Average Error: 33.9 → 8.8
Time: 10.1s
Precision: binary64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -3.4633822406296006 \cdot 10^{+113}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -2.535008326067869 \cdot 10^{-289}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{4 \cdot \left(c \cdot a\right)}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \leq 3.0185045942898314 \cdot 10^{+49}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array}\]
\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 -3.4633822406296006 \cdot 10^{+113}:\\
\;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\

\mathbf{elif}\;b \leq -2.535008326067869 \cdot 10^{-289}:\\
\;\;\;\;-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 3.0185045942898314 \cdot 10^{+49}:\\
\;\;\;\;-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 \left(\frac{b}{a} - \frac{c}{b}\right)\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 -3.4633822406296006e+113)
   (* -0.5 (* 2.0 (/ c b)))
   (if (<= b -2.535008326067869e-289)
     (*
      -0.5
      (/ (/ (* 4.0 (* c a)) a) (- b (sqrt (- (* b b) (* 4.0 (* c a)))))))
     (if (<= b 3.0185045942898314e+49)
       (* -0.5 (/ (+ b (sqrt (- (* b b) (* 4.0 (* c a))))) a))
       (* -0.5 (* 2.0 (- (/ b a) (/ 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 <= -3.4633822406296006e+113) {
		tmp = -0.5 * (2.0 * (c / b));
	} else if (b <= -2.535008326067869e-289) {
		tmp = -0.5 * (((4.0 * (c * a)) / a) / (b - sqrt((b * b) - (4.0 * (c * a)))));
	} else if (b <= 3.0185045942898314e+49) {
		tmp = -0.5 * ((b + sqrt((b * b) - (4.0 * (c * a)))) / a);
	} else {
		tmp = -0.5 * (2.0 * ((b / a) - (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

Original33.9
Target21.0
Herbie8.8
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.4633822406296006e113

    1. Initial program 60.7

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

    2. Simplified60.7

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

    3. Taylor expanded around -inf 1.8

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

    if -3.4633822406296006e113 < b < -2.5350083260678689e-289

    1. Initial program 33.3

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

    2. Simplified33.3

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

    4. Applied clear-num_binary64_75933.4

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

    6. Applied flip-+_binary64_73433.4

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

    7. Applied associate-/r/_binary64_70633.4

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

    8. Applied associate-/r*_binary64_70433.5

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

    9. Simplified14.9

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

    if -2.5350083260678689e-289 < b < 3.01850459428983145e49

    1. Initial program 10.1

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

    2. Simplified10.1

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

    if 3.01850459428983145e49 < b

    1. Initial program 37.0

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

    2. Simplified37.0

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

    3. Taylor expanded around inf 5.7

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

    4. Simplified5.7

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.4633822406296006 \cdot 10^{+113}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{c}{b}\right)\\ \mathbf{elif}\;b \leq -2.535008326067869 \cdot 10^{-289}:\\ \;\;\;\;-0.5 \cdot \frac{\frac{4 \cdot \left(c \cdot a\right)}{a}}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{elif}\;b \leq 3.0185045942898314 \cdot 10^{+49}:\\ \;\;\;\;-0.5 \cdot \frac{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021045 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))