The quadratic formula (r1)

Average Error: 34.2 → 9.6

Time: 5.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -4.73423627781717686 \cdot 10^{122}:\\ \;\;\;\;{\left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)}^{1}\\ \mathbf{elif}\;b \le 4.300666801536262 \cdot 10^{-42}:\\ \;\;\;\;{\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

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 <= -4.734236277817177e+122)) {
		temp = pow((1.0 * ((c / b) - (b / a))), 1.0);
	} else {
		double temp_1;
		if ((b <= 4.300666801536262e-42)) {
			temp_1 = pow(((-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)), 1.0);
		} else {
			temp_1 = (-1.0 * (c / b));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	20.8
Herbie	9.6

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

Derivation

1. Split input into 3 regimes

2. if b < -4.734236277817177e+122

   1. Initial program 52.5

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

   3. Applied div-inv 52.6

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

   5. Applied pow1 52.6

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

   6. Applied pow1 52.6

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

   7. Applied pow-prod-down 52.6

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

   8. Simplified 52.5

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

   9. Taylor expanded around -inf 3.4

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

   10. Simplified 3.4

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

   if -4.734236277817177e+122 < b < 4.300666801536262e-42

   1. Initial program 13.6

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

   3. Applied div-inv 13.7

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

   5. Applied pow1 13.7

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

   6. Applied pow1 13.7

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

   7. Applied pow-prod-down 13.7

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

   8. Simplified 13.6

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

   if 4.300666801536262e-42 < b

   1. Initial program 55.0

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

   2. Taylor expanded around inf 6.5

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.73423627781717686 \cdot 10^{122}:\\ \;\;\;\;{\left(1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\right)}^{1}\\ \mathbf{elif}\;b \le 4.300666801536262 \cdot 10^{-42}:\\ \;\;\;\;{\left(\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020053 
(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)))