quadm (p42, negative)

Average Error: 33.8 → 6.6

Time: 4.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.08129433547527851 \cdot 10^{156}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.419240425439938 \cdot 10^{-264}:\\ \;\;\;\;\left(4 \cdot c\right) \cdot \frac{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}\\ \mathbf{elif}\;b \le 1.06732918955854865 \cdot 10^{103}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

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 <= -3.0812943354752785e+156)) {
		VAR = (-1.0 * (c / b));
	} else {
		double VAR_1;
		if ((b <= 2.4192404254399383e-264)) {
			VAR_1 = ((4.0 * c) * ((1.0 / (sqrt(((b * b) - (4.0 * (a * c)))) - b)) / 2.0));
		} else {
			double VAR_2;
			if ((b <= 1.0673291895585486e+103)) {
				VAR_2 = ((-b - sqrt(((b * b) - (4.0 * (a * c))))) * (1.0 / (2.0 * a)));
			} else {
				VAR_2 = (-1.0 * (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	33.8
Target	20.6
Herbie	6.6

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

Derivation

1. Split input into 4 regimes

2. if b < -3.0812943354752785e+156

   1. Initial program 64.0

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

   2. Taylor expanded around -inf 1.3

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

   if -3.0812943354752785e+156 < b < 2.4192404254399383e-264

   1. Initial program 32.8

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

   3. Applied flip-- 32.8

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

   4. Simplified 15.5

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

   5. Simplified 15.5

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

   7. Applied *-commutative 15.5

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

   8. Applied div-inv 15.5

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

   9. Applied times-frac 13.9

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

   10. Taylor expanded around 0 8.2

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

   if 2.4192404254399383e-264 < b < 1.0673291895585486e+103

   1. Initial program 9.0

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

   3. Applied div-inv 9.1

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

   if 1.0673291895585486e+103 < b

   1. Initial program 47.7

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

   3. Applied flip-- 63.4

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

   4. Simplified 62.5

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

   5. Simplified 62.5

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

   7. Applied *-commutative 62.5

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

   8. Applied div-inv 62.5

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

   9. Applied times-frac 62.5

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

   10. Taylor expanded around 0 62.3

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

   11. Taylor expanded around 0 3.7

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

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.08129433547527851 \cdot 10^{156}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.419240425439938 \cdot 10^{-264}:\\ \;\;\;\;\left(4 \cdot c\right) \cdot \frac{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}{2}\\ \mathbf{elif}\;b \le 1.06732918955854865 \cdot 10^{103}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :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)))