Average Error: 34.3 → 9.8
Time: 12.2s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} t_0 := -0.5 \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \leq -3.566979873129798 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq -2.583305902750818 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq -1.677958789820394 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b_2 \leq 1.0611311445557777 \cdot 10^{+52}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array} \]
(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
↓
(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* -0.5 (/ c b_2))))
   (if (<= b_2 -3.566979873129798e+56)
     t_0
     (if (<= b_2 -2.583305902750818e-39)
       (/ (/ (* c a) a) (- (sqrt (- (* b_2 b_2) (* c a))) b_2))
       (if (<= b_2 -1.677958789820394e-73)
         t_0
         (if (<= b_2 1.0611311445557777e+52)
           (- (/ (- b_2) a) (/ (sqrt (fma b_2 b_2 (- (* c a)))) a))
           (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))))))))
double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}
↓
double code(double a, double b_2, double c) {
	double t_0 = -0.5 * (c / b_2);
	double tmp;
	if (b_2 <= -3.566979873129798e+56) {
		tmp = t_0;
	} else if (b_2 <= -2.583305902750818e-39) {
		tmp = ((c * a) / a) / (sqrt(((b_2 * b_2) - (c * a))) - b_2);
	} else if (b_2 <= -1.677958789820394e-73) {
		tmp = t_0;
	} else if (b_2 <= 1.0611311445557777e+52) {
		tmp = (-b_2 / a) - (sqrt(fma(b_2, b_2, -(c * a))) / a);
	} else {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	}
	return tmp;
}

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end
↓
function code(a, b_2, c)
	t_0 = Float64(-0.5 * Float64(c / b_2))
	tmp = 0.0
	if (b_2 <= -3.566979873129798e+56)
		tmp = t_0;
	elseif (b_2 <= -2.583305902750818e-39)
		tmp = Float64(Float64(Float64(c * a) / a) / Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2));
	elseif (b_2 <= -1.677958789820394e-73)
		tmp = t_0;
	elseif (b_2 <= 1.0611311445557777e+52)
		tmp = Float64(Float64(Float64(-b_2) / a) - Float64(sqrt(fma(b_2, b_2, Float64(-Float64(c * a)))) / a));
	else
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	end
	return tmp
end

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]
↓
code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b$95$2, -3.566979873129798e+56], t$95$0, If[LessEqual[b$95$2, -2.583305902750818e-39], N[(N[(N[(c * a), $MachinePrecision] / a), $MachinePrecision] / N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -1.677958789820394e-73], t$95$0, If[LessEqual[b$95$2, 1.0611311445557777e+52], N[(N[((-b$95$2) / a), $MachinePrecision] - N[(N[Sqrt[N[(b$95$2 * b$95$2 + (-N[(c * a), $MachinePrecision])), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
t_0 := -0.5 \cdot \frac{c}{b_2}\\
\mathbf{if}\;b_2 \leq -3.566979873129798 \cdot 10^{+56}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;b_2 \leq -2.583305902750818 \cdot 10^{-39}:\\
\;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\

\mathbf{elif}\;b_2 \leq -1.677958789820394 \cdot 10^{-73}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;b_2 \leq 1.0611311445557777 \cdot 10^{+52}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\


\end{array}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -3.56697987312979818e56 or -2.5833059027508179e-39 < b_2 < -1.67795878982039409e-73

    1. Initial program 55.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

    2. Taylor expanded in b_2 around -inf 6.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]

    if -3.56697987312979818e56 < b_2 < -2.5833059027508179e-39

    1. Initial program 43.2

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

    2. Applied div-inv_binary6443.2

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}} \]

    3. Applied flip--_binary6443.2

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{1}{a} \]

    4. Applied associate-*l/_binary6443.2

      \[\leadsto \color{blue}{\frac{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \]

    5. Simplified14.2

      \[\leadsto \frac{\color{blue}{\frac{c \cdot a}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}} \]

    if -1.67795878982039409e-73 < b_2 < 1.06113114455577771e52

    1. Initial program 13.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

    2. Applied div-sub_binary6413.8

      \[\leadsto \color{blue}{\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}} \]

    3. Applied fma-neg_binary6413.8

      \[\leadsto \frac{-b_2}{a} - \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b_2, b_2, -a \cdot c\right)}}}{a} \]

    4. Simplified13.8

      \[\leadsto \frac{-b_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, \color{blue}{-c \cdot a}\right)}}{a} \]

    if 1.06113114455577771e52 < b_2

    1. Initial program 39.0

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]

    2. Taylor expanded in b_2 around inf 5.6

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}} \]

    3. Simplified5.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -3.566979873129798 \cdot 10^{+56}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq -2.583305902750818 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}\\ \mathbf{elif}\;b_2 \leq -1.677958789820394 \cdot 10^{-73}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.0611311445557777 \cdot 10^{+52}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))