Average Error: 34.1 → 6.4
Time: 4.8s
Precision: binary64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \leq -2.0801679587946785 \cdot 10^{+111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq -8.299193755122384 \cdot 10^{-299}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.425599302719474 \cdot 10^{+102}:\\ \;\;\;\;\left(-4\right) \cdot \frac{\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -2.0801679587946785 \cdot 10^{+111}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \leq -8.299193755122384 \cdot 10^{-299}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\

\mathbf{elif}\;b \leq 2.425599302719474 \cdot 10^{+102}:\\
\;\;\;\;\left(-4\right) \cdot \frac{\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\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 -2.0801679587946785e+111)
   (* 1.0 (- (/ c b) (/ b a)))
   (if (<= b -8.299193755122384e-299)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (if (<= b 2.425599302719474e+102)
       (* (- 4.0) (/ (/ c (+ b (sqrt (- (* b b) (* c (* a 4.0)))))) 2.0))
       (* (/ c b) -1.0)))))
double code(double a, double b, double c) {
	return (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a)));
}
↓
double code(double a, double b, double c) {
	double tmp;
	if ((b <= -2.0801679587946785e+111)) {
		tmp = ((double) (1.0 * ((double) ((c / b) - (b / a)))));
	} else {
		double tmp_1;
		if ((b <= -8.299193755122384e-299)) {
			tmp_1 = (((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))) - b)) / ((double) (a * 2.0)));
		} else {
			double tmp_2;
			if ((b <= 2.425599302719474e+102)) {
				tmp_2 = ((double) (((double) -(4.0)) * ((c / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0))))))))))) / 2.0)));
			} else {
				tmp_2 = ((double) ((c / b) * -1.0));
			}
			tmp_1 = tmp_2;
		}
		tmp = tmp_1;
	}
	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
Target21.3
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.08016795879467846e111

    1. Initial program Error: 49.5 bits

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

    2. SimplifiedError: 49.5 bits

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

    3. Taylor expanded around -inf Error: 3.4 bits

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

    4. SimplifiedError: 3.4 bits

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

    if -2.08016795879467846e111 < b < -8.29919375512238394e-299

    1. Initial program Error: 8.9 bits

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

    2. SimplifiedError: 8.9 bits

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

    if -8.29919375512238394e-299 < b < 2.42559930271947385e102

    1. Initial program Error: 31.7 bits

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

    2. SimplifiedError: 31.7 bits

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

    4. Applied flip--Error: 31.7 bits

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

    5. SimplifiedError: 15.8 bits

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

    6. SimplifiedError: 15.8 bits

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

    8. Applied *-un-lft-identityError: 15.8 bits

      \[\leadsto \frac{\frac{-\left(4 \cdot a\right) \cdot c}{\color{blue}{1 \cdot \left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{a \cdot 2}\]

    9. Applied distribute-lft-neg-inError: 15.8 bits

      \[\leadsto \frac{\frac{\color{blue}{\left(-4 \cdot a\right) \cdot c}}{1 \cdot \left(b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{a \cdot 2}\]

    10. Applied times-fracError: 13.2 bits

      \[\leadsto \frac{\color{blue}{\frac{-4 \cdot a}{1} \cdot \frac{c}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2}\]

    11. Applied times-fracError: 8.6 bits

      \[\leadsto \color{blue}{\frac{\frac{-4 \cdot a}{1}}{a} \cdot \frac{\frac{c}{b + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2}}\]

    12. SimplifiedError: 8.6 bits

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

    if 2.42559930271947385e102 < b

    1. Initial program Error: 59.9 bits

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

    2. SimplifiedError: 59.9 bits

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

    3. Taylor expanded around inf Error: 2.3 bits

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplificationError: 6.4 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.0801679587946785 \cdot 10^{+111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq -8.299193755122384 \cdot 10^{-299}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 2.425599302719474 \cdot 10^{+102}:\\ \;\;\;\;\left(-4\right) \cdot \frac{\frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2020205 
(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)))