Average Error: 20.0 → 7.7
Time: 28.5s
Precision: 64
\[\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.6218828469248316 \cdot 10^{82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \end{array}\\ \mathbf{elif}\;b \le 2.276690291884487 \cdot 10^{80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}, -b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]
\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 -4.6218828469248316 \cdot 10^{82}:\\
\;\;\;\;\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}{-2 \cdot b}\\

\end{array}\\

\mathbf{elif}\;b \le 2.276690291884487 \cdot 10^{80}:\\
\;\;\;\;\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(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}, -b\right)}\\

\end{array}\\

\mathbf{elif}\;b \ge 0.0:\\
\;\;\;\;\frac{\left(-b\right) - \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}
double f(double a, double b, double c) {
        double r40450 = b;
        double r40451 = 0.0;
        bool r40452 = r40450 >= r40451;
        double r40453 = -r40450;
        double r40454 = r40450 * r40450;
        double r40455 = 4.0;
        double r40456 = a;
        double r40457 = r40455 * r40456;
        double r40458 = c;
        double r40459 = r40457 * r40458;
        double r40460 = r40454 - r40459;
        double r40461 = sqrt(r40460);
        double r40462 = r40453 - r40461;
        double r40463 = 2.0;
        double r40464 = r40463 * r40456;
        double r40465 = r40462 / r40464;
        double r40466 = r40463 * r40458;
        double r40467 = r40453 + r40461;
        double r40468 = r40466 / r40467;
        double r40469 = r40452 ? r40465 : r40468;
        return r40469;
}
↓
double f(double a, double b, double c) {
        double r40470 = b;
        double r40471 = -4.6218828469248316e+82;
        bool r40472 = r40470 <= r40471;
        double r40473 = 0.0;
        bool r40474 = r40470 >= r40473;
        double r40475 = -r40470;
        double r40476 = r40470 * r40470;
        double r40477 = 4.0;
        double r40478 = a;
        double r40479 = r40477 * r40478;
        double r40480 = c;
        double r40481 = r40479 * r40480;
        double r40482 = r40476 - r40481;
        double r40483 = sqrt(r40482);
        double r40484 = r40475 - r40483;
        double r40485 = 2.0;
        double r40486 = r40485 * r40478;
        double r40487 = r40484 / r40486;
        double r40488 = r40485 * r40480;
        double r40489 = -2.0;
        double r40490 = r40489 * r40470;
        double r40491 = r40488 / r40490;
        double r40492 = r40474 ? r40487 : r40491;
        double r40493 = 2.2766902918844874e+80;
        bool r40494 = r40470 <= r40493;
        double r40495 = sqrt(r40483);
        double r40496 = fma(r40495, r40495, r40475);
        double r40497 = r40488 / r40496;
        double r40498 = r40474 ? r40487 : r40497;
        double r40499 = r40478 * r40480;
        double r40500 = r40499 / r40470;
        double r40501 = r40485 * r40500;
        double r40502 = r40470 - r40501;
        double r40503 = r40475 - r40502;
        double r40504 = r40503 / r40486;
        double r40505 = r40483 - r40470;
        double r40506 = r40488 / r40505;
        double r40507 = r40474 ? r40504 : r40506;
        double r40508 = r40494 ? r40498 : r40507;
        double r40509 = r40472 ? r40492 : r40508;
        return r40509;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -4.6218828469248316e+82

    1. Initial program 29.0

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

    2. Simplified29.0

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

    4. Applied add-sqr-sqrt29.0

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

    5. Applied sqrt-prod29.1

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

    6. Applied fma-neg29.0

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

    7. Taylor expanded around -inf 2.7

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

    8. Simplified2.7

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

    if -4.6218828469248316e+82 < b < 2.2766902918844874e+80

    1. Initial program 8.9

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

    2. Simplified8.9

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

    4. Applied add-sqr-sqrt8.9

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

    5. Applied sqrt-prod9.0

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

    6. Applied fma-neg9.0

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

    if 2.2766902918844874e+80 < b

    1. Initial program 44.4

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

    2. Simplified44.4

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

    3. Taylor expanded around inf 11.0

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.6218828469248316 \cdot 10^{82}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{-2 \cdot b}\\ \end{array}\\ \mathbf{elif}\;b \le 2.276690291884487 \cdot 10^{80}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\mathsf{fma}\left(\sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}, \sqrt{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}, -b\right)}\\ \end{array}\\ \mathbf{elif}\;b \ge 0.0:\\ \;\;\;\;\frac{\left(-b\right) - \left(b - 2 \cdot \frac{a \cdot c}{b}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0.0) (/ (- (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)) (/ (* 2.0 c) (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))))))