Average Error: 33.7 → 10.2
Time: 7.6s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+113}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]
(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 -4.2e-82)
   (/ (- c) b)
   (if (<= b 2.7e+113)
     (/ (- (- b) (sqrt (fma a (* c -4.0) (* b b)))) (* a 2.0))
     (/ (- b) a))))
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 <= -4.2e-82) {
		tmp = -c / b;
	} else if (b <= 2.7e+113) {
		tmp = (-b - sqrt(fma(a, (c * -4.0), (b * b)))) / (a * 2.0);
	} else {
		tmp = -b / a;
	}
	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 <= -4.2e-82)
		tmp = Float64(Float64(-c) / b);
	elseif (b <= 2.7e+113)
		tmp = Float64(Float64(Float64(-b) - sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) / Float64(a * 2.0));
	else
		tmp = Float64(Float64(-b) / a);
	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, -4.2e-82], N[((-c) / b), $MachinePrecision], If[LessEqual[b, 2.7e+113], N[(N[((-b) - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[((-b) / a), $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 -4.2 \cdot 10^{-82}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \leq 2.7 \cdot 10^{+113}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-b}{a}\\


\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target20.9
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.2000000000000001e-82

    1. Initial program 53.1

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

    2. Taylor expanded in b around -inf 9.9

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]

    3. Simplified9.9

      \[\leadsto \color{blue}{\frac{-c}{b}} \]

    if -4.2000000000000001e-82 < b < 2.70000000000000011e113

    1. Initial program 12.5

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

    2. Taylor expanded in a around 0 12.5

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

    3. Simplified12.6

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

    4. Taylor expanded in b around 0 12.5

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{{b}^{2} + -4 \cdot \left(c \cdot a\right)}}}{2 \cdot a} \]

    5. Simplified12.6

      \[\leadsto \frac{\left(-b\right) - \sqrt{\color{blue}{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}}{2 \cdot a} \]

    if 2.70000000000000011e113 < b

    1. Initial program 50.0

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

    2. Taylor expanded in b around inf 3.6

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]

    3. Simplified3.6

      \[\leadsto \color{blue}{\frac{-b}{a}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \leq 2.7 \cdot 10^{+113}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022172 
(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)))