Average Error: 34.1 → 10.1
Time: 9.5s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -5.597250644341572 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 5.690509076716312 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
(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 -5.597250644341572e+143)
   (* (/ b a) -0.6666666666666666)
   (if (<= b 5.690509076716312e-61)
     (/ 1.0 (/ (* a 3.0) (- (sqrt (fma b b (* c (* a -3.0)))) b)))
     (* -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 <= -5.597250644341572e+143) {
		tmp = (b / a) * -0.6666666666666666;
	} else if (b <= 5.690509076716312e-61) {
		tmp = 1.0 / ((a * 3.0) / (sqrt(fma(b, b, (c * (a * -3.0)))) - b));
	} 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 <= -5.597250644341572e+143)
		tmp = Float64(Float64(b / a) * -0.6666666666666666);
	elseif (b <= 5.690509076716312e-61)
		tmp = Float64(1.0 / Float64(Float64(a * 3.0) / Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b)));
	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, -5.597250644341572e+143], N[(N[(b / a), $MachinePrecision] * -0.6666666666666666), $MachinePrecision], If[LessEqual[b, 5.690509076716312e-61], N[(1.0 / N[(N[(a * 3.0), $MachinePrecision] / N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $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 -5.597250644341572 \cdot 10^{+143}:\\
\;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\

\mathbf{elif}\;b \leq 5.690509076716312 \cdot 10^{-61}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}}\\

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


\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 < -5.5972506443415722e143

    1. Initial program 59.9

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

    2. Applied fma-neg_binary6459.9

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

    3. Simplified59.9

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

    4. Taylor expanded in b around -inf 2.7

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

    5. Simplified2.7

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

    if -5.5972506443415722e143 < b < 5.6905090767163117e-61

    1. Initial program 13.2

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

    2. Applied clear-num_binary6413.2

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

    3. Applied fma-neg_binary6413.2

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

    if 5.6905090767163117e-61 < b

    1. Initial program 53.5

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

    2. Taylor expanded in b around inf 8.2

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.597250644341572 \cdot 10^{+143}:\\ \;\;\;\;\frac{b}{a} \cdot -0.6666666666666666\\ \mathbf{elif}\;b \leq 5.690509076716312 \cdot 10^{-61}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 3}{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))