quad2p (problem 3.2.1, positive)

Average Error: 34.6 → 17.6

Time: 7.6s

Precision: binary64

Math

\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b_2 \leq -1.3358971003603176 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{0} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.222250058320258 \cdot 10^{-250}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \leq 1.3298179802620744 \cdot 10^{+154}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{0}}\\ \end{array}\]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

↓

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.3358971003603176e+154)
   (/ (- (sqrt 0.0) b_2) a)
   (if (<= b_2 -1.222250058320258e-250)
     (* (- (sqrt (- (* b_2 b_2) (* a c))) b_2) (/ 1.0 a))
     (if (<= b_2 1.3298179802620744e+154)
       (/ (- c) (+ b_2 (sqrt (- (* b_2 b_2) (* a c)))))
       (/ (- c) (+ b_2 (sqrt 0.0)))))))

C

double code(double a, double b_2, double c) {
	return (-b_2 + sqrt((b_2 * b_2) - (a * c))) / a;
}

↓

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.3358971003603176e+154) {
		tmp = (sqrt(0.0) - b_2) / a;
	} else if (b_2 <= -1.222250058320258e-250) {
		tmp = (sqrt((b_2 * b_2) - (a * c)) - b_2) * (1.0 / a);
	} else if (b_2 <= 1.3298179802620744e+154) {
		tmp = -c / (b_2 + sqrt((b_2 * b_2) - (a * c)));
	} else {
		tmp = -c / (b_2 + sqrt(0.0));
	}
	return tmp;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b_2 < -1.33589710036031761e154

   1. Initial program 64.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

   3. Taylor expanded around 0 52.2

      \[\leadsto \frac{\sqrt{\color{blue}{0}} - b_2}{a}\]

   if -1.33589710036031761e154 < b_2 < -1.222250058320258e-250

   1. Initial program 8.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified 8.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied div-inv_binary64 8.3

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]

   if -1.222250058320258e-250 < b_2 < 1.329817980262074e154

   1. Initial program 33.2

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified 33.2

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied flip--_binary64 33.3

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}}{a}\]

   5. Simplified 15.2

      \[\leadsto \frac{\frac{\color{blue}{-a \cdot c}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}{a}\]

   6. Simplified 15.2

      \[\leadsto \frac{\frac{-a \cdot c}{\color{blue}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity_binary64 15.2

      \[\leadsto \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{1 \cdot a}}\]

   9. Applied *-un-lft-identity_binary64 15.2

      \[\leadsto \frac{\frac{-a \cdot c}{\color{blue}{1 \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{1 \cdot a}\]

   10. Applied *-un-lft-identity_binary64 15.2

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(-a \cdot c\right)}}{1 \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{1 \cdot a}\]

   11. Applied times-frac_binary64 15.2

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{1 \cdot a}\]

   12. Applied times-frac_binary64 15.2

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}}\]

   13. Simplified 15.2

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   14. Simplified 8.3

       \[\leadsto 1 \cdot \color{blue}{\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   if 1.329817980262074e154 < b_2

   1. Initial program 64.0

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

   2. Simplified 64.0

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
   3. Using strategy rm

   4. Applied flip--_binary64 64.0

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c} - b_2 \cdot b_2}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}}{a}\]

   5. Simplified 38.4

      \[\leadsto \frac{\frac{\color{blue}{-a \cdot c}}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}}{a}\]

   6. Simplified 38.4

      \[\leadsto \frac{\frac{-a \cdot c}{\color{blue}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity_binary64 38.4

      \[\leadsto \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{\color{blue}{1 \cdot a}}\]

   9. Applied *-un-lft-identity_binary64 38.4

      \[\leadsto \frac{\frac{-a \cdot c}{\color{blue}{1 \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}}{1 \cdot a}\]

   10. Applied *-un-lft-identity_binary64 38.4

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(-a \cdot c\right)}}{1 \cdot \left(b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}\right)}}{1 \cdot a}\]

   11. Applied times-frac_binary64 38.4

       \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{1 \cdot a}\]

   12. Applied times-frac_binary64 38.4

       \[\leadsto \color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}}\]

   13. Simplified 38.4

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{-a \cdot c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

   14. Simplified 38.3

       \[\leadsto 1 \cdot \color{blue}{\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

   15. Taylor expanded around 0 32.8

       \[\leadsto 1 \cdot \frac{-c}{b_2 + \sqrt{\color{blue}{0}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \leq -1.3358971003603176 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sqrt{0} - b_2}{a}\\ \mathbf{elif}\;b_2 \leq -1.222250058320258 \cdot 10^{-250}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{elif}\;b_2 \leq 1.3298179802620744 \cdot 10^{+154}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b_2 + \sqrt{0}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  :precision binary64
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))