quadp (p42, positive)

Average Error: 34.2 → 13.4

Time: 36.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.221171080833594834010969486844711355513 \cdot 10^{166}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\frac{2 \cdot a}{\frac{b}{c}}}}{a}}{2}\\ \mathbf{elif}\;b \le 3.415721894860446366180063963245171885827 \cdot 10^{-240}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), b \cdot b\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 1.333498001175712852871390031258076947683 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(\frac{4 \cdot c}{\sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot \left(-c\right), a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(4 \cdot \left(-c\right), a, b \cdot b\right)}} + b}} \cdot \frac{a}{a}\right) \cdot \frac{\frac{-1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, \left(-4\right) \cdot a, b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, \left(-4\right) \cdot a, b \cdot b\right)}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, -2, 2 \cdot b\right)}}{a}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r63387 = b;
        double r63388 = -r63387;
        double r63389 = r63387 * r63387;
        double r63390 = 4.0;
        double r63391 = a;
        double r63392 = c;
        double r63393 = r63391 * r63392;
        double r63394 = r63390 * r63393;
        double r63395 = r63389 - r63394;
        double r63396 = sqrt(r63395);
        double r63397 = r63388 + r63396;
        double r63398 = 2.0;
        double r63399 = r63398 * r63391;
        double r63400 = r63397 / r63399;
        return r63400;
}

↓

double f(double a, double b, double c) {
        double r63401 = b;
        double r63402 = -1.2211710808335948e+166;
        bool r63403 = r63401 <= r63402;
        double r63404 = 4.0;
        double r63405 = c;
        double r63406 = -r63405;
        double r63407 = a;
        double r63408 = r63406 * r63407;
        double r63409 = 0.0;
        double r63410 = fma(r63404, r63408, r63409);
        double r63411 = 2.0;
        double r63412 = r63411 * r63407;
        double r63413 = r63401 / r63405;
        double r63414 = r63412 / r63413;
        double r63415 = r63410 / r63414;
        double r63416 = r63415 / r63407;
        double r63417 = r63416 / r63411;
        double r63418 = 3.4157218948604464e-240;
        bool r63419 = r63401 <= r63418;
        double r63420 = 1.0;
        double r63421 = r63420 / r63407;
        double r63422 = r63404 * r63406;
        double r63423 = r63401 * r63401;
        double r63424 = fma(r63407, r63422, r63423);
        double r63425 = sqrt(r63424);
        double r63426 = r63425 - r63401;
        double r63427 = r63421 * r63426;
        double r63428 = r63427 / r63411;
        double r63429 = 1.3334980011757129e+154;
        bool r63430 = r63401 <= r63429;
        double r63431 = r63404 * r63405;
        double r63432 = fma(r63422, r63407, r63423);
        double r63433 = sqrt(r63432);
        double r63434 = sqrt(r63433);
        double r63435 = r63434 * r63434;
        double r63436 = r63435 + r63401;
        double r63437 = cbrt(r63436);
        double r63438 = r63431 / r63437;
        double r63439 = r63407 / r63407;
        double r63440 = r63438 * r63439;
        double r63441 = -1.0;
        double r63442 = -r63404;
        double r63443 = r63442 * r63407;
        double r63444 = fma(r63405, r63443, r63423);
        double r63445 = sqrt(r63444);
        double r63446 = r63401 + r63445;
        double r63447 = cbrt(r63446);
        double r63448 = r63441 / r63447;
        double r63449 = r63448 / r63447;
        double r63450 = r63440 * r63449;
        double r63451 = r63450 / r63411;
        double r63452 = r63407 / r63413;
        double r63453 = -r63411;
        double r63454 = 2.0;
        double r63455 = r63454 * r63401;
        double r63456 = fma(r63452, r63453, r63455);
        double r63457 = r63410 / r63456;
        double r63458 = r63457 / r63407;
        double r63459 = r63458 / r63411;
        double r63460 = r63430 ? r63451 : r63459;
        double r63461 = r63419 ? r63428 : r63460;
        double r63462 = r63403 ? r63417 : r63461;
        return r63462;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.2
Target	21.0
Herbie	13.4

\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if b < -1.2211710808335948e+166

