jeff quadratic root 2

Average Error: 19.8 → 7.5

Time: 17.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.350676057343211554491874986591947593393 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{\sqrt{b}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le -4.452231346955637703324616244948712027068 \cdot 10^{-267} \lor \neg \left(b \le 1.287594296476507417293820707954348336152 \cdot 10^{78}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\left|\sqrt[3]{b}\right|}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt{\sqrt[3]{b}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(4 \cdot a\right) \cdot c}{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r30439 = b;
        double r30440 = 0.0;
        bool r30441 = r30439 >= r30440;
        double r30442 = 2.0;
        double r30443 = c;
        double r30444 = r30442 * r30443;
        double r30445 = -r30439;
        double r30446 = r30439 * r30439;
        double r30447 = 4.0;
        double r30448 = a;
        double r30449 = r30447 * r30448;
        double r30450 = r30449 * r30443;
        double r30451 = r30446 - r30450;
        double r30452 = sqrt(r30451);
        double r30453 = r30445 - r30452;
        double r30454 = r30444 / r30453;
        double r30455 = r30445 + r30452;
        double r30456 = r30442 * r30448;
        double r30457 = r30455 / r30456;
        double r30458 = r30441 ? r30454 : r30457;
        return r30458;
}

↓

double f(double a, double b, double c) {
        double r30459 = b;
        double r30460 = -1.3506760573432116e+154;
        bool r30461 = r30459 <= r30460;
        double r30462 = 0.0;
        bool r30463 = r30459 >= r30462;
        double r30464 = 2.0;
        double r30465 = c;
        double r30466 = r30464 * r30465;
        double r30467 = a;
        double r30468 = sqrt(r30459);
        double r30469 = r30467 / r30468;
        double r30470 = r30465 / r30468;
        double r30471 = r30469 * r30470;
        double r30472 = r30464 * r30471;
        double r30473 = 2.0;
        double r30474 = r30473 * r30459;
        double r30475 = r30472 - r30474;
        double r30476 = r30466 / r30475;
        double r30477 = r30467 * r30465;
        double r30478 = r30477 / r30459;
        double r30479 = r30464 * r30478;
        double r30480 = r30479 - r30459;
        double r30481 = r30480 - r30459;
        double r30482 = r30464 * r30467;
        double r30483 = r30481 / r30482;
        double r30484 = r30463 ? r30476 : r30483;
        double r30485 = -4.452231346955638e-267;
        bool r30486 = r30459 <= r30485;
        double r30487 = 1.2875942964765074e+78;
        bool r30488 = r30459 <= r30487;
        double r30489 = !r30488;
        bool r30490 = r30486 || r30489;
        double r30491 = cbrt(r30465);
        double r30492 = r30491 * r30491;
        double r30493 = cbrt(r30459);
        double r30494 = fabs(r30493);
        double r30495 = r30492 / r30494;
        double r30496 = r30469 * r30495;
        double r30497 = sqrt(r30493);
        double r30498 = r30491 / r30497;
        double r30499 = r30496 * r30498;
        double r30500 = r30464 * r30499;
        double r30501 = r30500 - r30474;
        double r30502 = r30466 / r30501;
        double r30503 = r30459 * r30459;
        double r30504 = 4.0;
        double r30505 = r30504 * r30467;
        double r30506 = r30505 * r30465;
        double r30507 = r30503 - r30506;
        double r30508 = sqrt(r30507);
        double r30509 = r30508 - r30459;
        double r30510 = r30509 / r30482;
        double r30511 = r30463 ? r30502 : r30510;
        double r30512 = -r30459;
        double r30513 = r30512 - r30508;
        double r30514 = r30466 / r30513;
        double r30515 = -r30506;
        double r30516 = pow(r30459, r30473);
        double r30517 = r30516 - r30506;
        double r30518 = sqrt(r30517);
        double r30519 = r30518 + r30459;
        double r30520 = r30515 / r30519;
        double r30521 = r30520 / r30482;
        double r30522 = r30463 ? r30514 : r30521;
        double r30523 = r30490 ? r30511 : r30522;
        double r30524 = r30461 ? r30484 : r30523;
        return r30524;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -1.3506760573432116e+154

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Taylor expanded around inf 64.0

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 64.0

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

   6. Applied times-frac 64.0

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

   7. Taylor expanded around -inf 11.8

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

   if -1.3506760573432116e+154 < b < -4.452231346955638e-267 or 1.2875942964765074e+78 < b

   1. Initial program 16.9

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

   2. Simplified 16.9

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

   3. Taylor expanded around inf 7.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
   4. Using strategy rm

   5. Applied add-sqr-sqrt 7.5

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

   6. Applied times-frac 5.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \color{blue}{\left(\frac{a}{\sqrt{b}} \cdot \frac{c}{\sqrt{b}}\right)} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 5.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{\sqrt{\color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

   9. Applied sqrt-prod 5.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{\color{blue}{\sqrt{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt{\sqrt[3]{b}}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

   10. Applied add-cube-cbrt 5.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{\color{blue}{\left(\sqrt[3]{c} \cdot \sqrt[3]{c}\right) \cdot \sqrt[3]{c}}}{\sqrt{\sqrt[3]{b} \cdot \sqrt[3]{b}} \cdot \sqrt{\sqrt[3]{b}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

   11. Applied times-frac 5.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \color{blue}{\left(\frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt{\sqrt[3]{b} \cdot \sqrt[3]{b}}} \cdot \frac{\sqrt[3]{c}}{\sqrt{\sqrt[3]{b}}}\right)}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

   12. Applied associate-*r* 5.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \color{blue}{\left(\left(\frac{a}{\sqrt{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt{\sqrt[3]{b} \cdot \sqrt[3]{b}}}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt{\sqrt[3]{b}}}\right)} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

   13. Simplified 5.6

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\color{blue}{\left(\frac{a}{\sqrt{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\left|\sqrt[3]{b}\right|}\right)} \cdot \frac{\sqrt[3]{c}}{\sqrt{\sqrt[3]{b}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

   if -4.452231346955638e-267 < b < 1.2875942964765074e+78

   1. Initial program 9.5

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

   2. Simplified 9.5

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt 9.5

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

   5. Applied sqrt-prod 9.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} - b}{2 \cdot a}\\ \end{array}\]
   6. Using strategy rm

   7. Applied flip-- 9.5

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

   8. Simplified 9.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0 - \left(4 \cdot a\right) \cdot c}{\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} + b}}{2 \cdot a}\\ \end{array}\]

   9. Simplified 9.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0 - \left(4 \cdot a\right) \cdot c}{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \end{array}\]
3. Recombined 3 regimes into one program.

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.350676057343211554491874986591947593393 \cdot 10^{154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\frac{a}{\sqrt{b}} \cdot \frac{c}{\sqrt{b}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le -4.452231346955637703324616244948712027068 \cdot 10^{-267} \lor \neg \left(b \le 1.287594296476507417293820707954348336152 \cdot 10^{78}\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \left(\left(\frac{a}{\sqrt{b}} \cdot \frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\left|\sqrt[3]{b}\right|}\right) \cdot \frac{\sqrt[3]{c}}{\sqrt{\sqrt[3]{b}}}\right) - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-\left(4 \cdot a\right) \cdot c}{\sqrt{{b}^{2} - \left(4 \cdot a\right) \cdot c} + b}}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  :precision binary64
  (if (>= b 0.0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))