Quadratic roots, full range

Average Error: 34.5 → 6.8

Time: 4.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -3.3421727427471946 \cdot 10^{+111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq -2.8405866436241524 \cdot 10^{-198}:\\ \;\;\;\;\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.722862686819048 \cdot 10^{+59}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{c \cdot \frac{a \cdot 2}{b} + b \cdot -2}\\ \end{array}\]

C

double code(double a, double b, double c) {
	return (((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 <= -3.3421727427471946e+111)) {
		VAR = ((double) (1.0 * ((double) ((c / b) - (b / a)))));
	} else {
		double VAR_1;
		if ((b <= -2.8405866436241524e-198)) {
			VAR_1 = ((double) (((double) (((double) sqrt(((double) (((double) (b * b)) - ((double) (c * ((double) (a * 4.0)))))))) - b)) * (1.0 / ((double) (a * 2.0)))));
		} else {
			double VAR_2;
			if ((b <= 1.722862686819048e+59)) {
				VAR_2 = ((double) ((4.0 / 2.0) * (c / ((double) (((double) -(b)) - ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (c * a)))))))))))));
			} else {
				VAR_2 = ((double) ((4.0 / 2.0) * (c / ((double) (((double) (c * (((double) (a * 2.0)) / b))) + ((double) (b * -2.0)))))));
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -3.34217274274719463e111

   1. Initial program 51.0

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

   2. Taylor expanded around -inf 3.0

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

   3. Simplified 3.0

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

   if -3.34217274274719463e111 < b < -2.8405866436241524e-198

   1. Initial program 7.4

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

   3. Applied div-inv 7.5

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

   4. Simplified 7.5

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

   if -2.8405866436241524e-198 < b < 1.72286268681904807e59

   1. Initial program 27.1

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

   3. Applied flip-+ 27.2

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

   4. Simplified 16.8

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

   5. Simplified 16.8

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

   7. Applied *-un-lft-identity 16.8

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

   8. Applied times-frac 16.8

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

   9. Applied times-frac 16.8

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

   10. Simplified 16.8

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

   11. Simplified 10.4

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

   if 1.72286268681904807e59 < b

   1. Initial program 57.9

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

   3. Applied flip-+ 57.9

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

   4. Simplified 30.5

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

   5. Simplified 30.5

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

   7. Applied *-un-lft-identity 30.5

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

   8. Applied times-frac 30.5

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

   9. Applied times-frac 30.5

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

   10. Simplified 30.5

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

   11. Simplified 27.1

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

   12. Taylor expanded around inf 8.0

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

   13. Simplified 3.9

       \[\leadsto \frac{4}{2} \cdot \left(1 \cdot \frac{c}{\color{blue}{c \cdot \frac{a \cdot 2}{b} + b \cdot -2}}\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3421727427471946 \cdot 10^{+111}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \leq -2.8405866436241524 \cdot 10^{-198}:\\ \;\;\;\;\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.722862686819048 \cdot 10^{+59}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{c \cdot \frac{a \cdot 2}{b} + b \cdot -2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))