Average Error: 33.9 → 11.0
Time: 13.1s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(\frac{c}{b} \cdot a\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 35000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
(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 -1.28e+67)
   (/ (- (- (* 1.5 (* (/ c b) a)) b) b) (* a 3.0))
   (if (<= b 35000.0)
     (/ (- (sqrt (fma c (* a -3.0) (* b b))) b) (* 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 <= -1.28e+67) {
		tmp = (((1.5 * ((c / b) * a)) - b) - b) / (a * 3.0);
	} else if (b <= 35000.0) {
		tmp = (sqrt(fma(c, (a * -3.0), (b * b))) - b) / (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 <= -1.28e+67)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(Float64(c / b) * a)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 35000.0)
		tmp = Float64(Float64(sqrt(fma(c, Float64(a * -3.0), Float64(b * b))) - b) / 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, -1.28e+67], N[(N[(N[(N[(1.5 * N[(N[(c / b), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 35000.0], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $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 -1.28 \cdot 10^{+67}:\\
\;\;\;\;\frac{\left(1.5 \cdot \left(\frac{c}{b} \cdot a\right) - b\right) - b}{a \cdot 3}\\

\mathbf{elif}\;b \leq 35000:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - b}{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 < -1.28e67

    1. Initial program 40.5

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

    2. Taylor expanded in b around -inf 10.0

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

    3. Simplified4.9

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

    if -1.28e67 < b < 35000

    1. Initial program 16.9

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

    2. Taylor expanded in b around 0 17.0

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

    3. Simplified16.9

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

    if 35000 < b

    1. Initial program 55.5

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

    2. Taylor expanded in b around inf 5.5

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

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.28 \cdot 10^{+67}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(\frac{c}{b} \cdot a\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 35000:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -3, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Reproduce

herbie shell --seed 2022155 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))