Average Error: 34.0 → 10.6
Time: 7.0s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a} \]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.4155395926676587 \cdot 10^{+106}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4079437542076584 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \mathsf{fma}\left(a, c \cdot -4, 0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]
\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 -3.4155395926676587 \cdot 10^{+106}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 1.4079437542076584 \cdot 10^{-134}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + \mathsf{fma}\left(a, c \cdot -4, 0\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 -3.4155395926676587e+106)
   (- (/ c b) (/ b a))
   (if (<= b 1.4079437542076584e-134)
     (/ (- (sqrt (+ (* b b) (fma a (* c -4.0) 0.0))) 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 <= -3.4155395926676587e+106) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.4079437542076584e-134) {
		tmp = (sqrt(((b * b) + fma(a, (c * -4.0), 0.0))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target20.8
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.41553959266765869e106

    1. Initial program 48.5

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

    2. Taylor expanded in b around -inf 3.3

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

    if -3.41553959266765869e106 < b < 1.40794375420765844e-134

    1. Initial program 11.7

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

    2. Applied egg-rr11.7

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

    3. Taylor expanded in c around 0 11.7

      \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \left(a \cdot c\right) \cdot -4\right) + \color{blue}{0}}}{2 \cdot a} \]

    4. Applied egg-rr11.7

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(b \cdot b + \mathsf{fma}\left(a, c \cdot -4, 0\right)\right)} + 0}}{2 \cdot a} \]

    if 1.40794375420765844e-134 < b

    1. Initial program 50.8

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

    2. Taylor expanded in b around inf 12.0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4155395926676587 \cdot 10^{+106}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.4079437542076584 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \mathsf{fma}\left(a, c \cdot -4, 0\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array} \]

Reproduce

herbie shell --seed 2022129 
(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)))