The quadratic formula (r1)

Average Error: 33.8 → 6.4

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -6.87536045284350627 \cdot 10^{92}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.27767730821019174 \cdot 10^{-175}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.8636444360254825 \cdot 10^{121}:\\ \;\;\;\;\frac{\frac{1}{0.5} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

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 temp;
	if ((b <= -6.875360452843506e+92)) {
		temp = (1.0 * ((c / b) - (b / a)));
	} else {
		double temp_1;
		if ((b <= -8.277677308210192e-175)) {
			temp_1 = ((-b + sqrt(((b * b) - ((4.0 * a) * c)))) * (1.0 / (2.0 * a)));
		} else {
			double temp_2;
			if ((b <= 3.8636444360254825e+121)) {
				temp_2 = (((1.0 / 0.5) * c) / (-b - sqrt(((b * b) - ((4.0 * a) * c)))));
			} else {
				temp_2 = (-1.0 * (c / b));
			}
			temp_1 = temp_2;
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	20.5
Herbie	6.4

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -6.875360452843506e+92

   1. Initial program 45.0

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

   2. Taylor expanded around -inf 3.5

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

   3. Simplified 3.5

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

   if -6.875360452843506e+92 < b < -8.277677308210192e-175

   1. Initial program 7.0

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

   3. Applied div-inv 7.1

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

   if -8.277677308210192e-175 < b < 3.8636444360254825e+121

   1. Initial program 29.7

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

   3. Applied flip-+ 29.8

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

   4. Simplified 16.1

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 16.1

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

   7. Applied *-un-lft-identity 16.1

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

   8. Applied times-frac 16.1

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

   9. Applied associate-/l* 16.2

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{2 \cdot a}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}\]

   10. Simplified 15.2

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

   11. Taylor expanded around 0 10.0

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

   13. Applied associate-/r* 9.6

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

   14. Simplified 9.5

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

   if 3.8636444360254825e+121 < b

   1. Initial program 60.9

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

   2. Taylor expanded around inf 1.5

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

4. Final simplification 6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.87536045284350627 \cdot 10^{92}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -8.27767730821019174 \cdot 10^{-175}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.8636444360254825 \cdot 10^{121}:\\ \;\;\;\;\frac{\frac{1}{0.5} \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))