Quadratic roots, full range

Average Error: 33.3 → 10.5

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -6.31172991047764724 \cdot 10^{146}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.26883362658816239 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \end{array}\]

C

double code(double a, double b, double c) {
	return ((double) (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / ((double) (2.0 * a))));
}

↓

double code(double a, double b, double c) {
	double VAR;
	if ((b <= -6.311729910477647e+146)) {
		VAR = ((double) (1.0 * ((double) (((double) (c / b)) - ((double) (b / a))))));
	} else {
		double VAR_1;
		if ((b <= 1.2688336265881624e-51)) {
			VAR_1 = ((double) (((double) (1.0 / 2.0)) * ((double) (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (4.0 * a)) * c)))))))) / a))));
		} else {
			VAR_1 = ((double) (((double) (1.0 / 2.0)) * ((double) (-2.0 * ((double) (c / b))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -6.311729910477647e+146

   1. Initial program 61.3

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

   2. Taylor expanded around -inf 2.7

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

   3. Simplified 2.7

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

   if -6.311729910477647e+146 < b < 1.2688336265881624e-51

   1. Initial program 13.3

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

   3. Applied *-un-lft-identity 13.3

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

   4. Applied times-frac 13.3

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

   if 1.2688336265881624e-51 < b

   1. Initial program 53.1

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

   3. Applied *-un-lft-identity 53.1

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

   4. Applied times-frac 53.1

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

   5. Taylor expanded around inf 8.8

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

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.31172991047764724 \cdot 10^{146}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.26883362658816239 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(-2 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))