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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -9.934253510808862 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7658541670298186 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;b \leq -9.934253510808862 \cdot 10^{+120}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \leq 2.7658541670298186 \cdot 10^{-38}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2} - \frac{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 -9.934253510808862e+120)
   (- (/ c b) (/ b a))
   (if (<= b 2.7658541670298186e-38)
     (- (/ (sqrt (- (* b b) (* c (* a 4.0)))) (* a 2.0)) (/ 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 <= -9.934253510808862e+120) {
		tmp = (c / b) - (b / a);
	} else if (b <= 2.7658541670298186e-38) {
		tmp = (sqrt((b * b) - (c * (a * 4.0))) / (a * 2.0)) - (b / (a * 2.0));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Original	33.7
Target	20.3
Herbie	9.9

Derivation

Split input into 3 regimes
if b < -9.9342535108088619e120
1. Initial program 51.6
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Simplified51.6
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}}\]
3. Taylor expanded around -inf 2.8
  \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
if -9.9342535108088619e120 < b < 2.7658541670298186e-38
1. Initial program 13.5
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Simplified13.5
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}}\]
3. Using strategy rm
4. Applied div-sub_binary64_7613.5
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} - \frac{b}{a \cdot 2}}\]
if 2.7658541670298186e-38 < b
1. Initial program 54.8
  \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Simplified54.8
  \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a \cdot 2}}\]
3. Taylor expanded around inf 7.4
  \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
4. Simplified7.4
  \[\leadsto \color{blue}{-\frac{c}{b}}\]
Recombined 3 regimes into one program.
Final simplification9.9
\[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.934253510808862 \cdot 10^{+120}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 2.7658541670298186 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}{a \cdot 2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :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 < -9.9342535108088619e120`

`if -9.9342535108088619e120 < b < 2.7658541670298186e-38`

`if 2.7658541670298186e-38 < b`

Reproduce

In
Out