quadm (p42, negative)

Average Error: 34.2 → 6.6

Time: 6.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \leq -8.049570799378294 \cdot 10^{+77}:\\ \;\;\;\;\frac{-0.5 \cdot \left(4 \cdot c\right)}{b \cdot 2}\\ \mathbf{elif}\;b \leq 1.0118934187104876 \cdot 10^{-285}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\ \mathbf{elif}\;b \leq 1.2458172873965611 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a} - 2 \cdot \frac{c}{b}\right)\\ \end{array}\]

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 -8.049570799378294e+77)
   (/ (* -0.5 (* 4.0 c)) (* b 2.0))
   (if (<= b 1.0118934187104876e-285)
     (* -0.5 (* 4.0 (/ c (- b (sqrt (- (* b b) (* 4.0 (* c a))))))))
     (if (<= b 1.2458172873965611e+125)
       (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* c a))))) (* 2.0 a))
       (* -0.5 (- (* 2.0 (/ b a)) (* 2.0 (/ 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 <= -8.049570799378294e+77) {
		tmp = (-0.5 * (4.0 * c)) / (b * 2.0);
	} else if (b <= 1.0118934187104876e-285) {
		tmp = -0.5 * (4.0 * (c / (b - sqrt((b * b) - (4.0 * (c * a))))));
	} else if (b <= 1.2458172873965611e+125) {
		tmp = (-b - sqrt((b * b) - (4.0 * (c * a)))) / (2.0 * a);
	} else {
		tmp = -0.5 * ((2.0 * (b / a)) - (2.0 * (c / b)));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.3
Herbie	6.6

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

Derivation

1. Split input into 4 regimes

2. if b < -8.0495707993782937e77

   1. Initial program 58.8

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

   2. Simplified 58.8

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

   4. Applied flip-+_binary64 58.8

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

   5. Simplified 31.3

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

   7. Applied *-un-lft-identity_binary64 31.3

      \[\leadsto -0.5 \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{\color{blue}{1 \cdot a}}\]

   8. Applied *-un-lft-identity_binary64 31.3

      \[\leadsto -0.5 \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{1 \cdot a}\]

   9. Applied times-frac_binary64 31.3

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{4}{1} \cdot \frac{a \cdot c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{1 \cdot a}\]

   10. Applied times-frac_binary64 31.3

       \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{\frac{4}{1}}{1} \cdot \frac{\frac{a \cdot c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\right)}\]

   11. Simplified 31.3

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

   12. Simplified 28.6

       \[\leadsto -0.5 \cdot \left(4 \cdot \color{blue}{\left(1 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}\right)\]

   13. Taylor expanded around -inf 2.9

       \[\leadsto -0.5 \cdot \left(4 \cdot \left(1 \cdot \frac{c}{\color{blue}{2 \cdot b}}\right)\right)\]

   14. Simplified 2.9

       \[\leadsto -0.5 \cdot \left(4 \cdot \left(1 \cdot \frac{c}{\color{blue}{b \cdot 2}}\right)\right)\]
   15. Using strategy rm

   16. Applied associate-*r/_binary64 2.9

       \[\leadsto -0.5 \cdot \left(4 \cdot \color{blue}{\frac{1 \cdot c}{b \cdot 2}}\right)\]

   17. Applied associate-*r/_binary64 2.9

       \[\leadsto -0.5 \cdot \color{blue}{\frac{4 \cdot \left(1 \cdot c\right)}{b \cdot 2}}\]

   18. Applied associate-*r/_binary64 2.9

       \[\leadsto \color{blue}{\frac{-0.5 \cdot \left(4 \cdot \left(1 \cdot c\right)\right)}{b \cdot 2}}\]

   if -8.0495707993782937e77 < b < 1.01189341871048762e-285

   1. Initial program 30.6

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

   2. Simplified 30.6

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

   4. Applied flip-+_binary64 30.7

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

   5. Simplified 16.4

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

   7. Applied *-un-lft-identity_binary64 16.4

      \[\leadsto -0.5 \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{\color{blue}{1 \cdot a}}\]

   8. Applied *-un-lft-identity_binary64 16.4

      \[\leadsto -0.5 \cdot \frac{\frac{4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{1 \cdot a}\]

   9. Applied times-frac_binary64 16.4

      \[\leadsto -0.5 \cdot \frac{\color{blue}{\frac{4}{1} \cdot \frac{a \cdot c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}{1 \cdot a}\]

   10. Applied times-frac_binary64 16.4

       \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{\frac{4}{1}}{1} \cdot \frac{\frac{a \cdot c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{a}\right)}\]

   11. Simplified 16.4

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

   12. Simplified 9.5

       \[\leadsto -0.5 \cdot \left(4 \cdot \color{blue}{\left(1 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\right)}\right)\]

   if 1.01189341871048762e-285 < b < 1.2458172873965611e125

   1. Initial program 8.5

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

   if 1.2458172873965611e125 < b

   1. Initial program 54.3

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

   2. Simplified 54.3

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

   3. Taylor expanded around inf 3.0

      \[\leadsto -0.5 \cdot \color{blue}{\left(2 \cdot \frac{b}{a} - 2 \cdot \frac{c}{b}\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.049570799378294 \cdot 10^{+77}:\\ \;\;\;\;\frac{-0.5 \cdot \left(4 \cdot c\right)}{b \cdot 2}\\ \mathbf{elif}\;b \leq 1.0118934187104876 \cdot 10^{-285}:\\ \;\;\;\;-0.5 \cdot \left(4 \cdot \frac{c}{b - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}\right)\\ \mathbf{elif}\;b \leq 1.2458172873965611 \cdot 10^{+125}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(2 \cdot \frac{b}{a} - 2 \cdot \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021173 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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