Average Error: 33.6 → 29.4
Time: 28.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 5.719882815963251 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 5.719882815963251 \cdot 10^{+100}:\\
\;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b\right)\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double a, double b, double c) {
        double r2468495 = b;
        double r2468496 = -r2468495;
        double r2468497 = r2468495 * r2468495;
        double r2468498 = 4.0;
        double r2468499 = a;
        double r2468500 = c;
        double r2468501 = r2468499 * r2468500;
        double r2468502 = r2468498 * r2468501;
        double r2468503 = r2468497 - r2468502;
        double r2468504 = sqrt(r2468503);
        double r2468505 = r2468496 + r2468504;
        double r2468506 = 2.0;
        double r2468507 = r2468506 * r2468499;
        double r2468508 = r2468505 / r2468507;
        return r2468508;
}
↓
double f(double a, double b, double c) {
        double r2468509 = b;
        double r2468510 = 5.719882815963251e+100;
        bool r2468511 = r2468509 <= r2468510;
        double r2468512 = 0.5;
        double r2468513 = a;
        double r2468514 = r2468512 / r2468513;
        double r2468515 = -4.0;
        double r2468516 = c;
        double r2468517 = r2468515 * r2468516;
        double r2468518 = r2468509 * r2468509;
        double r2468519 = fma(r2468513, r2468517, r2468518);
        double r2468520 = sqrt(r2468519);
        double r2468521 = r2468520 - r2468509;
        double r2468522 = r2468514 * r2468521;
        double r2468523 = 0.0;
        double r2468524 = r2468511 ? r2468522 : r2468523;
        return r2468524;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.8
Herbie29.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < 5.719882815963251e+100

    1. Initial program 25.8

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

    2. Simplified25.8

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

    4. Applied *-un-lft-identity25.8

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

    5. Applied div-inv25.8

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

    6. Applied times-frac25.9

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

    7. Simplified25.9

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

    8. Simplified25.9

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

    if 5.719882815963251e+100 < b

    1. Initial program 58.7

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

    2. Simplified58.7

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

    4. Applied clear-num58.7

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

    6. Applied div-inv58.7

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

    7. Applied *-un-lft-identity58.7

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

    8. Applied times-frac58.7

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

    9. Applied associate-/r*58.7

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

    10. Simplified58.7

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

    11. Taylor expanded around 0 40.5

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification29.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 5.719882815963251 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, \left(-4 \cdot c\right), \left(b \cdot b\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 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)))