Average Error: 33.5 → 9.7
Time: 8.5s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \leq 8.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]
(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 -1.85e+128)
   (fma 0.5 (/ c b_2) (* (/ b_2 a) -2.0))
   (if (<= b_2 8.2e-116)
     (- (/ (sqrt (- (* b_2 b_2) (* c a))) 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 <= -1.85e+128) {
		tmp = fma(0.5, (c / b_2), ((b_2 / a) * -2.0));
	} else if (b_2 <= 8.2e-116) {
		tmp = (sqrt(((b_2 * b_2) - (c * a))) / 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 <= -1.85e+128)
		tmp = fma(0.5, Float64(c / b_2), Float64(Float64(b_2 / a) * -2.0));
	elseif (b_2 <= 8.2e-116)
		tmp = Float64(Float64(sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a))) / a) - Float64(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, -1.85e+128], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 8.2e-116], N[(N[(N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision] - N[(b$95$2 / a), $MachinePrecision]), $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 -1.85 \cdot 10^{+128}:\\
\;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\

\mathbf{elif}\;b_2 \leq 8.2 \cdot 10^{-116}:\\
\;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{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 < -1.85e128

    1. Initial program 53.3

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

    2. Simplified53.3

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

    3. Applied egg-rr53.3

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

    4. Applied egg-rr53.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}}{1}, \frac{{\left(b_2 \cdot b_2 - a \cdot c\right)}^{0.25}}{a}, \frac{-b_2}{a}\right)} \]

    5. Taylor expanded in b_2 around -inf 2.6

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

    6. Simplified2.6

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

    if -1.85e128 < b_2 < 8.1999999999999998e-116

    1. Initial program 10.8

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

    2. Simplified10.8

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

    3. Applied egg-rr10.8

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

    if 8.1999999999999998e-116 < b_2

    1. Initial program 52.1

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

    2. Simplified52.1

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

    3. Taylor expanded in b_2 around inf 10.6

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.85 \cdot 10^{+128}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b_2}, \frac{b_2}{a} \cdot -2\right)\\ \mathbf{elif}\;b_2 \leq 8.2 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b_2 \cdot b_2 - c \cdot a}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b_2} \cdot -0.5\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))