Average Error: 33.6 → 10.4
Time: 19.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -3.136683434005781 \cdot 10^{-32}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2968498 = b;
        double r2968499 = -r2968498;
        double r2968500 = r2968498 * r2968498;
        double r2968501 = 4.0;
        double r2968502 = a;
        double r2968503 = c;
        double r2968504 = r2968502 * r2968503;
        double r2968505 = r2968501 * r2968504;
        double r2968506 = r2968500 - r2968505;
        double r2968507 = sqrt(r2968506);
        double r2968508 = r2968499 - r2968507;
        double r2968509 = 2.0;
        double r2968510 = r2968509 * r2968502;
        double r2968511 = r2968508 / r2968510;
        return r2968511;
}
↓
double f(double a, double b, double c) {
        double r2968512 = b;
        double r2968513 = -3.136683434005781e-32;
        bool r2968514 = r2968512 <= r2968513;
        double r2968515 = c;
        double r2968516 = r2968515 / r2968512;
        double r2968517 = -r2968516;
        double r2968518 = 2.927598127340643e+124;
        bool r2968519 = r2968512 <= r2968518;
        double r2968520 = -r2968512;
        double r2968521 = -4.0;
        double r2968522 = a;
        double r2968523 = r2968522 * r2968515;
        double r2968524 = r2968512 * r2968512;
        double r2968525 = fma(r2968521, r2968523, r2968524);
        double r2968526 = sqrt(r2968525);
        double r2968527 = r2968520 - r2968526;
        double r2968528 = 2.0;
        double r2968529 = r2968528 * r2968522;
        double r2968530 = r2968527 / r2968529;
        double r2968531 = r2968512 / r2968522;
        double r2968532 = r2968516 - r2968531;
        double r2968533 = r2968519 ? r2968530 : r2968532;
        double r2968534 = r2968514 ? r2968517 : r2968533;
        return r2968534;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.6
Target20.8
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.136683434005781e-32

    1. Initial program 53.4

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

    2. Taylor expanded around 0 53.4

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

    3. Simplified53.4

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

    4. Taylor expanded around -inf 7.3

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

    5. Simplified7.3

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

    if -3.136683434005781e-32 < b < 2.927598127340643e+124

    1. Initial program 14.7

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

    2. Taylor expanded around 0 14.7

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

    3. Simplified14.7

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

    if 2.927598127340643e+124 < b

    1. Initial program 50.6

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

    2. Taylor expanded around inf 2.9

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.136683434005781 \cdot 10^{-32}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.927598127340643 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

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