Average Error: 34.5 → 10.1
Time: 9.3s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.9158395222175973 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \leq 1.5940605920453892 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, \mathsf{fma}\left(a, -c, \mathsf{fma}\left(a, -c, c \cdot a\right)\right)\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]
(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 -2.9158395222175973e+150)
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))
   (if (<= b_2 1.5940605920453892e-74)
     (/ (- (sqrt (fma b_2 b_2 (fma a (- c) (fma a (- c) (* c a))))) b_2) a)
     (* (/ c b_2) -0.5))))
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 <= -2.9158395222175973e+150) {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	} else if (b_2 <= 1.5940605920453892e-74) {
		tmp = (sqrt(fma(b_2, b_2, fma(a, -c, fma(a, -c, (c * a))))) - b_2) / a;
	} else {
		tmp = (c / b_2) * -0.5;
	}
	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 <= -2.9158395222175973e+150)
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	elseif (b_2 <= 1.5940605920453892e-74)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, fma(a, Float64(-c), fma(a, Float64(-c), Float64(c * a))))) - b_2) / a);
	else
		tmp = Float64(Float64(c / b_2) * -0.5);
	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, -2.9158395222175973e+150], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.5940605920453892e-74], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(a * (-c) + N[(a * (-c) + N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b$95$2), $MachinePrecision] * -0.5), $MachinePrecision]]]

\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -2.9158395222175973 \cdot 10^{+150}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\

\mathbf{elif}\;b_2 \leq 1.5940605920453892 \cdot 10^{-74}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, \mathsf{fma}\left(a, -c, \mathsf{fma}\left(a, -c, c \cdot a\right)\right)\right)} - b_2}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b_2} \cdot -0.5\\


\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 < -2.91583952221759727e150

    1. Initial program 63.0

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

    2. Simplified63.0

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

    3. Taylor expanded in b_2 around -inf 2.0

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

    4. Simplified2.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)} \]

    if -2.91583952221759727e150 < b_2 < 1.59406059204538924e-74

    1. Initial program 12.7

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

    2. Simplified12.7

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

    3. Applied egg-rr12.7

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

    if 1.59406059204538924e-74 < b_2

    1. Initial program 53.5

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

    2. Simplified53.5

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

    3. Taylor expanded in b_2 around inf 9.2

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b_2}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9158395222175973 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, -2 \cdot \frac{b_2}{a}\right)\\ \mathbf{elif}\;b_2 \leq 1.5940605920453892 \cdot 10^{-74}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, \mathsf{fma}\left(a, -c, \mathsf{fma}\left(a, -c, c \cdot a\right)\right)\right)} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Reproduce

herbie shell --seed 2022133 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))