quadp (p42, positive)

Average Error: 33.8 → 6.7

Time: 6.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -2.8122428251785394 \cdot 10^{+127}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5480753463744248 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.435883796381114 \cdot 10^{+111}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8122428251785394e+127)
   (- (/ c b) (/ b a))
   (if (<= b 1.5480753463744248e-296)
     (/ (- (sqrt (- (* b b) (* 4.0 (* c a)))) b) (* a 2.0))
     (if (<= b 1.435883796381114e+111)
       (/ (* c -2.0) (+ b (sqrt (- (* b b) (* 4.0 (* c a))))))
       (- (/ c b))))))

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 tmp;
	if (b <= -2.8122428251785394e+127) {
		tmp = (c / b) - (b / a);
	} else if (b <= 1.5480753463744248e-296) {
		tmp = (sqrt((b * b) - (4.0 * (c * a))) - b) / (a * 2.0);
	} else if (b <= 1.435883796381114e+111) {
		tmp = (c * -2.0) / (b + sqrt((b * b) - (4.0 * (c * a))));
	} else {
		tmp = -(c / b);
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.8
Target	20.9
Herbie	6.7

\[\begin{array}{l} \mathbf{if}\;b < 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -2.8122428251785394e127

   1. Initial program 53.4

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

   2. Simplified 53.4

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

   3. Taylor expanded around -inf 2.8

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

   if -2.8122428251785394e127 < b < 1.5480753463744248e-296

   1. Initial program 9.5

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

   2. Simplified 9.5

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

   if 1.5480753463744248e-296 < b < 1.43588379638111404e111

   1. Initial program 33.2

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

   2. Simplified 33.2

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

   4. Applied flip--_binary64 33.3

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

   5. Simplified 15.7

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

   6. Simplified 15.7

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

   8. Applied div-inv_binary64 15.8

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

   9. Applied times-frac_binary64 15.0

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

   10. Simplified 8.1

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

   11. Simplified 8.1

       \[\leadsto \left(c \cdot -4\right) \cdot \color{blue}{\frac{0.5}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]
   12. Using strategy rm

   13. Applied associate-*r/_binary64 8.0

       \[\leadsto \color{blue}{\frac{\left(c \cdot -4\right) \cdot 0.5}{b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

   14. Simplified 8.0

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

   if 1.43588379638111404e111 < b

   1. Initial program 60.9

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

   2. Simplified 60.9

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

   3. Taylor expanded around inf 2.8

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

   4. Simplified 2.8

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8122428251785394 \cdot 10^{+127}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 1.5480753463744248 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{elif}\;b \leq 1.435883796381114 \cdot 10^{+111}:\\ \;\;\;\;\frac{c \cdot -2}{b + \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64
  :herbie-expected #f

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))