Average Error: 33.8 → 9.9
Time: 12.5s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -7.5 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{0.5}{b_2} \cdot c\right)\\ \mathbf{elif}\;b_2 \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
(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 -7.5e+127)
   (fma (/ b_2 a) -2.0 (* (/ 0.5 b_2) c))
   (if (<= b_2 2.3e-42)
     (/ (- b_2 (sqrt (- (* b_2 b_2) (* a c)))) (- 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 <= -7.5e+127) {
		tmp = fma((b_2 / a), -2.0, ((0.5 / b_2) * c));
	} else if (b_2 <= 2.3e-42) {
		tmp = (b_2 - sqrt(((b_2 * b_2) - (a * c)))) / -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 <= -7.5e+127)
		tmp = fma(Float64(b_2 / a), -2.0, Float64(Float64(0.5 / b_2) * c));
	elseif (b_2 <= 2.3e-42)
		tmp = Float64(Float64(b_2 - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / Float64(-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, -7.5e+127], 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, 2.3e-42], 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], 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 -7.5 \cdot 10^{+127}:\\
\;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{0.5}{b_2} \cdot c\right)\\

\mathbf{elif}\;b_2 \leq 2.3 \cdot 10^{-42}:\\
\;\;\;\;\frac{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}{-a}\\

\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 < -7.4999999999999996e127

    1. Initial program 53.7

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

    2. Simplified53.7

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

    3. Applied egg-rr53.7

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

    4. Taylor expanded in b_2 around -inf 3.1

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

    5. Simplified3.1

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

    if -7.4999999999999996e127 < b_2 < 2.30000000000000004e-42

    1. Initial program 13.4

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

    2. Simplified13.4

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

    3. Applied egg-rr13.5

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

    4. Applied egg-rr13.4

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

    if 2.30000000000000004e-42 < b_2

    1. Initial program 54.0

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

    2. Simplified54.0

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

    3. Taylor expanded in b_2 around inf 7.8

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

    4. Simplified7.8

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -7.5 \cdot 10^{+127}:\\ \;\;\;\;\mathsf{fma}\left(\frac{b_2}{a}, -2, \frac{0.5}{b_2} \cdot c\right)\\ \mathbf{elif}\;b_2 \leq 2.3 \cdot 10^{-42}:\\ \;\;\;\;\frac{b_2 - \sqrt{b_2 \cdot b_2 - a \cdot c}}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022152 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))