Average Error: 33.5 → 10.1
Time: 5.3s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b_2} \cdot a, b_2 \cdot -2\right)}{a}\\ \end{array} \]
(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 -6.8e-121)
   (* -0.5 (/ c b_2))
   (if (<= b_2 1.45e+117)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* c a)))) a)
     (/ (fma 0.5 (* (/ c b_2) a) (* b_2 -2.0)) a))))
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 <= -6.8e-121) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.45e+117) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (c * a)))) / a;
	} else {
		tmp = fma(0.5, ((c / b_2) * a), (b_2 * -2.0)) / a;
	}
	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 <= -6.8e-121)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.45e+117)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a);
	else
		tmp = Float64(fma(0.5, Float64(Float64(c / b_2) * a), Float64(b_2 * -2.0)) / a);
	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, -6.8e-121], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.45e+117], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(0.5 * N[(N[(c / b$95$2), $MachinePrecision] * a), $MachinePrecision] + N[(b$95$2 * -2.0), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]]]

\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-121}:\\
\;\;\;\;-0.5 \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+117}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b_2} \cdot a, b_2 \cdot -2\right)}{a}\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -6.80000000000000003e-121

    1. Initial program 51.2

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

    2. Taylor expanded in b_2 around -inf 11.4

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

    if -6.80000000000000003e-121 < b_2 < 1.45000000000000014e117

    1. Initial program 11.2

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

    if 1.45000000000000014e117 < b_2

    1. Initial program 51.7

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

    2. Taylor expanded in b_2 around inf 10.2

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

    3. Simplified2.9

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b_2} \cdot a, b_2 \cdot -2\right)}}{a} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.8 \cdot 10^{-121}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.45 \cdot 10^{+117}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{c}{b_2} \cdot a, b_2 \cdot -2\right)}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022192 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))