The quadratic formula (r2)

Average Error: 34.6 → 10.4

Time: 4.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.37866586320142018 \cdot 10^{-92}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.3130237661631722 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le -9.09785312290987894 \cdot 10^{-156}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.9701546350182474 \cdot 10^{129}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

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 <= -1.3786658632014202e-92)) {
		VAR = (-1.0 * (c / b));
	} else {
		double VAR_1;
		if ((b <= -6.313023766163172e-125)) {
			VAR_1 = ((fma(b, b, -((b * b) - (4.0 * (a * c)))) / (sqrt(((b * b) - (4.0 * (a * c)))) - b)) * (1.0 / (2.0 * a)));
		} else {
			double VAR_2;
			if ((b <= -9.097853122909879e-156)) {
				VAR_2 = (-1.0 * (c / b));
			} else {
				double VAR_3;
				if ((b <= 3.9701546350182474e+129)) {
					VAR_3 = (1.0 / ((2.0 * a) / (-b - sqrt(((b * b) - (4.0 * (a * c)))))));
				} else {
					VAR_3 = (1.0 * ((c / b) - (b / a)));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.6
Target	21.1
Herbie	10.4

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

Derivation

1. Split input into 4 regimes

2. if b < -1.3786658632014202e-92 or -6.313023766163172e-125 < b < -9.097853122909879e-156

   1. Initial program 51.3

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

   2. Taylor expanded around -inf 11.5

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

   if -1.3786658632014202e-92 < b < -6.313023766163172e-125

   1. Initial program 26.2

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

   3. Applied div-inv 26.2

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

   5. Applied flip-- 26.2

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

   6. Simplified 26.1

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

   7. Simplified 26.1

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

   if -9.097853122909879e-156 < b < 3.9701546350182474e+129

   1. Initial program 11.0

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

   3. Applied clear-num 11.1

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

   if 3.9701546350182474e+129 < b

   1. Initial program 54.3

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

   2. Taylor expanded around inf 2.0

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

   3. Simplified 2.0

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.37866586320142018 \cdot 10^{-92}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.3130237661631722 \cdot 10^{-125}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, b, -\left(b \cdot b - 4 \cdot \left(a \cdot c\right)\right)\right)}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le -9.09785312290987894 \cdot 10^{-156}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 3.9701546350182474 \cdot 10^{129}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 +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)))