Average Error: 33.9 → 10.9
Time: 5.3s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{a}{\frac{\frac{b}{-3}}{c}} \cdot -0.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -6.6e+133)
   (/ (fma b -2.0 (* (/ a (/ (/ b -3.0) c)) -0.5)) (* a 3.0))
   (if (<= b 4.1e-141)
     (/ (- (sqrt (+ (* b b) (* c (* a -3.0)))) b) (* a 3.0))
     (* -0.5 (/ c b)))))
double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e+133) {
		tmp = fma(b, -2.0, ((a / ((b / -3.0) / c)) * -0.5)) / (a * 3.0);
	} else if (b <= 4.1e-141) {
		tmp = (sqrt(((b * b) + (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end
↓
function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e+133)
		tmp = Float64(fma(b, -2.0, Float64(Float64(a / Float64(Float64(b / -3.0) / c)) * -0.5)) / Float64(a * 3.0));
	elseif (b <= 4.1e-141)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) + Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, c_] := If[LessEqual[b, -6.6e+133], N[(N[(b * -2.0 + N[(N[(a / N[(N[(b / -3.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.1e-141], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]]

\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -6.6 \cdot 10^{+133}:\\
\;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{a}{\frac{\frac{b}{-3}}{c}} \cdot -0.5\right)}{a \cdot 3}\\

\mathbf{elif}\;b \leq 4.1 \cdot 10^{-141}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b}\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if b < -6.6e133

    1. Initial program 56.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

    2. Applied egg-rr56.1

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -3, \mathsf{fma}\left(c, a \cdot -3, 3 \cdot \left(a \cdot c\right)\right)\right)\right)}}}{3 \cdot a} \]

    3. Taylor expanded in b around -inf 11.5

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \frac{3 \cdot \left(c \cdot a\right) + -6 \cdot \left(c \cdot a\right)}{b} + -2 \cdot b}}{3 \cdot a} \]

    4. Simplified3.2

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \frac{a}{\frac{\frac{b}{-3}}{c}} \cdot -0.5\right)}}{3 \cdot a} \]

    if -6.6e133 < b < 4.10000000000000002e-141

    1. Initial program 11.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

    if 4.10000000000000002e-141 < b

    1. Initial program 50.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]

    2. Taylor expanded in b around inf 13.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \frac{a}{\frac{\frac{b}{-3}}{c}} \cdot -0.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022213 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))