Average Error: 34.6 → 10.2
Time: 6.5s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{b_2}{\frac{a}{-2}}\right)\\ \mathbf{elif}\;b_2 \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \end{array} \]
(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 -2.9e+95)
   (fma 0.5 (/ c b_2) (/ b_2 (/ a -2.0)))
   (if (<= b_2 1.4e-58)
     (/ (- (sqrt (- (* b_2 b_2) (* 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 <= -2.9e+95) {
		tmp = fma(0.5, (c / b_2), (b_2 / (a / -2.0)));
	} else if (b_2 <= 1.4e-58) {
		tmp = (sqrt(((b_2 * b_2) - (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 <= -2.9e+95)
		tmp = fma(0.5, Float64(c / b_2), Float64(b_2 / Float64(a / -2.0)));
	elseif (b_2 <= 1.4e-58)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(c / Float64(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, -2.9e+95], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(b$95$2 / N[(a / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.4e-58], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision] / a), $MachinePrecision], N[(c / N[(b$95$2 / -0.5), $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 -2.9 \cdot 10^{+95}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{b_2}{\frac{a}{-2}}\right)\\

\mathbf{elif}\;b_2 \leq 1.4 \cdot 10^{-58}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

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


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -2.90000000000000013e95

    1. Initial program 44.8

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

    2. Simplified44.8

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

    3. Taylor expanded in b_2 around -inf 4.2

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

    4. Simplified4.2

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

    if -2.90000000000000013e95 < b_2 < 1.4e-58

    1. Initial program 14.2

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

    2. Simplified14.2

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

    if 1.4e-58 < b_2

    1. Initial program 54.1

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

    2. Simplified54.1

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

    3. Taylor expanded in b_2 around inf 8.0

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

    4. Simplified8.0

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.9 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{b_2}{\frac{a}{-2}}\right)\\ \mathbf{elif}\;b_2 \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{b_2}{-0.5}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))