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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -6.676265206129343 \cdot 10^{+63}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.996853827496981 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

\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 -6.676265206129343 \cdot 10^{+63}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 3.996853827496981 \cdot 10^{-57}:\\
\;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\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 -6.676265206129343e+63)
   (- (/ c b) (/ b a))
   (if (<= b 3.996853827496981e-57)
     (/ (- (sqrt (+ (* b b) (* a (* c -4.0)))) 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 <= -6.676265206129343e+63) {
		tmp = (c / b) - (b / a);
	} else if (b <= 3.996853827496981e-57) {
		tmp = (sqrt((b * b) + (a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Original	33.8
Target	20.6
Herbie	10.3

Derivation

Split input into 3 regimes
if b < -6.676265206129343e63
1. Initial program 38.7
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Simplified38.7
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]
3. Taylor expanded around -inf 5.2
  \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
if -6.676265206129343e63 < b < 3.9968538274969811e-57
1. Initial program 14.3
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Simplified14.3
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]
3. Using strategy rm
4. Applied sub-neg_binary64_75314.3
  \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(-4 \cdot \left(a \cdot c\right)\right)}} - b}{a \cdot 2}\]
5. Simplified14.3
  \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{a \cdot \left(c \cdot -4\right)}} - b}{a \cdot 2}\]
if 3.9968538274969811e-57 < b
1. Initial program 53.5
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
2. Simplified53.5
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}{a \cdot 2}}\]
3. Taylor expanded around inf 8.1
  \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
Recombined 3 regimes into one program.
Final simplification10.3
\[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.676265206129343 \cdot 10^{+63}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 3.996853827496981 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021015 
(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)))

Error

Try it out

Target

Derivation

`if b < -6.676265206129343e63`

`if -6.676265206129343e63 < b < 3.9968538274969811e-57`

`if 3.9968538274969811e-57 < b`

Reproduce

In
Out