The quadratic formula (r1)

Average Error: 33.6 → 8.6

Time: 6.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6236541306877014 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 7.131198222427231 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 5.525835260688754 \cdot 10^{+121}:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}\\ \end{array}\]

FPCore

(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 -1.6236541306877014e+56)
   (/ (* 2.0 (- (/ (* a c) b) b)) (* 2.0 a))
   (if (<= b 7.131198222427231e-227)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* 2.0 a))
     (if (<= b 5.525835260688754e+121)
       (* -2.0 (/ c (+ b (sqrt (- (* b b) (* c (* a 4.0)))))))
       (* -2.0 (/ c (* 2.0 (- b (/ (* a c) b)))))))))

C

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 <= -1.6236541306877014e+56) {
		tmp = (2.0 * (((a * c) / b) - b)) / (2.0 * a);
	} else if (b <= 7.131198222427231e-227) {
		tmp = (sqrt((b * b) - (c * (a * 4.0))) - b) / (2.0 * a);
	} else if (b <= 5.525835260688754e+121) {
		tmp = -2.0 * (c / (b + sqrt((b * b) - (c * (a * 4.0)))));
	} else {
		tmp = -2.0 * (c / (2.0 * (b - ((a * c) / b))));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.6
Target	20.5
Herbie	8.6

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

Derivation

1. Split input into 4 regimes

2. if b < -1.6236541306877014e56

   1. Initial program 38.1

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

   2. Simplified 38.1

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

   3. Taylor expanded around -inf 11.2

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

   4. Simplified 11.2

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

   if -1.6236541306877014e56 < b < 7.131198222427231e-227

   1. Initial program 10.2

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

   2. Simplified 10.2

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

   if 7.131198222427231e-227 < b < 5.52583526068875372e121

   1. Initial program 36.8

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

   2. Simplified 36.8

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

   4. Applied flip--_binary64_1430 36.8

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

   5. Simplified 17.2

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

   6. Simplified 17.2

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

   8. Applied *-un-lft-identity_binary64_1455 17.2

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

   9. Applied times-frac_binary64_1461 14.4

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

   10. Applied times-frac_binary64_1461 7.1

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

   11. Simplified 7.1

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

   12. Simplified 7.1

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

   if 5.52583526068875372e121 < b

   1. Initial program 61.3

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

   2. Simplified 61.3

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

   4. Applied flip--_binary64_1430 61.3

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

   5. Simplified 34.3

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

   6. Simplified 34.3

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

   8. Applied *-un-lft-identity_binary64_1455 34.3

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

   9. Applied times-frac_binary64_1461 34.2

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

   10. Applied times-frac_binary64_1461 32.6

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

   11. Simplified 32.6

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

   12. Simplified 32.6

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

   13. Taylor expanded around inf 5.5

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

   14. Simplified 5.5

       \[\leadsto 1 \cdot \left(-2 \cdot \frac{c}{\color{blue}{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 8.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6236541306877014 \cdot 10^{+56}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{a \cdot c}{b} - b\right)}{2 \cdot a}\\ \mathbf{elif}\;b \leq 7.131198222427231 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\\ \mathbf{elif}\;b \leq 5.525835260688754 \cdot 10^{+121}:\\ \;\;\;\;-2 \cdot \frac{c}{b + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{c}{2 \cdot \left(b - \frac{a \cdot c}{b}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020281 
(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)))