Average Error: 34.2 → 13.4
Time: 8.5s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.2117472535674635 \cdot 10^{+127}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.0996677579824965 \cdot 10^{-97}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -3, \mathsf{fma}\left(c, a \cdot -3, \left(3 \cdot a\right) \cdot c\right)\right)\right)\right)}^{0.5} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \frac{c}{b}\right) \cdot -1.5}{3 \cdot a}\\ \end{array} \]
(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 -2.2117472535674635e+127)
   (/ (* b -2.0) (* 3.0 a))
   (if (<= b 1.0996677579824965e-97)
     (/
      (-
       (pow
        (fma b b (fma c (* a -3.0) (fma c (* a -3.0) (* (* 3.0 a) c))))
        0.5)
       b)
      (* 3.0 a))
     (/ (* (* a (/ c b)) -1.5) (* 3.0 a)))))
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 <= -2.2117472535674635e+127) {
		tmp = (b * -2.0) / (3.0 * a);
	} else if (b <= 1.0996677579824965e-97) {
		tmp = (pow(fma(b, b, fma(c, (a * -3.0), fma(c, (a * -3.0), ((3.0 * a) * c)))), 0.5) - b) / (3.0 * a);
	} else {
		tmp = ((a * (c / b)) * -1.5) / (3.0 * a);
	}
	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 <= -2.2117472535674635e+127)
		tmp = Float64(Float64(b * -2.0) / Float64(3.0 * a));
	elseif (b <= 1.0996677579824965e-97)
		tmp = Float64(Float64((fma(b, b, fma(c, Float64(a * -3.0), fma(c, Float64(a * -3.0), Float64(Float64(3.0 * a) * c)))) ^ 0.5) - b) / Float64(3.0 * a));
	else
		tmp = Float64(Float64(Float64(a * Float64(c / b)) * -1.5) / Float64(3.0 * a));
	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, -2.2117472535674635e+127], N[(N[(b * -2.0), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.0996677579824965e-97], N[(N[(N[Power[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision] + N[(c * N[(a * -3.0), $MachinePrecision] + N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision] * -1.5), $MachinePrecision] / N[(3.0 * a), $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 -2.2117472535674635 \cdot 10^{+127}:\\
\;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\

\mathbf{elif}\;b \leq 1.0996677579824965 \cdot 10^{-97}:\\
\;\;\;\;\frac{{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -3, \mathsf{fma}\left(c, a \cdot -3, \left(3 \cdot a\right) \cdot c\right)\right)\right)\right)}^{0.5} - b}{3 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(a \cdot \frac{c}{b}\right) \cdot -1.5}{3 \cdot a}\\


\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 < -2.21174725356746348e127

    1. Initial program 55.0

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

    2. Taylor expanded in b around -inf 3.6

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

    if -2.21174725356746348e127 < b < 1.0996677579824965e-97

    1. Initial program 12.5

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

    2. Applied egg-rr12.5

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

    3. Applied egg-rr12.5

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

    if 1.0996677579824965e-97 < 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 20.7

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

    3. Simplified17.6

      \[\leadsto \frac{\color{blue}{\left(a \cdot \frac{c}{b}\right) \cdot -1.5}}{3 \cdot a} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification13.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2117472535674635 \cdot 10^{+127}:\\ \;\;\;\;\frac{b \cdot -2}{3 \cdot a}\\ \mathbf{elif}\;b \leq 1.0996677579824965 \cdot 10^{-97}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(b, b, \mathsf{fma}\left(c, a \cdot -3, \mathsf{fma}\left(c, a \cdot -3, \left(3 \cdot a\right) \cdot c\right)\right)\right)\right)}^{0.5} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \frac{c}{b}\right) \cdot -1.5}{3 \cdot a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022151 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))