Average Error: 33.9 → 10.2
Time: 4.8s
Precision: binary64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \leq -6.800448154136341 \cdot 10^{-99}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.1253871947171223 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b_2}{a}, \frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \end{array} \]
(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 -6.800448154136341e-99)
   (/ (* -0.5 c) b_2)
   (if (<= b_2 1.1253871947171223e+114)
     (fma -1.0 (/ b_2 a) (/ (- (sqrt (- (* b_2 b_2) (* c a)))) a))
     (+ (* 0.5 (/ c b_2)) (* (/ b_2 a) -2.0)))))
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 <= -6.800448154136341e-99) {
		tmp = (-0.5 * c) / b_2;
	} else if (b_2 <= 1.1253871947171223e+114) {
		tmp = fma(-1.0, (b_2 / a), (-sqrt(((b_2 * b_2) - (c * a))) / a));
	} else {
		tmp = (0.5 * (c / b_2)) + ((b_2 / a) * -2.0);
	}
	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 <= -6.800448154136341e-99)
		tmp = Float64(Float64(-0.5 * c) / b_2);
	elseif (b_2 <= 1.1253871947171223e+114)
		tmp = fma(-1.0, Float64(b_2 / a), Float64(Float64(-sqrt(Float64(Float64(b_2 * b_2) - Float64(c * a)))) / a));
	else
		tmp = Float64(Float64(0.5 * Float64(c / b_2)) + Float64(Float64(b_2 / a) * -2.0));
	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, -6.800448154136341e-99], N[(N[(-0.5 * c), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.1253871947171223e+114], N[(-1.0 * N[(b$95$2 / a), $MachinePrecision] + N[((-N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / a), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(b$95$2 / a), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]]]

\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \leq -6.800448154136341 \cdot 10^{-99}:\\
\;\;\;\;\frac{-0.5 \cdot c}{b_2}\\

\mathbf{elif}\;b_2 \leq 1.1253871947171223 \cdot 10^{+114}:\\
\;\;\;\;\mathsf{fma}\left(-1, \frac{b_2}{a}, \frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\right)\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -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 < -6.8004481541363413e-99

    1. Initial program 52.0

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

    2. Taylor expanded in b_2 around -inf 10.5

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

    3. Applied egg-rr10.4

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

    if -6.8004481541363413e-99 < b_2 < 1.1253871947171223e114

    1. Initial program 12.4

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

    2. Applied egg-rr12.4

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

    if 1.1253871947171223e114 < b_2

    1. Initial program 50.5

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

    2. Taylor expanded in b_2 around inf 2.9

      \[\leadsto \color{blue}{0.5 \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -6.800448154136341 \cdot 10^{-99}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b_2}\\ \mathbf{elif}\;b_2 \leq 1.1253871947171223 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{b_2}{a}, \frac{-\sqrt{b_2 \cdot b_2 - c \cdot a}}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b_2} + \frac{b_2}{a} \cdot -2\\ \end{array} \]

Reproduce

herbie shell --seed 2022153 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))