2-ancestry mixing, positive discriminant

?

Percentage Accurate: 43.4% → 95.9%
Time: 39.8s
Precision: binary64
Cost: 39876

?

\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} \]
↓
\[\begin{array}{l} t_0 := g + \mathsf{hypot}\left(g, h\right)\\ t_1 := \mathsf{hypot}\left(g, h\right) - g\\ t_2 := \sqrt[3]{\frac{-0.5}{a}}\\ \mathbf{if}\;\frac{1}{a \cdot 2} \leq -5 \cdot 10^{-301}:\\ \;\;\;\;t_2 \cdot \sqrt[3]{t_0} + t_2 \cdot \sqrt[3]{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_1}}{\sqrt[3]{a}} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{t_0}}}\\ \end{array} \]
(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 (+ g (hypot g h)))
        (t_1 (- (hypot g h) g))
        (t_2 (cbrt (/ -0.5 a))))
   (if (<= (/ 1.0 (* a 2.0)) -5e-301)
     (+ (* t_2 (cbrt t_0)) (* t_2 (cbrt t_1)))
     (+ (/ (cbrt (* 0.5 t_1)) (cbrt a)) (/ 1.0 (cbrt (/ (* a -2.0) t_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 = g + hypot(g, h);
	double t_1 = hypot(g, h) - g;
	double t_2 = cbrt((-0.5 / a));
	double tmp;
	if ((1.0 / (a * 2.0)) <= -5e-301) {
		tmp = (t_2 * cbrt(t_0)) + (t_2 * cbrt(t_1));
	} else {
		tmp = (cbrt((0.5 * t_1)) / cbrt(a)) + (1.0 / cbrt(((a * -2.0) / t_0)));
	}
	return tmp;
}

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 = g + Math.hypot(g, h);
	double t_1 = Math.hypot(g, h) - g;
	double t_2 = Math.cbrt((-0.5 / a));
	double tmp;
	if ((1.0 / (a * 2.0)) <= -5e-301) {
		tmp = (t_2 * Math.cbrt(t_0)) + (t_2 * Math.cbrt(t_1));
	} else {
		tmp = (Math.cbrt((0.5 * t_1)) / Math.cbrt(a)) + (1.0 / Math.cbrt(((a * -2.0) / t_0)));
	}
	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(g + hypot(g, h))
	t_1 = Float64(hypot(g, h) - g)
	t_2 = cbrt(Float64(-0.5 / a))
	tmp = 0.0
	if (Float64(1.0 / Float64(a * 2.0)) <= -5e-301)
		tmp = Float64(Float64(t_2 * cbrt(t_0)) + Float64(t_2 * cbrt(t_1)));
	else
		tmp = Float64(Float64(cbrt(Float64(0.5 * t_1)) / cbrt(a)) + Float64(1.0 / cbrt(Float64(Float64(a * -2.0) / t_0))));
	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[(g + N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[g ^ 2 + h ^ 2], $MachinePrecision] - g), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(-0.5 / a), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(1.0 / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], -5e-301], N[(N[(t$95$2 * N[Power[t$95$0, 1/3], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[N[(0.5 * t$95$1), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[a, 1/3], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Power[N[(N[(a * -2.0), $MachinePrecision] / t$95$0), $MachinePrecision], 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 := g + \mathsf{hypot}\left(g, h\right)\\
t_1 := \mathsf{hypot}\left(g, h\right) - g\\
t_2 := \sqrt[3]{\frac{-0.5}{a}}\\
\mathbf{if}\;\frac{1}{a \cdot 2} \leq -5 \cdot 10^{-301}:\\
\;\;\;\;t_2 \cdot \sqrt[3]{t_0} + t_2 \cdot \sqrt[3]{t_1}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_1}}{\sqrt[3]{a}} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{t_0}}}\\


\end{array}

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 7 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (/.f64 1 (*.f64 2 a)) < -5.00000000000000013e-301

    1. Initial program 50.0%

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

    2. Simplified50.0%

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

      [Start]50.0%

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

      associate-/r* [=>]50.0%

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

      metadata-eval [=>]50.0%

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

      +-commutative [=>]50.0%

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

      unsub-neg [=>]50.0%

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

      fma-neg [=>]50.0%

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

      sub-neg [=>]50.0%

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

      distribute-neg-out [=>]50.0%

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

      neg-mul-1 [=>]50.0%

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

      associate-*r* [=>]50.0%

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

    3. Applied egg-rr57.2%

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

      [Start]50.0%

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

      div-inv [=>]50.0%

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

      clear-num [<=]50.0%

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

      cbrt-prod [=>]56.2%

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

    4. Simplified57.2%

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

      [Start]57.2%

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

      *-commutative [=>]57.2%

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

    5. Applied egg-rr96.0%

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

      [Start]57.2%

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

      *-commutative [=>]57.2%

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

      cbrt-prod [=>]56.1%

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

      fma-udef [=>]56.1%

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

      add-sqr-sqrt [=>]31.2%

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

      hypot-def [=>]43.6%

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

      add-sqr-sqrt [=>]43.6%

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

      sqrt-unprod [=>]81.9%

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

      sqr-neg [=>]81.9%

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

      sqrt-unprod [<=]89.6%

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

      add-sqr-sqrt [<=]89.6%

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

      sqrt-prod [=>]42.8%

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

      add-sqr-sqrt [<=]95.4%

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

      add-sqr-sqrt [=>]0.0%

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

    6. Simplified96.0%

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

      [Start]96.0%

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

      *-commutative [=>]96.0%

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

    if -5.00000000000000013e-301 < (/.f64 1 (*.f64 2 a))

