Average Error: 33.2 → 9.9
Time: 2.9m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.303853124735619 \cdot 10^{+50}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.2295616480632551 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 -3.303853124735619 \cdot 10^{+50}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.2295616480632551 \cdot 10^{-79}:\\
\;\;\;\;\frac{\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r8319398 = b;
        double r8319399 = -r8319398;
        double r8319400 = r8319398 * r8319398;
        double r8319401 = 4.0;
        double r8319402 = a;
        double r8319403 = c;
        double r8319404 = r8319402 * r8319403;
        double r8319405 = r8319401 * r8319404;
        double r8319406 = r8319400 - r8319405;
        double r8319407 = sqrt(r8319406);
        double r8319408 = r8319399 + r8319407;
        double r8319409 = 2.0;
        double r8319410 = r8319409 * r8319402;
        double r8319411 = r8319408 / r8319410;
        return r8319411;
}
↓
double f(double a, double b, double c) {
        double r8319412 = b;
        double r8319413 = -3.303853124735619e+50;
        bool r8319414 = r8319412 <= r8319413;
        double r8319415 = c;
        double r8319416 = r8319415 / r8319412;
        double r8319417 = a;
        double r8319418 = r8319412 / r8319417;
        double r8319419 = r8319416 - r8319418;
        double r8319420 = 1.2295616480632551e-79;
        bool r8319421 = r8319412 <= r8319420;
        double r8319422 = r8319415 * r8319417;
        double r8319423 = -4.0;
        double r8319424 = r8319412 * r8319412;
        double r8319425 = fma(r8319422, r8319423, r8319424);
        double r8319426 = sqrt(r8319425);
        double r8319427 = r8319426 - r8319412;
        double r8319428 = 2.0;
        double r8319429 = r8319427 / r8319428;
        double r8319430 = r8319429 / r8319417;
        double r8319431 = -r8319415;
        double r8319432 = r8319431 / r8319412;
        double r8319433 = r8319421 ? r8319430 : r8319432;
        double r8319434 = r8319414 ? r8319419 : r8319433;
        return r8319434;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.4
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.303853124735619e+50

    1. Initial program 35.2

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

    2. Simplified35.2

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

    3. Taylor expanded around -inf 5.2

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]

    if -3.303853124735619e+50 < b < 1.2295616480632551e-79

    1. Initial program 12.7

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

    2. Simplified12.7

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

    3. Taylor expanded around -inf 12.7

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

    4. Simplified12.7

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

    if 1.2295616480632551e-79 < b

    1. Initial program 52.3

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

    2. Simplified52.3

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

    3. Taylor expanded around inf 9.6

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

    4. Simplified9.6

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.303853124735619 \cdot 10^{+50}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.2295616480632551 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 +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)))