quadp (p42, positive)

Average Error: 34.0 → 10.3

Time: 20.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -3.7622181721724994 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\left(\frac{a}{b} \cdot c - b\right) \cdot 2}{a}}{2}\\ \mathbf{elif}\;b \le 7.296044290893796 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r1907313 = b;
        double r1907314 = -r1907313;
        double r1907315 = r1907313 * r1907313;
        double r1907316 = 4.0;
        double r1907317 = a;
        double r1907318 = c;
        double r1907319 = r1907317 * r1907318;
        double r1907320 = r1907316 * r1907319;
        double r1907321 = r1907315 - r1907320;
        double r1907322 = sqrt(r1907321);
        double r1907323 = r1907314 + r1907322;
        double r1907324 = 2.0;
        double r1907325 = r1907324 * r1907317;
        double r1907326 = r1907323 / r1907325;
        return r1907326;
}

↓

double f(double a, double b, double c) {
        double r1907327 = b;
        double r1907328 = -3.7622181721724994e+94;
        bool r1907329 = r1907327 <= r1907328;
        double r1907330 = a;
        double r1907331 = r1907330 / r1907327;
        double r1907332 = c;
        double r1907333 = r1907331 * r1907332;
        double r1907334 = r1907333 - r1907327;
        double r1907335 = 2.0;
        double r1907336 = r1907334 * r1907335;
        double r1907337 = r1907336 / r1907330;
        double r1907338 = r1907337 / r1907335;
        double r1907339 = 7.296044290893796e-114;
        bool r1907340 = r1907327 <= r1907339;
        double r1907341 = r1907327 * r1907327;
        double r1907342 = r1907332 * r1907330;
        double r1907343 = -4.0;
        double r1907344 = r1907342 * r1907343;
        double r1907345 = r1907341 + r1907344;
        double r1907346 = sqrt(r1907345);
        double r1907347 = r1907346 - r1907327;
        double r1907348 = r1907347 / r1907330;
        double r1907349 = r1907348 / r1907335;
        double r1907350 = r1907332 / r1907327;
        double r1907351 = -2.0;
        double r1907352 = r1907350 * r1907351;
        double r1907353 = r1907352 / r1907335;
        double r1907354 = r1907340 ? r1907349 : r1907353;
        double r1907355 = r1907329 ? r1907338 : r1907354;
        return r1907355;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.0
Target	20.9
Herbie	10.3

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

Derivation

1. Split input into 3 regimes

2. if b < -3.7622181721724994e+94

   1. Initial program 43.6

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

   2. Simplified 43.6

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

   3. Taylor expanded around -inf 9.8

      \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{a}}{2}\]

   4. Simplified 3.8

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

   if -3.7622181721724994e+94 < b < 7.296044290893796e-114

   1. Initial program 12.3

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

   2. Simplified 12.4

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

   4. Applied *-un-lft-identity 12.4

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

   5. Applied *-un-lft-identity 12.4

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

   6. Applied distribute-lft-out-- 12.4

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

   7. Applied associate-/l* 12.5

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

   9. Applied div-inv 12.5

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

   10. Applied *-un-lft-identity 12.5

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

   11. Applied times-frac 12.6

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

   12. Simplified 12.5

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

   14. Applied associate-*l/ 12.4

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

   15. Simplified 12.3

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

   if 7.296044290893796e-114 < b

   1. Initial program 51.6

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

   2. Simplified 51.6

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

   4. Applied *-un-lft-identity 51.6

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

   5. Applied *-un-lft-identity 51.6

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

   6. Applied distribute-lft-out-- 51.6

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

   7. Applied associate-/l* 51.6

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

   9. Applied div-inv 51.6

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

   10. Applied *-un-lft-identity 51.6

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

   11. Applied times-frac 51.6

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

   12. Simplified 51.6

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

   14. Applied associate-*l/ 51.6

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

   15. Simplified 51.6

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

   16. Taylor expanded around inf 10.8

       \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.7622181721724994 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\left(\frac{a}{b} \cdot c - b\right) \cdot 2}{a}}{2}\\ \mathbf{elif}\;b \le 7.296044290893796 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b + \left(c \cdot a\right) \cdot -4} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b} \cdot -2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(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)))