The quadratic formula (r2)

Average Error: 34.1 → 9.3

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -0.04248736169027556:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -7.2720221422685763 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}{\frac{\frac{1}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{\sqrt[3]{\frac{1}{2 \cdot a}}}}\\ \mathbf{elif}\;b \le 5.3362850167303113 \cdot 10^{150}:\\ \;\;\;\;\frac{\frac{1}{2 \cdot a}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

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 VAR;
	if ((b <= -0.04248736169027556)) {
		VAR = (-1.0 * (c / b));
	} else {
		double VAR_1;
		if ((b <= -7.272022142268576e-101)) {
			VAR_1 = ((cbrt((1.0 / (2.0 * a))) * cbrt((1.0 / (2.0 * a)))) / ((1.0 / ((0.0 + (4.0 * (a * c))) / (sqrt(((b * b) - (4.0 * (a * c)))) - b))) / cbrt((1.0 / (2.0 * a)))));
		} else {
			double VAR_2;
			if ((b <= 5.336285016730311e+150)) {
				VAR_2 = ((1.0 / (2.0 * a)) / (1.0 / (-b - sqrt(((b * b) - (4.0 * (a * c)))))));
			} else {
				VAR_2 = (1.0 * ((c / b) - (b / 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	34.1
Target	20.9
Herbie	9.3

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

Derivation

1. Split input into 4 regimes

2. if b < -0.04248736169027556

   1. Initial program 56.1

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

   2. Taylor expanded around -inf 5.6

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

   if -0.04248736169027556 < b < -7.272022142268576e-101

   1. Initial program 37.3

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

   3. Applied clear-num 37.3

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

   5. Applied div-inv 37.3

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

   6. Applied associate-/r* 37.3

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

   8. Applied add-cube-cbrt 37.7

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   9. Applied associate-/l* 37.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}{\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{\sqrt[3]{\frac{1}{2 \cdot a}}}}}\]
   10. Using strategy rm

   11. Applied flip-- 37.7

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}{\frac{\frac{1}{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}}{\sqrt[3]{\frac{1}{2 \cdot a}}}}\]

   12. Simplified 17.4

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}{\frac{\frac{1}{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{\sqrt[3]{\frac{1}{2 \cdot a}}}}\]

   13. Simplified 17.4

       \[\leadsto \frac{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}{\frac{\frac{1}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}}{\sqrt[3]{\frac{1}{2 \cdot a}}}}\]

   if -7.272022142268576e-101 < b < 5.336285016730311e+150

   1. Initial program 11.9

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

   3. Applied clear-num 12.0

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

   5. Applied div-inv 12.0

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

   6. Applied associate-/r* 12.0

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

   if 5.336285016730311e+150 < b

   1. Initial program 62.8

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

   2. Taylor expanded around inf 2.2

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

   3. Simplified 2.2

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

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -0.04248736169027556:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -7.2720221422685763 \cdot 10^{-101}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}{\frac{\frac{1}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}{\sqrt[3]{\frac{1}{2 \cdot a}}}}\\ \mathbf{elif}\;b \le 5.3362850167303113 \cdot 10^{150}:\\ \;\;\;\;\frac{\frac{1}{2 \cdot a}}{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

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

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