Average Error: 33.4 → 29.8
Time: 24.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 4.307246706626464 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 4.307246706626464 \cdot 10^{+94}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2004548 = b;
        double r2004549 = -r2004548;
        double r2004550 = r2004548 * r2004548;
        double r2004551 = 4.0;
        double r2004552 = a;
        double r2004553 = c;
        double r2004554 = r2004552 * r2004553;
        double r2004555 = r2004551 * r2004554;
        double r2004556 = r2004550 - r2004555;
        double r2004557 = sqrt(r2004556);
        double r2004558 = r2004549 + r2004557;
        double r2004559 = 2.0;
        double r2004560 = r2004559 * r2004552;
        double r2004561 = r2004558 / r2004560;
        return r2004561;
}
↓
double f(double a, double b, double c) {
        double r2004562 = b;
        double r2004563 = 4.307246706626464e+94;
        bool r2004564 = r2004562 <= r2004563;
        double r2004565 = a;
        double r2004566 = c;
        double r2004567 = r2004565 * r2004566;
        double r2004568 = -4.0;
        double r2004569 = r2004562 * r2004562;
        double r2004570 = fma(r2004567, r2004568, r2004569);
        double r2004571 = sqrt(r2004570);
        double r2004572 = sqrt(r2004571);
        double r2004573 = -r2004562;
        double r2004574 = fma(r2004572, r2004572, r2004573);
        double r2004575 = r2004574 / r2004565;
        double r2004576 = 2.0;
        double r2004577 = r2004575 / r2004576;
        double r2004578 = 0.0;
        double r2004579 = r2004564 ? r2004577 : r2004578;
        return r2004579;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.3
Herbie29.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < 4.307246706626464e+94

    1. Initial program 25.3

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

    2. Simplified25.3

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

    4. Applied add-sqr-sqrt25.3

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

    5. Applied sqrt-prod25.8

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

    6. Applied fma-neg26.2

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

    if 4.307246706626464e+94 < b

    1. Initial program 58.0

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

    2. Simplified58.0

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

    4. Applied div-sub58.8

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

    6. Applied div-inv59.7

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

    7. Applied fma-neg61.8

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

    8. Taylor expanded around 0 40.7

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

  4. Final simplification29.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.307246706626464 \cdot 10^{+94}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, \sqrt{\sqrt{\mathsf{fma}\left(a \cdot c, -4, b \cdot b\right)}}, -b\right)}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +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)))