jeff quadratic root 2

Average Error: 19.8 → 7.5

Time: 19.4s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r33430 = b;
        double r33431 = 0.0;
        bool r33432 = r33430 >= r33431;
        double r33433 = 2.0;
        double r33434 = c;
        double r33435 = r33433 * r33434;
        double r33436 = -r33430;
        double r33437 = r33430 * r33430;
        double r33438 = 4.0;
        double r33439 = a;
        double r33440 = r33438 * r33439;
        double r33441 = r33440 * r33434;
        double r33442 = r33437 - r33441;
        double r33443 = sqrt(r33442);
        double r33444 = r33436 - r33443;
        double r33445 = r33435 / r33444;
        double r33446 = r33436 + r33443;
        double r33447 = r33433 * r33439;
        double r33448 = r33446 / r33447;
        double r33449 = r33432 ? r33445 : r33448;
        return r33449;
}

↓

double f(double a, double b, double c) {
        double r33450 = b;
        double r33451 = -1.3506760573432116e+154;
        bool r33452 = r33450 <= r33451;
        double r33453 = 0.0;
        bool r33454 = r33450 >= r33453;
        double r33455 = 2.0;
        double r33456 = c;
        double r33457 = r33455 * r33456;
        double r33458 = -r33450;
        double r33459 = a;
        double r33460 = sqrt(r33450);
        double r33461 = r33459 / r33460;
        double r33462 = r33456 / r33460;
        double r33463 = r33461 * r33462;
        double r33464 = r33455 * r33463;
        double r33465 = r33450 - r33464;
        double r33466 = r33458 - r33465;
        double r33467 = r33457 / r33466;
        double r33468 = r33459 * r33456;
        double r33469 = r33468 / r33450;
        double r33470 = r33469 * r33455;
        double r33471 = r33470 - r33450;
        double r33472 = r33471 - r33450;
        double r33473 = r33455 * r33459;
        double r33474 = r33472 / r33473;
        double r33475 = r33454 ? r33467 : r33474;
        double r33476 = -4.452231346955638e-267;
        bool r33477 = r33450 <= r33476;
        double r33478 = r33458 - r33450;
        double r33479 = r33455 * r33469;
        double r33480 = r33478 + r33479;
        double r33481 = cbrt(r33480);
        double r33482 = r33481 * r33481;
        double r33483 = r33457 / r33482;
        double r33484 = cbrt(r33466);
        double r33485 = r33483 / r33484;
        double r33486 = r33450 * r33450;
        double r33487 = 4.0;
        double r33488 = r33487 * r33459;
        double r33489 = r33488 * r33456;
        double r33490 = r33486 - r33489;
        double r33491 = sqrt(r33490);
        double r33492 = r33491 - r33450;
        double r33493 = r33492 / r33473;
        double r33494 = r33454 ? r33485 : r33493;
        double r33495 = 1.2875942964765074e+78;
        bool r33496 = r33450 <= r33495;
        double r33497 = r33458 - r33491;
        double r33498 = r33457 / r33497;
        double r33499 = -r33489;
        double r33500 = 2.0;
        double r33501 = pow(r33450, r33500);
        double r33502 = r33501 - r33489;
        double r33503 = sqrt(r33502);
        double r33504 = r33503 + r33450;
        double r33505 = r33499 / r33504;
        double r33506 = r33505 / r33473;
        double r33507 = r33454 ? r33498 : r33506;
        double r33508 = r33496 ? r33507 : r33475;
        double r33509 = r33477 ? r33494 : r33508;
        double r33510 = r33452 ? r33475 : r33509;
        return r33510;
}

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 or 1.2875942964765074e+78 < b

   1. Initial program 38.5

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

   2. Simplified 38.5

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

   3. Taylor expanded around inf 24.0

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

   5. Applied add-sqr-sqrt 24.0

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

   6. Applied times-frac 21.2

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

   7. Taylor expanded around -inf 5.7

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

   8. Simplified 5.7

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

   if -1.3506760573432116e+154 < b < -4.452231346955638e-267

   1. Initial program 7.7

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

   2. Simplified 7.7

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

   3. Taylor expanded around inf 7.7

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

   5. Applied add-sqr-sqrt 7.7

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

   6. Applied times-frac 7.7

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

   8. Applied add-cube-cbrt 7.7

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

   9. Applied associate-/r* 7.7

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

   10. Simplified 7.7

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

   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

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

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))))