Average Error: 34.1 → 10.5
Time: 17.9s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -9.75764015235097 \cdot 10^{+32}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a} - \frac{b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 7.843535803153518 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot -2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -9.75764015235097 \cdot 10^{+32}:\\
\;\;\;\;-0.5 \cdot \frac{b}{a} - \frac{b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 7.843535803153518 \cdot 10^{-50}:\\
\;\;\;\;\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\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 -9.75764015235097e+32)
   (- (* -0.5 (/ b a)) (/ b (* a 2.0)))
   (if (<= b 7.843535803153518e-50)
     (- (/ (- (sqrt (- (* b b) (* 4.0 (* a c))))) (* 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 <= -9.75764015235097e+32) {
		tmp = (-0.5 * (b / a)) - (b / (a * 2.0));
	} else if (b <= 7.843535803153518e-50) {
		tmp = (-sqrt((b * b) - (4.0 * (a * c))) / (a * -2.0)) - (b / (a * 2.0));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.1
Target20.9
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.7576401523509693e32

    1. Initial program 36.0

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

    2. Simplified36.0

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

    4. Applied div-sub_binary64_76536.0

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

    5. Taylor expanded around -inf 5.9

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b}{a}} - \frac{b}{a \cdot 2}\]

    if -9.7576401523509693e32 < b < 7.8435358031535177e-50

    1. Initial program 15.4

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

    2. Simplified15.4

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

    4. Applied div-sub_binary64_76515.4

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

    6. Applied frac-2neg_binary64_77115.4

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

    7. Simplified15.4

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

    if 7.8435358031535177e-50 < b

    1. Initial program 54.1

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

    2. Simplified54.1

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

    3. Taylor expanded around inf 7.8

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

    4. Simplified7.8

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -9.75764015235097 \cdot 10^{+32}:\\ \;\;\;\;-0.5 \cdot \frac{b}{a} - \frac{b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 7.843535803153518 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot -2} - \frac{b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2021077 
(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)))