Average Error: 34.9 → 10.1
Time: 7.8s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b_2, -2, \frac{0.5}{\frac{\frac{b_2}{c}}{a}}\right)}{a}\\ \mathbf{elif}\;b_2 \leq 3.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]
(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 -2.1e+148)
   (/ (fma b_2 -2.0 (/ 0.5 (/ (/ b_2 c) a))) a)
   (if (<= b_2 3.2e-27)
     (/ (- (sqrt (- (* b_2 b_2) (* c a))) b_2) a)
     (* -0.5 (/ c b_2)))))
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 <= -2.1e+148) {
		tmp = fma(b_2, -2.0, (0.5 / ((b_2 / c) / a))) / a;
	} else if (b_2 <= 3.2e-27) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) - b_2) / a;
	} else {
		tmp = -0.5 * (c / b_2);
	}
	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 <= -2.1e+148)
		tmp = Float64(fma(b_2, -2.0, Float64(0.5 / Float64(Float64(b_2 / c) / a))) / a);
	elseif (b_2 <= 3.2e-27)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) - b_2) / a);
	else
		tmp = Float64(-0.5 * Float64(c / b_2));
	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, -2.1e+148], N[(N[(b$95$2 * -2.0 + N[(0.5 / N[(N[(b$95$2 / c), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 3.2e-27], 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[(-0.5 * N[(c / b$95$2), $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 -2.1 \cdot 10^{+148}:\\
\;\;\;\;\frac{\mathsf{fma}\left(b_2, -2, \frac{0.5}{\frac{\frac{b_2}{c}}{a}}\right)}{a}\\

\mathbf{elif}\;b_2 \leq 3.2 \cdot 10^{-27}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\

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


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -2.09999999999999999e148

    1. Initial program 61.4

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

    2. Simplified61.4

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

    3. Taylor expanded in b_2 around -inf 10.4

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

    4. Simplified2.1

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

    if -2.09999999999999999e148 < b_2 < 3.19999999999999991e-27

    1. Initial program 14.6

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

    2. Simplified14.6

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

    if 3.19999999999999991e-27 < b_2

    1. Initial program 55.0

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

    2. Simplified55.0

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

    3. Taylor expanded in b_2 around inf 6.5

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.1 \cdot 10^{+148}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b_2, -2, \frac{0.5}{\frac{\frac{b_2}{c}}{a}}\right)}{a}\\ \mathbf{elif}\;b_2 \leq 3.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022206 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))