Cubic critical

Average Error: 34.0 → 11.2

Time: 5.0s

Precision: 64

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le -34801737241079.82:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -1.4167290353563065 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 1.3695280426588302 \cdot 10^{154}:\\ \;\;\;\;1 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r119352 = b;
        double r119353 = -r119352;
        double r119354 = r119352 * r119352;
        double r119355 = 3.0;
        double r119356 = a;
        double r119357 = r119355 * r119356;
        double r119358 = c;
        double r119359 = r119357 * r119358;
        double r119360 = r119354 - r119359;
        double r119361 = sqrt(r119360);
        double r119362 = r119353 + r119361;
        double r119363 = r119362 / r119357;
        return r119363;
}

↓

double f(double a, double b, double c) {
        double r119364 = b;
        double r119365 = -34801737241079.82;
        bool r119366 = r119364 <= r119365;
        double r119367 = -r119364;
        double r119368 = 1.5;
        double r119369 = a;
        double r119370 = c;
        double r119371 = r119369 * r119370;
        double r119372 = r119371 / r119364;
        double r119373 = r119368 * r119372;
        double r119374 = r119373 - r119364;
        double r119375 = r119367 + r119374;
        double r119376 = 3.0;
        double r119377 = r119376 * r119369;
        double r119378 = r119375 / r119377;
        double r119379 = -1.4167290353563065e-64;
        bool r119380 = r119364 <= r119379;
        double r119381 = 1.0;
        double r119382 = r119376 * r119371;
        double r119383 = r119377 / r119382;
        double r119384 = 2.0;
        double r119385 = pow(r119364, r119384);
        double r119386 = r119385 - r119385;
        double r119387 = r119386 + r119382;
        double r119388 = r119364 * r119364;
        double r119389 = r119377 * r119370;
        double r119390 = r119388 - r119389;
        double r119391 = sqrt(r119390);
        double r119392 = r119391 - r119364;
        double r119393 = r119387 / r119392;
        double r119394 = r119383 * r119393;
        double r119395 = r119381 / r119394;
        double r119396 = 1.3695280426588302e+154;
        bool r119397 = r119364 <= r119396;
        double r119398 = r119367 - r119391;
        double r119399 = r119370 / r119398;
        double r119400 = r119381 * r119399;
        double r119401 = r119370 / r119381;
        double r119402 = r119381 / r119401;
        double r119403 = r119384 * r119364;
        double r119404 = r119373 - r119403;
        double r119405 = r119402 * r119404;
        double r119406 = r119381 / r119405;
        double r119407 = r119397 ? r119400 : r119406;
        double r119408 = r119380 ? r119395 : r119407;
        double r119409 = r119366 ? r119378 : r119408;
        return r119409;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 regimes

2. if b < -34801737241079.82

   1. Initial program 34.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

   2. Taylor expanded around -inf 12.9

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a}\]

   if -34801737241079.82 < b < -1.4167290353563065e-64

   1. Initial program 4.5

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

   3. Applied flip-+ 40.3

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

   4. Simplified 40.4

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 40.4

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

   7. Applied *-un-lft-identity 40.4

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

   8. Applied times-frac 40.4

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

   9. Applied associate-/l* 40.4

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

   10. Simplified 40.4

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}\]
   11. Using strategy rm

   12. Applied flip-- 40.5

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

   13. Simplified 14.6

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

   14. Simplified 14.6

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

   if -1.4167290353563065e-64 < b < 1.3695280426588302e+154

   1. Initial program 27.4

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

   3. Applied flip-+ 30.2

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

   4. Simplified 16.8

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 16.8

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

   7. Applied *-un-lft-identity 16.8

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

   8. Applied times-frac 16.8

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

   9. Applied associate-/l* 17.0

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

   10. Simplified 16.0

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}\]
   11. Using strategy rm

   12. Applied clear-num 16.0

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{1}{\frac{3 \cdot \left(a \cdot c\right)}{3 \cdot a}}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]

   13. Simplified 11.8

       \[\leadsto \frac{\frac{1}{1}}{\frac{1}{\color{blue}{\frac{c}{1}}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]
   14. Using strategy rm

   15. Applied div-inv 11.8

       \[\leadsto \color{blue}{\frac{1}{1} \cdot \frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}\]

   16. Simplified 11.4

       \[\leadsto \frac{1}{1} \cdot \color{blue}{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}\]

   if 1.3695280426588302e+154 < b

   1. Initial program 64.0

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

   3. Applied flip-+ 64.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

   4. Simplified 38.1

      \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
   5. Using strategy rm

   6. Applied *-un-lft-identity 38.1

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

   7. Applied *-un-lft-identity 38.1

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

   8. Applied times-frac 38.1

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

   9. Applied associate-/l* 38.1

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

   10. Simplified 38.1

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}\]
   11. Using strategy rm

   12. Applied clear-num 38.1

       \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{1}{\frac{3 \cdot \left(a \cdot c\right)}{3 \cdot a}}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]

   13. Simplified 38.0

       \[\leadsto \frac{\frac{1}{1}}{\frac{1}{\color{blue}{\frac{c}{1}}} \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]

   14. Taylor expanded around inf 7.5

       \[\leadsto \frac{\frac{1}{1}}{\frac{1}{\frac{c}{1}} \cdot \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -34801737241079.82:\\ \;\;\;\;\frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}{3 \cdot a}\\ \mathbf{elif}\;b \le -1.4167290353563065 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{3 \cdot \left(a \cdot c\right)} \cdot \frac{\left({b}^{2} - {b}^{2}\right) + 3 \cdot \left(a \cdot c\right)}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 1.3695280426588302 \cdot 10^{154}:\\ \;\;\;\;1 \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{c}{1}} \cdot \left(1.5 \cdot \frac{a \cdot c}{b} - 2 \cdot b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))