Average Error: 34.0 → 10.3
Time: 7.0s
Precision: binary64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]
(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 -1e+145)
   (/ (* b_2 -2.0) a)
   (if (<= b_2 2.1e-83)
     (- (/ (sqrt (fma b_2 b_2 (* c (- a)))) 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 <= -1e+145) {
		tmp = (b_2 * -2.0) / a;
	} else if (b_2 <= 2.1e-83) {
		tmp = (sqrt(fma(b_2, b_2, (c * -a))) / 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 <= -1e+145)
		tmp = Float64(Float64(b_2 * -2.0) / a);
	elseif (b_2 <= 2.1e-83)
		tmp = Float64(Float64(sqrt(fma(b_2, b_2, Float64(c * Float64(-a)))) / a) - 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, -1e+145], N[(N[(b$95$2 * -2.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b$95$2, 2.1e-83], N[(N[(N[Sqrt[N[(b$95$2 * b$95$2 + N[(c * (-a)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 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 -1 \cdot 10^{+145}:\\
\;\;\;\;\frac{b_2 \cdot -2}{a}\\

\mathbf{elif}\;b_2 \leq 2.1 \cdot 10^{-83}:\\
\;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a} - \frac{b_2}{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 < -9.9999999999999999e144

    1. Initial program 60.3

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

    2. Simplified60.3

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

    3. Taylor expanded in b_2 around -inf 2.9

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

    if -9.9999999999999999e144 < b_2 < 2.0999999999999999e-83

    1. Initial program 12.7

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

    2. Simplified12.7

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

    3. Applied egg-rr12.8

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

    4. Applied egg-rr12.7

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

    if 2.0999999999999999e-83 < b_2

    1. Initial program 52.4

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

    2. Simplified52.4

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

    3. Taylor expanded in b_2 around inf 9.7

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

    4. Simplified9.7

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\frac{b_2 \cdot -2}{a}\\ \mathbf{elif}\;b_2 \leq 2.1 \cdot 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b_2, b_2, c \cdot \left(-a\right)\right)}}{a} - \frac{b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b_2}\\ \end{array} \]

Reproduce

herbie shell --seed 2022160 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))