Average Error: 33.6 → 10.6
Time: 6.2s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, c \cdot \frac{a}{b}, b \cdot -2\right)}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
(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 -3.5e+152)
   (/ (fma 1.5 (* c (/ a b)) (* b -2.0)) (* 3.0 a))
   (if (<= b 6.5e-18)
     (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a))
     (* -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 <= -3.5e+152) {
		tmp = fma(1.5, (c * (a / b)), (b * -2.0)) / (3.0 * a);
	} else if (b <= 6.5e-18) {
		tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
	} 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 <= -3.5e+152)
		tmp = Float64(fma(1.5, Float64(c * Float64(a / b)), Float64(b * -2.0)) / Float64(3.0 * a));
	elseif (b <= 6.5e-18)
		tmp = Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a));
	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, -3.5e+152], N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-18], 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], 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 -3.5 \cdot 10^{+152}:\\
\;\;\;\;\frac{\mathsf{fma}\left(1.5, c \cdot \frac{a}{b}, b \cdot -2\right)}{3 \cdot a}\\

\mathbf{elif}\;b \leq 6.5 \cdot 10^{-18}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\

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


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if b < -3.49999999999999981e152

    1. Initial program 63.4

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

    2. Taylor expanded in b around -inf 10.1

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{c \cdot a}{b} + -2 \cdot b}}{3 \cdot a} \]

    3. Simplified2.6

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5, c \cdot \frac{a}{b}, b \cdot -2\right)}}{3 \cdot a} \]

    if -3.49999999999999981e152 < b < 6.50000000000000008e-18

    1. Initial program 14.6

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

    if 6.50000000000000008e-18 < b

    1. Initial program 54.8

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

    2. Taylor expanded in b around inf 6.5

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

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.5 \cdot 10^{+152}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1.5, c \cdot \frac{a}{b}, b \cdot -2\right)}{3 \cdot a}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022203 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))