Average Error: 34.6 → 10.3
Time: 4.6s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -9.582839230886112 \cdot 10^{-49}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.8527115384289745 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array} \]
(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))
↓
(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -9.582839230886112e-49)
   (* -0.5 (/ c b_2))
   (if (<= b_2 3.8527115384289745e+111)
     (/ (- (- b_2) (sqrt (- (* 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 tmp;
	if (b_2 <= -9.582839230886112e-49) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 3.8527115384289745e+111) {
		tmp = (-b_2 - sqrt(((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)
	tmp = 0.0
	if (b_2 <= -9.582839230886112e-49)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 3.8527115384289745e+111)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - 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_] := If[LessEqual[b$95$2, -9.582839230886112e-49], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.8527115384289745e+111], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $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}
\mathbf{if}\;b_2 \leq -9.582839230886112 \cdot 10^{-49}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \leq 3.8527115384289745 \cdot 10^{+111}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{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 3 regimes

  2. if b_2 < -9.58283923088611151e-49

    1. Initial program 53.8

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

    2. Taylor expanded in b_2 around -inf 8.5

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

    if -9.58283923088611151e-49 < b_2 < 3.85271153842897446e111

    1. Initial program 14.0

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

    if 3.85271153842897446e111 < b_2

    1. Initial program 50.3

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

    2. Taylor expanded in b_2 around inf 3.4

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

    3. Simplified3.4

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -9.582839230886112 \cdot 10^{-49}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 3.8527115384289745 \cdot 10^{+111}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022150 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))