Average Error: 33.8 → 10.1
Time: 7.5s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -2.3813446637561993 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{0.5}{b_2} \cdot c\right)\\ \mathbf{elif}\;b_2 \leq 3.5516450204745413 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{b_2 \cdot b_2 - a \cdot c}, \frac{1}{a}, \frac{-b_2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
(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 -2.3813446637561993e+60)
   (fma (/ b_2 a) -2.0 (* (/ 0.5 b_2) c))
   (if (<= b_2 3.5516450204745413e-95)
     (fma (sqrt (- (* b_2 b_2) (* a c))) (/ 1.0 a) (/ (- b_2) a))
     (/ (* c -0.5) 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 <= -2.3813446637561993e+60) {
		tmp = fma((b_2 / a), -2.0, ((0.5 / b_2) * c));
	} else if (b_2 <= 3.5516450204745413e-95) {
		tmp = fma(sqrt(((b_2 * b_2) - (a * c))), (1.0 / a), (-b_2 / a));
	} else {
		tmp = (c * -0.5) / 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 <= -2.3813446637561993e+60)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(Float64(0.5 / b_2) * c));
	elseif (b_2 <= 3.5516450204745413e-95)
		tmp = fma(sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c))), Float64(1.0 / a), Float64(Float64(-b_2) / a));
	else
		tmp = Float64(Float64(c * -0.5) / 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, -2.3813446637561993e+60], N[(N[(b$95$2 / a), $MachinePrecision] * -2.0 + N[(N[(0.5 / b$95$2), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 3.5516450204745413e-95], N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / a), $MachinePrecision] + N[((-b$95$2) / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision]]]

\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -2.3813446637561993 \cdot 10^{+60}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{0.5}{b_2} \cdot c\right)\\

\mathbf{elif}\;b_2 \leq 3.5516450204745413 \cdot 10^{-95}:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{b_2 \cdot b_2 - a \cdot c}, \frac{1}{a}, \frac{-b_2}{a}\right)\\

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


\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 < -2.3813446637561993e60

    1. Initial program 39.4

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

    2. Simplified39.4

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

    3. Applied egg-rr39.5

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

    4. Taylor expanded in b_2 around -inf 5.0

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

    5. Simplified5.0

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

    if -2.3813446637561993e60 < b_2 < 3.5516450204745413e-95

    1. Initial program 12.5

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

    2. Simplified12.5

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

    3. Applied egg-rr12.5

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

    if 3.5516450204745413e-95 < b_2

    1. Initial program 52.0

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

    2. Simplified52.0

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

    3. Taylor expanded in b_2 around inf 10.1

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

    4. Simplified10.1

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -2.3813446637561993 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{0.5}{b_2} \cdot c\right)\\ \mathbf{elif}\;b_2 \leq 3.5516450204745413 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{b_2 \cdot b_2 - a \cdot c}, \frac{1}{a}, \frac{-b_2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))