quadp (p42, positive)

Average Error: 33.0 → 8.9

Time: 20.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.374834642760219 \cdot 10^{+154}:\\ \;\;\;\;\frac{c \cdot \frac{a \cdot -4}{a}}{\left(2 \cdot \frac{c \cdot a}{b}\right) \cdot 2}\\ \mathbf{elif}\;b \le -7.16975135916936 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{elif}\;b \le 5.966643100467746 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \cdot \left(c \cdot \frac{a \cdot -4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a \cdot -4}{a}}{\left(\left(b - \frac{a}{b} \cdot c\right) \cdot 2\right) \cdot 2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r3184301 = b;
        double r3184302 = -r3184301;
        double r3184303 = r3184301 * r3184301;
        double r3184304 = 4.0;
        double r3184305 = a;
        double r3184306 = c;
        double r3184307 = r3184305 * r3184306;
        double r3184308 = r3184304 * r3184307;
        double r3184309 = r3184303 - r3184308;
        double r3184310 = sqrt(r3184309);
        double r3184311 = r3184302 + r3184310;
        double r3184312 = 2.0;
        double r3184313 = r3184312 * r3184305;
        double r3184314 = r3184311 / r3184313;
        return r3184314;
}

↓

double f(double a, double b, double c) {
        double r3184315 = b;
        double r3184316 = -1.374834642760219e+154;
        bool r3184317 = r3184315 <= r3184316;
        double r3184318 = c;
        double r3184319 = a;
        double r3184320 = -4.0;
        double r3184321 = r3184319 * r3184320;
        double r3184322 = r3184321 / r3184319;
        double r3184323 = r3184318 * r3184322;
        double r3184324 = 2.0;
        double r3184325 = r3184318 * r3184319;
        double r3184326 = r3184325 / r3184315;
        double r3184327 = r3184324 * r3184326;
        double r3184328 = r3184327 * r3184324;
        double r3184329 = r3184323 / r3184328;
        double r3184330 = -7.16975135916936e-173;
        bool r3184331 = r3184315 <= r3184330;
        double r3184332 = r3184315 * r3184315;
        double r3184333 = fma(r3184325, r3184320, r3184332);
        double r3184334 = sqrt(r3184333);
        double r3184335 = sqrt(r3184334);
        double r3184336 = -r3184315;
        double r3184337 = fma(r3184335, r3184335, r3184336);
        double r3184338 = r3184337 / r3184319;
        double r3184339 = r3184338 / r3184324;
        double r3184340 = 5.966643100467746e+48;
        bool r3184341 = r3184315 <= r3184340;
        double r3184342 = 1.0;
        double r3184343 = fma(r3184318, r3184321, r3184332);
        double r3184344 = sqrt(r3184343);
        double r3184345 = r3184315 + r3184344;
        double r3184346 = r3184324 * r3184345;
        double r3184347 = r3184342 / r3184346;
        double r3184348 = r3184347 * r3184323;
        double r3184349 = r3184319 / r3184315;
        double r3184350 = r3184349 * r3184318;
        double r3184351 = r3184315 - r3184350;
        double r3184352 = r3184351 * r3184324;
        double r3184353 = r3184352 * r3184324;
        double r3184354 = r3184323 / r3184353;
        double r3184355 = r3184341 ? r3184348 : r3184354;
        double r3184356 = r3184331 ? r3184339 : r3184355;
        double r3184357 = r3184317 ? r3184329 : r3184356;
        return r3184357;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	33.0
Target	20.1
Herbie	8.9

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

Derivation

1. Split input into 4 regimes

2. if b < -1.374834642760219e+154

   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

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

   4. Applied flip-- 62.3

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

   5. Simplified 62.5

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, 0\right)}}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 62.5

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{\color{blue}{1 \cdot a}}}{2}\]

   8. Applied *-un-lft-identity 62.5

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}}{1 \cdot a}}{2}\]

   9. Applied *-un-lft-identity 62.5

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(a \cdot c, -4, 0\right)}}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}{1 \cdot a}}{2}\]

   10. Applied times-frac 62.5

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}}{1 \cdot a}}{2}\]

   11. Applied times-frac 62.5

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}}{2}\]

   12. Simplified 62.5

       \[\leadsto \frac{\color{blue}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]