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied flip-- 64.0

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

   5. Simplified 62.8

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

   6. Simplified 62.8

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

   7. Taylor expanded around -inf 24.9

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

   8. Simplified 37.3

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

   if -1.2211710808335948e+166 < b < 3.4157218948604464e-240

   1. Initial program 10.9

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

   2. Simplified 10.9

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

   4. Applied div-inv 11.0

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

   if 3.4157218948604464e-240 < b < 1.3334980011757129e+154

   1. Initial program 37.6

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

   2. Simplified 37.6

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

   4. Applied flip-- 37.6

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

   5. Simplified 16.4

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

   6. Simplified 16.4

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

   8. Applied *-un-lft-identity 16.4

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

   9. Applied add-cube-cbrt 17.0

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\color{blue}{\left(\sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}} \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}\right) \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}}}}{1 \cdot a}}{2}\]

   10. Applied *-un-lft-identity 17.0

       \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}}{\left(\sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}} \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}\right) \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}}}{1 \cdot a}}{2}\]

   11. Applied times-frac 17.0

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}} \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}} \cdot \frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}}}}{1 \cdot a}}{2}\]

   12. Applied times-frac 15.9

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}} \cdot \sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}}}{1} \cdot \frac{\frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}}}{a}}}{2}\]

   13. Simplified 15.9

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}} \cdot \frac{\frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(-a \cdot 4, c, b \cdot b\right)}}}}{a}}{2}\]

   14. Simplified 8.5

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}} \cdot \color{blue}{\left(\frac{4 \cdot \left(-c\right)}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}} \cdot \frac{a}{a}\right)}}{2}\]
   15. Using strategy rm

   16. Applied add-sqr-sqrt 8.5

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}} \cdot \left(\frac{4 \cdot \left(-c\right)}{\sqrt[3]{b + \sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}}} \cdot \frac{a}{a}\right)}{2}\]

   17. Applied sqrt-prod 8.5

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}} \cdot \left(\frac{4 \cdot \left(-c\right)}{\sqrt[3]{b + \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}}} \cdot \frac{a}{a}\right)}{2}\]

   18. Simplified 8.5

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}} \cdot \left(\frac{4 \cdot \left(-c\right)}{\sqrt[3]{b + \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}} \cdot \frac{a}{a}\right)}{2}\]

   19. Simplified 8.5

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, 4 \cdot \left(-a\right), b \cdot b\right)}}} \cdot \left(\frac{4 \cdot \left(-c\right)}{\sqrt[3]{b + \sqrt{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}} \cdot \color{blue}{\sqrt{\sqrt{\mathsf{fma}\left(c \cdot \left(-4\right), a, b \cdot b\right)}}}}} \cdot \frac{a}{a}\right)}{2}\]

   if 1.3334980011757129e+154 < b

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied flip-- 64.0

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

   5. Simplified 38.3

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

   6. Simplified 38.3

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

   7. Taylor expanded around inf 14.6

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

   8. Simplified 14.6

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\color{blue}{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, -2, 2 \cdot b\right)}}}{a}}{2}\]
3. Recombined 4 regimes into one program.

4. Final simplification 13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.221171080833594834010969486844711355513 \cdot 10^{166}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\frac{2 \cdot a}{\frac{b}{c}}}}{a}}{2}\\ \mathbf{elif}\;b \le 3.415721894860446366180063963245171885827 \cdot 10^{-240}:\\ \;\;\;\;\frac{\frac{1}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, 4 \cdot \left(-c\right), b \cdot b\right)} - b\right)}{2}\\ \mathbf{elif}\;b \le 1.333498001175712852871390031258076947683 \cdot 10^{154}:\\ \;\;\;\;\frac{\left(\frac{4 \cdot c}{\sqrt[3]{\sqrt{\sqrt{\mathsf{fma}\left(4 \cdot \left(-c\right), a, b \cdot b\right)}} \cdot \sqrt{\sqrt{\mathsf{fma}\left(4 \cdot \left(-c\right), a, b \cdot b\right)}} + b}} \cdot \frac{a}{a}\right) \cdot \frac{\frac{-1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, \left(-4\right) \cdot a, b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(c, \left(-4\right) \cdot a, b \cdot b\right)}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(4, \left(-c\right) \cdot a, 0\right)}{\mathsf{fma}\left(\frac{a}{\frac{b}{c}}, -2, 2 \cdot b\right)}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))