jeff quadratic root 1

Average Error: 20.1 → 7.3

Time: 17.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -2.185192396950695672251895649876633904734 \cdot 10^{101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{a}{\frac{b}{c}}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \le -3.200573489739562245809246032581319999588 \cdot 10^{-310} \lor \neg \left(b \le 87537227540251800037021545535125898395650\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \left(\frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt{b}} \cdot \frac{a}{\frac{\sqrt{b}}{\sqrt[3]{c}}}\right)\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r36343 = b;
        double r36344 = 0.0;
        bool r36345 = r36343 >= r36344;
        double r36346 = -r36343;
        double r36347 = r36343 * r36343;
        double r36348 = 4.0;
        double r36349 = a;
        double r36350 = r36348 * r36349;
        double r36351 = c;
        double r36352 = r36350 * r36351;
        double r36353 = r36347 - r36352;
        double r36354 = sqrt(r36353);
        double r36355 = r36346 - r36354;
        double r36356 = 2.0;
        double r36357 = r36356 * r36349;
        double r36358 = r36355 / r36357;
        double r36359 = r36356 * r36351;
        double r36360 = r36346 + r36354;
        double r36361 = r36359 / r36360;
        double r36362 = r36345 ? r36358 : r36361;
        return r36362;
}

↓

double f(double a, double b, double c) {
        double r36363 = b;
        double r36364 = -2.1851923969506957e+101;
        bool r36365 = r36363 <= r36364;
        double r36366 = 0.0;
        bool r36367 = r36363 >= r36366;
        double r36368 = -r36363;
        double r36369 = 2.0;
        double r36370 = a;
        double r36371 = c;
        double r36372 = r36363 / r36371;
        double r36373 = r36370 / r36372;
        double r36374 = r36369 * r36373;
        double r36375 = r36363 - r36374;
        double r36376 = r36368 - r36375;
        double r36377 = r36369 * r36370;
        double r36378 = r36376 / r36377;
        double r36379 = r36369 * r36371;
        double r36380 = -2.0;
        double r36381 = r36363 * r36380;
        double r36382 = fma(r36369, r36373, r36381);
        double r36383 = r36379 / r36382;
        double r36384 = r36367 ? r36378 : r36383;
        double r36385 = -3.20057348973956e-310;
        bool r36386 = r36363 <= r36385;
        double r36387 = 8.75372275402518e+40;
        bool r36388 = r36363 <= r36387;
        double r36389 = !r36388;
        bool r36390 = r36386 || r36389;
        double r36391 = cbrt(r36371);
        double r36392 = r36391 * r36391;
        double r36393 = sqrt(r36363);
        double r36394 = r36392 / r36393;
        double r36395 = r36393 / r36391;
        double r36396 = r36370 / r36395;
        double r36397 = r36394 * r36396;
        double r36398 = r36369 * r36397;
        double r36399 = r36363 - r36398;
        double r36400 = r36368 - r36399;
        double r36401 = r36400 / r36377;
        double r36402 = r36363 * r36363;
        double r36403 = 4.0;
        double r36404 = r36403 * r36370;
        double r36405 = r36404 * r36371;
        double r36406 = r36402 - r36405;
        double r36407 = sqrt(r36406);
        double r36408 = r36407 - r36363;
        double r36409 = r36379 / r36408;
        double r36410 = r36367 ? r36401 : r36409;
        double r36411 = r36368 - r36407;
        double r36412 = r36411 / r36377;
        double r36413 = r36370 * r36371;
        double r36414 = r36413 / r36363;
        double r36415 = r36380 * r36363;
        double r36416 = fma(r36369, r36414, r36415);
        double r36417 = r36379 / r36416;
        double r36418 = r36367 ? r36412 : r36417;
        double r36419 = r36390 ? r36410 : r36418;
        double r36420 = r36365 ? r36384 : r36419;
        return r36420;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 3 regimes

2. if b < -2.1851923969506957e+101

   1. Initial program 30.7

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

   2. Simplified 30.7

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

   3. Taylor expanded around inf 30.7

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

   5. Applied associate-/l* 30.7

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

   6. Taylor expanded around -inf 6.7

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

   7. Simplified 2.4

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

   if -2.1851923969506957e+101 < b < -3.20057348973956e-310 or 8.75372275402518e+40 < b

   1. Initial program 20.2

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

   2. Simplified 20.2

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

   3. Taylor expanded around inf 10.0

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

   5. Applied associate-/l* 8.1

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

   7. Applied add-cube-cbrt 8.1

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

   8. Applied add-sqr-sqrt 8.1

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

   9. Applied times-frac 8.1

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

   10. Applied *-un-lft-identity 8.1

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{\color{blue}{1 \cdot a}}{\frac{\sqrt{b}}{\sqrt[3]{c} \cdot \sqrt[3]{c}} \cdot \frac{\sqrt{b}}{\sqrt[3]{c}}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   11. Applied times-frac 8.1

       \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \color{blue}{\left(\frac{1}{\frac{\sqrt{b}}{\sqrt[3]{c} \cdot \sqrt[3]{c}}} \cdot \frac{a}{\frac{\sqrt{b}}{\sqrt[3]{c}}}\right)}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

   12. Simplified 8.1

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

   if -3.20057348973956e-310 < b < 8.75372275402518e+40

   1. Initial program 9.9

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

   2. Simplified 9.9

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

   3. Taylor expanded around -inf 9.9

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

   4. Simplified 9.9

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

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.185192396950695672251895649876633904734 \cdot 10^{101}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a}{\frac{b}{c}}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{a}{\frac{b}{c}}, b \cdot -2\right)}\\ \end{array}\\ \mathbf{elif}\;b \le -3.200573489739562245809246032581319999588 \cdot 10^{-310} \lor \neg \left(b \le 87537227540251800037021545535125898395650\right):\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \left(\frac{\sqrt[3]{c} \cdot \sqrt[3]{c}}{\sqrt{b}} \cdot \frac{a}{\frac{\sqrt{b}}{\sqrt[3]{c}}}\right)\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(2, \frac{a \cdot c}{b}, -2 \cdot b\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  :precision binary64
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))