   13. Simplified 62.4

       \[\leadsto \frac{1 \cdot \color{blue}{\left(\frac{c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{a \cdot -4}{a}\right)}}{2}\]
   14. Using strategy rm

   15. Applied associate-*l/ 62.4

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

   16. Applied associate-*r/ 62.4

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

   17. Applied associate-/l/ 62.4

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

   18. Taylor expanded around -inf 22.0

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

   if -1.374834642760219e+154 < b < -7.16975135916936e-173

   1. Initial program 5.9

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

   2. Simplified 6.0

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

   4. Applied add-sqr-sqrt 6.0

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

   5. Applied sqrt-prod 6.2

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

   6. Applied fma-neg 6.1

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

   if -7.16975135916936e-173 < b < 5.966643100467746e+48

   1. Initial program 25.8

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

   2. Simplified 25.8

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

   4. Applied flip-- 26.0

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

   5. Simplified 16.6

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, 0\right)}}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 16.6

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{\color{blue}{1 \cdot a}}}{2}\]

   8. Applied *-un-lft-identity 16.6

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}}{1 \cdot a}}{2}\]

   9. Applied *-un-lft-identity 16.6

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(a \cdot c, -4, 0\right)}}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}{1 \cdot a}}{2}\]

   10. Applied times-frac 16.6

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}}{1 \cdot a}}{2}\]

   11. Applied times-frac 16.6

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}}{2}\]

   12. Simplified 16.6

       \[\leadsto \frac{\color{blue}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]

   13. Simplified 11.1

       \[\leadsto \frac{1 \cdot \color{blue}{\left(\frac{c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{a \cdot -4}{a}\right)}}{2}\]
   14. Using strategy rm

   15. Applied associate-*l/ 11.1

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

   16. Applied associate-*r/ 11.1

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

   17. Applied associate-/l/ 11.1

       \[\leadsto \color{blue}{\frac{1 \cdot \left(c \cdot \frac{a \cdot -4}{a}\right)}{2 \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)}}\]
   18. Using strategy rm

   19. Applied div-inv 11.2

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

   if 5.966643100467746e+48 < b

   1. Initial program 56.7

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

   2. Simplified 56.7

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

   4. Applied flip-- 56.7

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

   5. Simplified 28.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(a \cdot c, -4, 0\right)}}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]
   6. Using strategy rm

   7. Applied *-un-lft-identity 28.4

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{\color{blue}{1 \cdot a}}}{2}\]

   8. Applied *-un-lft-identity 28.4

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}}{1 \cdot a}}{2}\]

   9. Applied *-un-lft-identity 28.4

      \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(a \cdot c, -4, 0\right)}}{1 \cdot \left(\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b\right)}}{1 \cdot a}}{2}\]

   10. Applied times-frac 28.4

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{1} \cdot \frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}}{1 \cdot a}}{2}\]

   11. Applied times-frac 28.4

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{1}}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}}{2}\]

   12. Simplified 28.4

       \[\leadsto \frac{\color{blue}{1} \cdot \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)} + b}}{a}}{2}\]

   13. Simplified 25.0

       \[\leadsto \frac{1 \cdot \color{blue}{\left(\frac{c}{b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}} \cdot \frac{a \cdot -4}{a}\right)}}{2}\]
   14. Using strategy rm

   15. Applied associate-*l/ 25.0

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

   16. Applied associate-*r/ 25.0

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

   17. Applied associate-/l/ 25.0

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

   18. Taylor expanded around inf 6.9

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

   19. Simplified 3.6

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.374834642760219 \cdot 10^{+154}:\\ \;\;\;\;\frac{c \cdot \frac{a \cdot -4}{a}}{\left(2 \cdot \frac{c \cdot a}{b}\right) \cdot 2}\\ \mathbf{elif}\;b \le -7.16975135916936 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{elif}\;b \le 5.966643100467746 \cdot 10^{+48}:\\ \;\;\;\;\frac{1}{2 \cdot \left(b + \sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}\right)} \cdot \left(c \cdot \frac{a \cdot -4}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \frac{a \cdot -4}{a}}{\left(\left(b - \frac{a}{b} \cdot c\right) \cdot 2\right) \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))