr31.4

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

    4. Taylor expanded in g around -inf 31.0

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

    5. Simplified31.0

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

    if -1.60000000000000008e-180 < g

    1. Initial program 36.7

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

    2. Simplified36.6

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

    3. Applied egg-rr33.2

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

    4. Taylor expanded in g around inf 32.2

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

    5. Simplified32.2

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{\left(g + \color{blue}{\mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right)}\right) \cdot -0.5}}{\sqrt[3]{a}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification31.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \leq -1.6 \cdot 10^{-180}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\left(\left(h \cdot h\right) \cdot \frac{0.5}{g}\right) \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} - g\right) \cdot \frac{0.5}{a}} + \frac{\sqrt[3]{-0.5 \cdot \left(g + \mathsf{fma}\left(-0.5, \frac{h}{\frac{g}{h}}, g\right)\right)}}{\sqrt[3]{a}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022178 
(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))))))))