The quadratic formula (r1)

Average Error: 34.6 → 16.2

Time: 22.9s

Precision: 64

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\begin{array}{l} \mathbf{if}\;b \le 2.49445624012960862396084940365110205816 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 3.224491050532555179035846228386712352959 \cdot 10^{112}:\\ \;\;\;\;\frac{\left(\frac{a}{a} \cdot \frac{4 \cdot c}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}\right) \cdot \frac{\frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot \left(-c\right), 4, 0\right)}{b + b}}{a}}{2}\\ \end{array}\]

C

double f(double a, double b, double c) {
        double r121456 = b;
        double r121457 = -r121456;
        double r121458 = r121456 * r121456;
        double r121459 = 4.0;
        double r121460 = a;
        double r121461 = r121459 * r121460;
        double r121462 = c;
        double r121463 = r121461 * r121462;
        double r121464 = r121458 - r121463;
        double r121465 = sqrt(r121464);
        double r121466 = r121457 + r121465;
        double r121467 = 2.0;
        double r121468 = r121467 * r121460;
        double r121469 = r121466 / r121468;
        return r121469;
}

↓

double f(double a, double b, double c) {
        double r121470 = b;
        double r121471 = 2.4944562401296086e-289;
        bool r121472 = r121470 <= r121471;
        double r121473 = a;
        double r121474 = -r121473;
        double r121475 = 4.0;
        double r121476 = r121474 * r121475;
        double r121477 = c;
        double r121478 = r121470 * r121470;
        double r121479 = fma(r121476, r121477, r121478);
        double r121480 = sqrt(r121479);
        double r121481 = r121480 - r121470;
        double r121482 = r121481 / r121473;
        double r121483 = 2.0;
        double r121484 = r121482 / r121483;
        double r121485 = 3.224491050532555e+112;
        bool r121486 = r121470 <= r121485;
        double r121487 = r121473 / r121473;
        double r121488 = r121475 * r121477;
        double r121489 = -r121477;
        double r121490 = r121473 * r121489;
        double r121491 = fma(r121475, r121490, r121478);
        double r121492 = sqrt(r121491);
        double r121493 = r121492 + r121470;
        double r121494 = cbrt(r121493);
        double r121495 = r121488 / r121494;
        double r121496 = r121487 * r121495;
        double r121497 = -1.0;
        double r121498 = r121497 / r121494;
        double r121499 = r121498 / r121494;
        double r121500 = r121496 * r121499;
        double r121501 = r121500 / r121483;
        double r121502 = 0.0;
        double r121503 = fma(r121490, r121475, r121502);
        double r121504 = r121470 + r121470;
        double r121505 = r121503 / r121504;
        double r121506 = r121505 / r121473;
        double r121507 = r121506 / r121483;
        double r121508 = r121486 ? r121501 : r121507;
        double r121509 = r121472 ? r121484 : r121508;
        return r121509;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original	34.6
Target	21.0
Herbie	16.2

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

Derivation

1. Split input into 3 regimes

2. if b < 2.4944562401296086e-289

   1. Initial program 22.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Simplified 22.5

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b}{a}}{2}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 22.5

      \[\leadsto \frac{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b}{\color{blue}{1 \cdot a}}}{2}\]

   5. Applied *-un-lft-identity 22.5

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b\right)}}{1 \cdot a}}{2}\]

   6. Applied times-frac 22.5

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b}{a}}}{2}\]

   7. Simplified 22.5

      \[\leadsto \frac{\color{blue}{1} \cdot \frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b}{a}}{2}\]

   8. Simplified 22.5

      \[\leadsto \frac{1 \cdot \color{blue}{\frac{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} - b}{a}}}{2}\]

   if 2.4944562401296086e-289 < b < 3.224491050532555e+112

   1. Initial program 33.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Simplified 33.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b}{a}}{2}}\]
   3. Using strategy rm

   4. Applied flip-- 33.8

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} + b}}}{a}}{2}\]

   5. Simplified 15.8

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}}{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} + b}}{a}}{2}\]

   6. Simplified 15.8

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{a}}{2}\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 15.8

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}{\color{blue}{1 \cdot a}}}{2}\]

   9. Applied add-cube-cbrt 16.5

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\color{blue}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}}{1 \cdot a}}{2}\]

   10. Applied *-un-lft-identity 16.5

       \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot \mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}}{\left(\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}\right) \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{1 \cdot a}}{2}\]

   11. Applied times-frac 16.5

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}} \cdot \frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}}{1 \cdot a}}{2}\]

   12. Applied times-frac 15.8

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b} \cdot \sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{1} \cdot \frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{a}}}{2}\]

   13. Simplified 15.8

       \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}} \cdot \frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\sqrt[3]{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{a}}{2}\]

   14. Simplified 9.1

       \[\leadsto \frac{\frac{\frac{1}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}} \cdot \color{blue}{\left(\frac{-a}{a} \cdot \frac{4 \cdot c}{\sqrt[3]{b + \sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)}}}\right)}}{2}\]

   if 3.224491050532555e+112 < b

   1. Initial program 60.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

   2. Simplified 60.8

      \[\leadsto \color{blue}{\frac{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b}{a}}{2}}\]
   3. Using strategy rm

   4. Applied flip-- 60.8

      \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} + b}}}{a}}{2}\]

   5. Simplified 32.5

      \[\leadsto \frac{\frac{\frac{\color{blue}{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}}{\sqrt{\mathsf{fma}\left(-a, 4 \cdot c, b \cdot b\right)} + b}}{a}}{2}\]

   6. Simplified 32.5

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} + b}}}{a}}{2}\]

   7. Taylor expanded around 0 13.1

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(-a\right) \cdot c, 4, 0\right)}{\color{blue}{b} + b}}{a}}{2}\]
3. Recombined 3 regimes into one program.

4. Final simplification 16.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.49445624012960862396084940365110205816 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(-a\right) \cdot 4, c, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{elif}\;b \le 3.224491050532555179035846228386712352959 \cdot 10^{112}:\\ \;\;\;\;\frac{\left(\frac{a}{a} \cdot \frac{4 \cdot c}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}\right) \cdot \frac{\frac{-1}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}}{\sqrt[3]{\sqrt{\mathsf{fma}\left(4, a \cdot \left(-c\right), b \cdot b\right)} + b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\mathsf{fma}\left(a \cdot \left(-c\right), 4, 0\right)}{b + b}}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))