Average Error: 33.5 → 11.6
Time: 10.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.478406535480561669630649836752919254545 \cdot 10^{60}:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.175674080408202904737521150531781836244 \cdot 10^{-169} \lor \neg \left(b \le 4.731827132847258030366645007940380497628 \cdot 10^{-110}\right) \land b \le 1.846537578115044209159378851836663670838 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.478406535480561669630649836752919254545 \cdot 10^{60}:\\
\;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.175674080408202904737521150531781836244 \cdot 10^{-169} \lor \neg \left(b \le 4.731827132847258030366645007940380497628 \cdot 10^{-110}\right) \land b \le 1.846537578115044209159378851836663670838 \cdot 10^{-4}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r79514 = b;
        double r79515 = -r79514;
        double r79516 = r79514 * r79514;
        double r79517 = 4.0;
        double r79518 = a;
        double r79519 = c;
        double r79520 = r79518 * r79519;
        double r79521 = r79517 * r79520;
        double r79522 = r79516 - r79521;
        double r79523 = sqrt(r79522);
        double r79524 = r79515 + r79523;
        double r79525 = 2.0;
        double r79526 = r79525 * r79518;
        double r79527 = r79524 / r79526;
        return r79527;
}
↓
double f(double a, double b, double c) {
        double r79528 = b;
        double r79529 = -1.4784065354805617e+60;
        bool r79530 = r79528 <= r79529;
        double r79531 = 1.0;
        double r79532 = 2.0;
        double r79533 = r79531 / r79532;
        double r79534 = c;
        double r79535 = r79534 / r79528;
        double r79536 = -2.0;
        double r79537 = a;
        double r79538 = r79528 / r79537;
        double r79539 = r79536 * r79538;
        double r79540 = fma(r79532, r79535, r79539);
        double r79541 = r79533 * r79540;
        double r79542 = 2.175674080408203e-169;
        bool r79543 = r79528 <= r79542;
        double r79544 = 4.731827132847258e-110;
        bool r79545 = r79528 <= r79544;
        double r79546 = !r79545;
        double r79547 = 0.00018465375781150442;
        bool r79548 = r79528 <= r79547;
        bool r79549 = r79546 && r79548;
        bool r79550 = r79543 || r79549;
        double r79551 = 2.0;
        double r79552 = pow(r79528, r79551);
        double r79553 = 4.0;
        double r79554 = r79537 * r79534;
        double r79555 = r79553 * r79554;
        double r79556 = r79552 - r79555;
        double r79557 = sqrt(r79556);
        double r79558 = r79557 - r79528;
        double r79559 = r79558 / r79537;
        double r79560 = r79533 * r79559;
        double r79561 = -1.0;
        double r79562 = r79561 * r79535;
        double r79563 = r79550 ? r79560 : r79562;
        double r79564 = r79530 ? r79541 : r79563;
        return r79564;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.5
Target20.5
Herbie11.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.4784065354805617e+60

    1. Initial program 38.9

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

    2. Simplified38.9

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

    4. Applied clear-num39.0

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity39.0

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

    7. Applied times-frac39.0

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

    8. Applied add-cube-cbrt39.0

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

    9. Applied times-frac39.1

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

    10. Simplified39.1

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

    11. Simplified39.0

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

    12. Taylor expanded around -inf 5.5

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

    13. Simplified5.5

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

    if -1.4784065354805617e+60 < b < 2.175674080408203e-169 or 4.731827132847258e-110 < b < 0.00018465375781150442

    1. Initial program 15.2

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

    2. Simplified15.2

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

    4. Applied clear-num15.3

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity15.3

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

    7. Applied times-frac15.3

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

    8. Applied add-cube-cbrt15.3

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

    9. Applied times-frac15.3

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

    10. Simplified15.3

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

    11. Simplified15.2

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

    if 2.175674080408203e-169 < b < 4.731827132847258e-110 or 0.00018465375781150442 < b

    1. Initial program 51.7

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

    2. Simplified51.7

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

    3. Taylor expanded around inf 10.5

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.478406535480561669630649836752919254545 \cdot 10^{60}:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(2, \frac{c}{b}, -2 \cdot \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.175674080408202904737521150531781836244 \cdot 10^{-169} \lor \neg \left(b \le 4.731827132847258030366645007940380497628 \cdot 10^{-110}\right) \land b \le 1.846537578115044209159378851836663670838 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))