Average Error: 34.0 → 10.4
Time: 7.6s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.62263644777804 \cdot 10^{-15}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 2.1532958080945476 \cdot 10^{+95}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array} \]
(FPCore (a b c)
 :precision binary64
 (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))
↓
(FPCore (a b c)
 :precision binary64
 (if (<= b -3.62263644777804e-15)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b 2.1532958080945476e+95)
     (* -0.5 (/ (+ b (sqrt (fma a (* c -4.0) (* b b)))) a))
     (* -0.5 (* 2.0 (- (/ b a) (/ c b)))))))
double code(double a, double b, double c) {
	return (-b - sqrt(((b * b) - (4.0 * (a * c))))) / (2.0 * a);
}
↓
double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.62263644777804e-15) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 2.1532958080945476e+95) {
		tmp = -0.5 * ((b + sqrt(fma(a, (c * -4.0), (b * b)))) / a);
	} else {
		tmp = -0.5 * (2.0 * ((b / a) - (c / b)));
	}
	return tmp;
}

function code(a, b, c)
	return Float64(Float64(Float64(-b) - sqrt(Float64(Float64(b * b) - Float64(4.0 * Float64(a * c))))) / Float64(2.0 * a))
end
↓
function code(a, b, c)
	tmp = 0.0
	if (b <= -3.62263644777804e-15)
		tmp = Float64(-0.5 * Float64(2.0 * Float64(c / b)));
	elseif (b <= 2.1532958080945476e+95)
		tmp = Float64(-0.5 * Float64(Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) / a));
	else
		tmp = Float64(-0.5 * Float64(2.0 * Float64(Float64(b / a) - Float64(c / b))));
	end
	return tmp
end

code[a_, b_, c_] := N[(N[((-b) - N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(4.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, c_] := If[LessEqual[b, -3.62263644777804e-15], N[(-0.5 * N[(2.0 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1532958080945476e+95], N[(-0.5 * N[(N[(b + N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(2.0 * N[(N[(b / a), $MachinePrecision] - N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \leq -3.62263644777804 \cdot 10^{-15}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq 2.1532958080945476 \cdot 10^{+95}:\\
\;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\


\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target21.7
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.62263644777803983e-15

    1. Initial program 54.9

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Simplified55.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]

    3. Taylor expanded in b around -inf 6.6

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{c}{b}\right)} \]

    if -3.62263644777803983e-15 < b < 2.15329580809454757e95

    1. Initial program 15.3

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Simplified15.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]

    3. Applied *-un-lft-identity_binary6415.3

      \[\leadsto -0.5 \cdot \frac{b + \sqrt{\color{blue}{1 \cdot \mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{a} \]

    if 2.15329580809454757e95 < b

    1. Initial program 46.0

      \[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]

    2. Simplified46.0

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}} \]

    3. Taylor expanded in b around inf 3.8

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{b}{a} - 2 \cdot \frac{c}{b}\right)} \]

    4. Simplified3.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.62263644777804 \cdot 10^{-15}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 2.1532958080945476 \cdot 10^{+95}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))