The quadratic formula (r2)

Average Error: 34.5 → 7.2

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6342944889415786 \cdot 10^{+91}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{\left(a \cdot \frac{2}{\frac{b}{c}} - b\right) - b}\\ \mathbf{elif}\;b \leq 8.663143669049511 \cdot 10^{-296}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{1}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{c}}\\ \mathbf{elif}\;b \leq 2.0984248321180817 \cdot 10^{+70}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{-1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{2 \cdot a}\\ \end{array}\]

C

double code(double a, double b, double c) {
	return (((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c)))))))))) / ((double) (2.0 * a)));
}

↓

double code(double a, double b, double c) {
	double VAR;
	if ((b <= -1.6342944889415786e+91)) {
		VAR = ((double) ((4.0 / 2.0) * (c / ((double) (((double) (((double) (a * (2.0 / (b / c)))) - b)) - b)))));
	} else {
		double VAR_1;
		if ((b <= 8.663143669049511e-296)) {
			VAR_1 = ((double) ((4.0 / 2.0) * (1.0 / (((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (c * a)))))))) - b)) / c))));
		} else {
			double VAR_2;
			if ((b <= 2.0984248321180817e+70)) {
				VAR_2 = ((double) (((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (c * a)))))))))) * (-1.0 / ((double) (2.0 * a)))));
			} else {
				VAR_2 = (((double) (((double) (((double) (2.0 * ((double) (c * (a / b))))) - b)) - b)) / ((double) (2.0 * a)));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.5
Target	21.4
Herbie	7.2

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

Derivation

1. Split input into 4 regimes

2. if b < -1.6342944889415786e91

   1. Initial program 58.9

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

   3. Applied flip-- 58.9

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

   4. Simplified 31.5

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

   5. Simplified 31.5

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

   7. Applied *-un-lft-identity 31.5

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

   8. Applied times-frac 31.5

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

   9. Applied times-frac 31.5

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

   10. Simplified 31.5

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

   11. Simplified 29.2

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

   12. Taylor expanded around -inf 6.8

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

   13. Simplified 2.7

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

   if -1.6342944889415786e91 < b < 8.66314366904951067e-296

   1. Initial program 32.3

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

   3. Applied flip-- 32.3

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

   4. Simplified 17.1

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

   5. Simplified 17.1

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

   7. Applied *-un-lft-identity 17.1

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

   8. Applied times-frac 17.1

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

   9. Applied times-frac 17.1

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

   10. Simplified 17.1

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

   11. Simplified 9.3

       \[\leadsto \frac{4}{2} \cdot \color{blue}{\left(1 \cdot \frac{c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\right)}\]
   12. Using strategy rm

   13. Applied clear-num 9.5

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

   if 8.66314366904951067e-296 < b < 2.0984248321180817e70

   1. Initial program 9.9

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

   3. Applied div-inv 10.0

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

   4. Simplified 10.0

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

   if 2.0984248321180817e70 < b

   1. Initial program 41.6

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

   2. Taylor expanded around inf 10.8

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

   3. Simplified 4.9

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6342944889415786 \cdot 10^{+91}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{\left(a \cdot \frac{2}{\frac{b}{c}} - b\right) - b}\\ \mathbf{elif}\;b \leq 8.663143669049511 \cdot 10^{-296}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{1}{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{c}}\\ \mathbf{elif}\;b \leq 2.0984248321180817 \cdot 10^{+70}:\\ \;\;\;\;\left(b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}\right) \cdot \frac{-1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020199 
(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)))