jeff quadratic root 1

Average Error: 20.1 → 7.3

Time: 18.2s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r36411 = b;
        double r36412 = 0.0;
        bool r36413 = r36411 >= r36412;
        double r36414 = -r36411;
        double r36415 = r36411 * r36411;
        double r36416 = 4.0;
        double r36417 = a;
        double r36418 = r36416 * r36417;
        double r36419 = c;
        double r36420 = r36418 * r36419;
        double r36421 = r36415 - r36420;
        double r36422 = sqrt(r36421);
        double r36423 = r36414 - r36422;
        double r36424 = 2.0;
        double r36425 = r36424 * r36417;
        double r36426 = r36423 / r36425;
        double r36427 = r36424 * r36419;
        double r36428 = r36414 + r36422;
        double r36429 = r36427 / r36428;
        double r36430 = r36413 ? r36426 : r36429;
        return r36430;
}

↓

double f(double a, double b, double c) {
        double r36431 = b;
        double r36432 = -2.1851923969506957e+101;
        bool r36433 = r36431 <= r36432;
        double r36434 = 0.0;
        bool r36435 = r36431 >= r36434;
        double r36436 = 2.0;
        double r36437 = a;
        double r36438 = c;
        double r36439 = r36431 / r36438;
        double r36440 = r36437 / r36439;
        double r36441 = -2.0;
        double r36442 = r36441 * r36431;
        double r36443 = fma(r36436, r36440, r36442);
        double r36444 = r36436 * r36437;
        double r36445 = r36443 / r36444;
        double r36446 = r36436 * r36438;
        double r36447 = r36446 / r36443;
        double r36448 = r36435 ? r36445 : r36447;
        double r36449 = -3.20057348973956e-310;
        bool r36450 = r36431 <= r36449;
        double r36451 = r36437 * r36438;
        double r36452 = r36451 / r36431;
        double r36453 = fma(r36436, r36452, r36442);
        double r36454 = r36453 / r36444;
        double r36455 = log(r36454);
        double r36456 = exp(r36455);
        double r36457 = -r36431;
        double r36458 = r36431 * r36431;
        double r36459 = 4.0;
        double r36460 = r36459 * r36437;
        double r36461 = r36460 * r36438;
        double r36462 = r36458 - r36461;
        double r36463 = sqrt(r36462);
        double r36464 = r36457 + r36463;
        double r36465 = r36446 / r36464;
        double r36466 = r36435 ? r36456 : r36465;
        double r36467 = 8.75372275402518e+40;
        bool r36468 = r36431 <= r36467;
        double r36469 = r36457 - r36463;
        double r36470 = r36469 / r36444;
        double r36471 = r36446 / r36453;
        double r36472 = r36435 ? r36470 : r36471;
        double r36473 = cbrt(r36438);
        double r36474 = r36473 * r36473;
        double r36475 = sqrt(r36431);
        double r36476 = r36474 / r36475;
        double r36477 = r36475 / r36473;
        double r36478 = r36437 / r36477;
        double r36479 = r36476 * r36478;
        double r36480 = fma(r36436, r36479, r36442);
        double r36481 = r36480 / r36444;
        double r36482 = r36435 ? r36481 : r36465;
        double r36483 = r36468 ? r36472 : r36482;
        double r36484 = r36450 ? r36466 : r36483;
        double r36485 = r36433 ? r36448 : r36484;
        return r36485;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Split input into 4 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. Taylor expanded around inf 30.7

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

   3. Simplified 30.7

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

   5. Applied associate-/l* 30.7

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

   6. Taylor expanded around -inf 6.7

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

   7. Simplified 2.4

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

   if -2.1851923969506957e+101 < b < -3.20057348973956e-310

   1. Initial program 9.3

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

   2. Taylor expanded around inf 9.3

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

   3. Simplified 9.3

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

   5. Applied add-exp-log 9.3

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

   6. Applied add-exp-log 9.3

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

   7. Applied prod-exp 9.3

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

   8. Applied add-exp-log 9.3

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

   9. Applied div-exp 9.3

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

   10. Simplified 9.3

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

   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. Taylor expanded around -inf 9.9

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

   3. Simplified 9.9

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

   if 8.75372275402518e+40 < b

   1. Initial program 36.5

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

   2. Taylor expanded around inf 11.0

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

   3. Simplified 11.0

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

   5. Applied associate-/l* 6.5

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

   7. Applied add-cube-cbrt 6.5

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

   8. Applied add-sqr-sqrt 6.5

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

   9. Applied times-frac 6.5

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

   10. Applied *-un-lft-identity 6.5

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

   11. Applied times-frac 6.5

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

   12. Simplified 6.5

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

4. Final simplification 7.3

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

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)))))))