    1. Initial program 45.1%

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

    2. Simplified45.1%

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

      [Start]45.1%

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

      associate-/r* [=>]45.1%

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

      metadata-eval [=>]45.1%

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

      +-commutative [=>]45.1%

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

      unsub-neg [=>]45.1%

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

      fma-neg [=>]45.1%

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

      sub-neg [=>]45.1%

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

      distribute-neg-out [=>]45.1%

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

      neg-mul-1 [=>]45.1%

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

      associate-*r* [=>]45.1%

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

    3. Applied egg-rr44.8%

      \[\leadsto \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \color{blue}{\frac{1}{\sqrt[3]{\frac{a \cdot -2}{g + \mathsf{hypot}\left(g, h\right)}}}} \]
      Step-by-step derivation

      [Start]45.1%

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

      clear-num [=>]45.1%

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

      cbrt-div [=>]45.1%

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

      metadata-eval [=>]45.1%

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

      div-inv [=>]45.1%

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

      metadata-eval [=>]45.1%

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

      fma-udef [=>]45.1%

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

      add-sqr-sqrt [=>]21.5%

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

      hypot-def [=>]22.0%

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

      add-sqr-sqrt [=>]22.0%

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

      sqrt-unprod [=>]43.7%

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

      sqr-neg [=>]43.7%

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

      sqrt-unprod [<=]44.8%

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

      add-sqr-sqrt [<=]44.8%

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

      sqrt-prod [=>]19.3%

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

      add-sqr-sqrt [<=]44.8%

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

    4. Applied egg-rr94.0%

      \[\leadsto \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}}{\sqrt[3]{a}}} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{g + \mathsf{hypot}\left(g, h\right)}}} \]
      Step-by-step derivation

      [Start]44.8%

      \[ \sqrt[3]{\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{g + \mathsf{hypot}\left(g, h\right)}}} \]

      associate-*l/ [=>]44.9%

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

      cbrt-div [=>]48.1%

      \[ \color{blue}{\frac{\sqrt[3]{0.5 \cdot \left(\sqrt{\mathsf{fma}\left(g, g, -h \cdot h\right)} - g\right)}}{\sqrt[3]{a}}} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{g + \mathsf{hypot}\left(g, h\right)}}} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification95.1%

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{a \cdot 2} \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{\mathsf{hypot}\left(g, h\right) - g}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}}{\sqrt[3]{a}} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{g + \mathsf{hypot}\left(g, h\right)}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy95.9%
Cost39876
\[\begin{array}{l} t_0 := g + \mathsf{hypot}\left(g, h\right)\\ t_1 := \mathsf{hypot}\left(g, h\right) - g\\ t_2 := \sqrt[3]{\frac{-0.5}{a}}\\ \mathbf{if}\;\frac{1}{a \cdot 2} \leq -5 \cdot 10^{-301}:\\ \;\;\;\;t_2 \cdot \sqrt[3]{t_0} + t_2 \cdot \sqrt[3]{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_1}}{\sqrt[3]{a}} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{t_0}}}\\ \end{array} \]
Alternative 2
Accuracy95.7%
Cost39488
\[\frac{\sqrt[3]{0.5 \cdot \left(\mathsf{hypot}\left(g, h\right) - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} \]
Alternative 3
Accuracy95.7%
Cost33604
\[\begin{array}{l} t_0 := g + \mathsf{hypot}\left(g, h\right)\\ t_1 := \mathsf{hypot}\left(g, h\right) - g\\ \mathbf{if}\;\frac{1}{a \cdot 2} \leq -5 \cdot 10^{-301}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{t_0} + \sqrt[3]{t_1 \cdot \frac{-0.5}{a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{0.5 \cdot t_1}}{\sqrt[3]{a}} + \frac{1}{\sqrt[3]{\frac{a \cdot -2}{t_0}}}\\ \end{array} \]
Alternative 4
Accuracy84.2%
Cost33220
\[\begin{array}{l} \mathbf{if}\;g \leq 3.05 \cdot 10^{-248}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} + \sqrt[3]{\left(\mathsf{hypot}\left(g, h\right) - g\right) \cdot \frac{-0.5}{a}}\\ \end{array} \]
Alternative 5
Accuracy81.6%
Cost27012
\[\begin{array}{l} \mathbf{if}\;g \leq 1.15 \cdot 10^{-215}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} + \sqrt[3]{\frac{0.5}{a} \cdot \left(-0.5 \cdot \frac{h \cdot h}{g}\right)}\\ \end{array} \]
Alternative 6
Accuracy79.7%
Cost26884
\[\begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-296}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a}} \cdot \sqrt[3]{g + \mathsf{hypot}\left(g, h\right)} + \sqrt[3]{-0.25 \cdot \frac{h \cdot h}{g \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{0.5 \cdot \left(g \cdot -2\right)}{a}}\\ \end{array} \]
Alternative 7
Accuracy73.1%
Cost13632
\[\sqrt[3]{\frac{-0.5}{a} \cdot 0} + \sqrt[3]{\frac{-0.5}{a} \cdot \left(g + g\right)} \]

Reproduce?

herbie shell --seed 2023271 
(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))))))))