Average Error: 34.3 → 10.3
Time: 9.7s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -1.3690582298938998 \cdot 10^{+151}:\\ \;\;\;\;-\mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 3.964279773171437 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
\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 -1.3690582298938998 \cdot 10^{+151}:\\
\;\;\;\;-\mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \frac{b}{a}\right)\\

\mathbf{elif}\;b \leq 3.964279773171437 \cdot 10^{-50}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\

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


\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 -1.3690582298938998e+151)
   (- (fma 0.25 (/ (* c -4.0) b) (/ b a)))
   (if (<= b 3.964279773171437e-50)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (- (/ 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 <= -1.3690582298938998e+151) {
		tmp = -fma(0.25, ((c * -4.0) / b), (b / a));
	} else if (b <= 3.964279773171437e-50) {
		tmp = (sqrt((b * b) - (4.0 * (c * a))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.3
Target21.2
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.3690582298938998e151

    1. Initial program 62.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
    2. Using strategy rm

    3. Applied add-exp-log_binary6463.0

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

    4. Applied add-exp-log_binary6463.0

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

    5. Applied prod-exp_binary6463.0

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

    6. Applied add-exp-log_binary6463.0

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

    7. Applied div-exp_binary6463.0

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

    8. Applied exp-diff_binary6463.0

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

    9. Simplified49.1

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

    10. Simplified34.2

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

    11. Taylor expanded around -inf 34.3

      \[\leadsto \color{blue}{-\left(0.25 \cdot \frac{{\left(\sqrt{-4 \cdot \left(c \cdot a\right)}\right)}^{2}}{a \cdot b} + \frac{b}{a}\right)} \]

    12. Simplified2.0

      \[\leadsto \color{blue}{-\mathsf{fma}\left(0.25, \frac{c \cdot -4}{b} \cdot 1, \frac{b}{a}\right)} \]

    if -1.3690582298938998e151 < b < 3.964279773171437e-50

    1. Initial program 14.1

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
    2. Using strategy rm

    3. Applied +-commutative_binary6414.1

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

    4. Applied unsub-neg_binary6414.1

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

    if 3.964279773171437e-50 < b

    1. Initial program 53.9

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

    2. Taylor expanded around inf 7.4

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

    3. Simplified7.4

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3690582298938998 \cdot 10^{+151}:\\ \;\;\;\;-\mathsf{fma}\left(0.25, \frac{c \cdot -4}{b}, \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq 3.964279773171437 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2021211 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

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

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