Average Error: 33.8 → 9.7
Time: 8.9s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -6.828974244409841 \cdot 10^{-68}:\\ \;\;\;\;-0.5 \cdot {\left(0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)\right)}^{-1}\\ \mathbf{elif}\;b \leq 1.3933076843620099 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array} \]
(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 -6.828974244409841e-68)
   (* -0.5 (pow (* 0.5 (- (/ b c) (/ a b))) -1.0))
   (if (<= b 1.3933076843620099e+140)
     (* 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 <= -6.828974244409841e-68) {
		tmp = -0.5 * pow((0.5 * ((b / c) - (a / b))), -1.0);
	} else if (b <= 1.3933076843620099e+140) {
		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 <= -6.828974244409841e-68)
		tmp = Float64(-0.5 * (Float64(0.5 * Float64(Float64(b / c) - Float64(a / b))) ^ -1.0));
	elseif (b <= 1.3933076843620099e+140)
		tmp = Float64(0.5 * Float64(Float64(b + sqrt(fma(a, Float64(c * -4.0), Float64(b * b)))) / Float64(-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, -6.828974244409841e-68], N[(-0.5 * N[Power[N[(0.5 * N[(N[(b / c), $MachinePrecision] - N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.3933076843620099e+140], 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 -6.828974244409841 \cdot 10^{-68}:\\
\;\;\;\;-0.5 \cdot {\left(0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)\right)}^{-1}\\

\mathbf{elif}\;b \leq 1.3933076843620099 \cdot 10^{+140}:\\
\;\;\;\;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

Original33.8
Target21.2
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.82897424440984089e-68

    1. Initial program 53.5

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

    2. Simplified53.6

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

    3. Applied egg-rr53.6

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

    4. Taylor expanded in b around -inf 9.0

      \[\leadsto -0.5 \cdot {\color{blue}{\left(0.5 \cdot \frac{b}{c} - 0.5 \cdot \frac{a}{b}\right)}}^{-1} \]

    5. Simplified9.0

      \[\leadsto -0.5 \cdot {\color{blue}{\left(0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)\right)}}^{-1} \]

    if -6.82897424440984089e-68 < b < 1.39330768436200987e140

    1. Initial program 12.3

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

    2. Simplified12.4

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

    3. Applied egg-rr12.5

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

    4. Applied egg-rr12.4

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

    if 1.39330768436200987e140 < b

    1. Initial program 58.1

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

    2. Simplified58.1

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

    3. Taylor expanded in b around inf 1.3

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

    4. Simplified1.3

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.828974244409841 \cdot 10^{-68}:\\ \;\;\;\;-0.5 \cdot {\left(0.5 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)\right)}^{-1}\\ \mathbf{elif}\;b \leq 1.3933076843620099 \cdot 10^{+140}:\\ \;\;\;\;0.5 \cdot \frac{b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}}{-a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022145 
(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)))