Average Error: 33.7 → 10.0
Time: 6.7s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.492330097195095 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, b \cdot \frac{-0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.7454011737234576 \cdot 10^{-87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
(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 -6.492330097195095e+142)
   (fma 0.5 (/ c b) (* b (/ -0.6666666666666666 a)))
   (if (<= b 1.7454011737234576e-87)
     (/ (fma -1.0 b (sqrt (fma b b (* c (* a -3.0))))) (* a 3.0))
     (* (/ c b) -0.5))))
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 <= -6.492330097195095e+142) {
		tmp = fma(0.5, (c / b), (b * (-0.6666666666666666 / a)));
	} else if (b <= 1.7454011737234576e-87) {
		tmp = fma(-1.0, b, sqrt(fma(b, b, (c * (a * -3.0))))) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= -6.492330097195095e+142)
		tmp = fma(0.5, Float64(c / b), Float64(b * Float64(-0.6666666666666666 / a)));
	elseif (b <= 1.7454011737234576e-87)
		tmp = Float64(fma(-1.0, b, sqrt(fma(b, b, Float64(c * Float64(a * -3.0))))) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	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, -6.492330097195095e+142], N[(0.5 * N[(c / b), $MachinePrecision] + N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.7454011737234576e-87], N[(N[(-1.0 * b + N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $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 -6.492330097195095 \cdot 10^{+142}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, b \cdot \frac{-0.6666666666666666}{a}\right)\\

\mathbf{elif}\;b \leq 1.7454011737234576 \cdot 10^{-87}:\\
\;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a \cdot 3}\\

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


\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -6.492330097195095e142

    1. Initial program 58.9

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

    2. Taylor expanded in b around -inf 3.0

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b} - 0.6666666666666666 \cdot \frac{b}{a}} \]

    3. Simplified2.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b}, \frac{-0.6666666666666666}{a} \cdot b\right)} \]

    if -6.492330097195095e142 < b < 1.74540117372345764e-87

    1. Initial program 11.7

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

    2. Applied egg-rr11.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}}{3 \cdot a} \]

    if 1.74540117372345764e-87 < b

    1. Initial program 52.1

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

    2. Taylor expanded in b around inf 10.1

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.492330097195095 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b}, b \cdot \frac{-0.6666666666666666}{a}\right)\\ \mathbf{elif}\;b \leq 1.7454011737234576 \cdot 10^{-87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))