Average Error: 33.8 → 9.8
Time: 8.6s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.7823652310513766 \cdot 10^{+65}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -7524134179.167663:\\ \;\;\;\;-0.5 \cdot \frac{\frac{4 \cdot \left(c \cdot a\right)}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \leq -3.559620564434074 \cdot 10^{-74}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 3.685989243441257 \cdot 10^{+139}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a}\right)\\ \end{array}\]
\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 -2.7823652310513766 \cdot 10^{+65}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq -7524134179.167663:\\
\;\;\;\;-0.5 \cdot \frac{\frac{4 \cdot \left(c \cdot a\right)}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\

\mathbf{elif}\;b \leq -3.559620564434074 \cdot 10^{-74}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq 3.685989243441257 \cdot 10^{+139}:\\
\;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\

\mathbf{else}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a}\right)\\

\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 -2.7823652310513766e+65)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b -7524134179.167663)
     (*
      -0.5
      (/ (/ (* 4.0 (* c a)) a) (- b (sqrt (- (* b b) (* 4.0 (* c a)))))))
     (if (<= b -3.559620564434074e-74)
       (* -0.5 (* 2.0 (/ c b)))
       (if (<= b 3.685989243441257e+139)
         (* -0.5 (/ (+ b (sqrt (- (* b b) (* 4.0 (* c a))))) a))
         (* -0.5 (* 2.0 (/ b a))))))))
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 <= -2.7823652310513766e+65) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= -7524134179.167663) {
		tmp = -0.5 * (((4.0 * (c * a)) / a) / (b - sqrt((b * b) - (4.0 * (c * a)))));
	} else if (b <= -3.559620564434074e-74) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= 3.685989243441257e+139) {
		tmp = -0.5 * ((b + sqrt((b * b) - (4.0 * (c * a)))) / a);
	} else {
		tmp = -0.5 * (2.0 * (b / a));
	}
	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

Original33.8
Target20.9
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.7823652310513766e65 or -7524134179.16766262 < b < -3.559620564434074e-74

    1. Initial program 53.7

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

    2. Simplified53.7

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

    3. Taylor expanded around -inf 8.3

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{c}{b}\right)}\]

    if -2.7823652310513766e65 < b < -7524134179.16766262

    1. Initial program 46.3

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

    2. Simplified46.3

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]
    3. Using strategy rm

    4. Applied div-inv_binary64_143946.3

      \[\leadsto -0.5 \cdot \color{blue}{\left(\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a}\right)}\]
    5. Using strategy rm

    6. Applied flip-+_binary64_141646.3

      \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}} \cdot \frac{1}{a}\right)\]

    7. Applied associate-*l/_binary64_138546.3

      \[\leadsto -0.5 \cdot \color{blue}{\frac{\left(b \cdot b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    8. Simplified11.0

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{4 \cdot \left(a \cdot c\right)}{a}}}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\]

    if -3.559620564434074e-74 < b < 3.6859892434412571e139

    1. Initial program 12.5

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

    2. Simplified12.5

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]

    if 3.6859892434412571e139 < b

    1. Initial program 57.1

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

    2. Simplified57.1

      \[\leadsto \color{blue}{-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a}}\]
    3. Using strategy rm

    4. Applied clear-num_binary64_144157.1

      \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{a}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    5. Taylor expanded around 0 2.5

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7823652310513766 \cdot 10^{+65}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -7524134179.167663:\\ \;\;\;\;-0.5 \cdot \frac{\frac{4 \cdot \left(c \cdot a\right)}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \leq -3.559620564434074 \cdot 10^{-74}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq 3.685989243441257 \cdot 10^{+139}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020308 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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