Quadratic roots, narrow range

Average Error: 28.4 → 16.3

Time: 6.6s

Precision: binary64

\[1.0536712127723509 \cdot 10^{-08} < a \land a < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < b \land b < 94906265.62425156 \land 1.0536712127723509 \cdot 10^{-08} < c \land c < 94906265.62425156\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq 207.26574717373114:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot -2\right) \cdot \frac{1}{a \cdot \left(b \cdot 2\right)}\\ \end{array}\]

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 <= 207.26574717373114)) {
		VAR = ((((double) (((double) (b * b)) - ((double) (((double) (b * b)) + ((double) (4.0 * ((double) (a * c)))))))) / ((double) (b + ((double) sqrt(((double) (((double) (b * b)) - ((double) (4.0 * ((double) (a * c))))))))))) / ((double) (a * 2.0)));
	} else {
		VAR = ((double) (((double) (((double) (a * c)) * -2.0)) * (1.0 / ((double) (a * ((double) (b * 2.0)))))));
	}
	return VAR;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 2 regimes

2. if b < 207.2657471737311

   1. Initial program 15.3

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

   2. Simplified 15.3

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

   4. Applied flip-- 15.3

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

   5. Simplified 14.4

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

   6. Simplified 14.4

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

   if 207.2657471737311 < b

   1. Initial program 35.0

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

   2. Simplified 35.0

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

   3. Taylor expanded around inf 17.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2}\]

   4. Simplified 17.3

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

   6. Applied associate-*l/ 17.3

      \[\leadsto \frac{-2 \cdot \color{blue}{\frac{a \cdot c}{b}}}{a \cdot 2}\]

   7. Applied associate-*r/ 17.3

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

   8. Applied associate-/l/ 17.3

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

   9. Simplified 17.3

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

   11. Applied div-inv 17.3

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

4. Final simplification 16.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 207.26574717373114:\\ \;\;\;\;\frac{\frac{b \cdot b - \left(b \cdot b + 4 \cdot \left(a \cdot c\right)\right)}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(a \cdot c\right) \cdot -2\right) \cdot \frac{1}{a \cdot \left(b \cdot 2\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))