Average Error: 34.2 → 11.4
Time: 17.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.247157674878585888858757389039773391247 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{1 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \le 9.027398388687083073747117877445020640893 \cdot 10^{77}:\\ \;\;\;\;\frac{-1}{a \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(-a \cdot c, 4, b \cdot b\right)} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\frac{b}{a} - \frac{c}{b}\right)\right) \cdot 1\\ \end{array}\]
\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 -1.247157674878585888858757389039773391247 \cdot 10^{-136}:\\
\;\;\;\;\frac{-1}{1 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\

\mathbf{elif}\;b \le 9.027398388687083073747117877445020640893 \cdot 10^{77}:\\
\;\;\;\;\frac{-1}{a \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(-a \cdot c, 4, b \cdot b\right)} + b\right)\\

\mathbf{else}:\\
\;\;\;\;\left(-\left(\frac{b}{a} - \frac{c}{b}\right)\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r92681 = b;
        double r92682 = -r92681;
        double r92683 = r92681 * r92681;
        double r92684 = 4.0;
        double r92685 = a;
        double r92686 = c;
        double r92687 = r92685 * r92686;
        double r92688 = r92684 * r92687;
        double r92689 = r92683 - r92688;
        double r92690 = sqrt(r92689);
        double r92691 = r92682 - r92690;
        double r92692 = 2.0;
        double r92693 = r92692 * r92685;
        double r92694 = r92691 / r92693;
        return r92694;
}
↓
double f(double a, double b, double c) {
        double r92695 = b;
        double r92696 = -1.2471576748785859e-136;
        bool r92697 = r92695 <= r92696;
        double r92698 = -1.0;
        double r92699 = 1.0;
        double r92700 = c;
        double r92701 = r92695 / r92700;
        double r92702 = a;
        double r92703 = r92702 / r92695;
        double r92704 = r92701 - r92703;
        double r92705 = r92699 * r92704;
        double r92706 = r92698 / r92705;
        double r92707 = 9.027398388687083e+77;
        bool r92708 = r92695 <= r92707;
        double r92709 = 2.0;
        double r92710 = r92702 * r92709;
        double r92711 = r92698 / r92710;
        double r92712 = r92702 * r92700;
        double r92713 = -r92712;
        double r92714 = 4.0;
        double r92715 = r92695 * r92695;
        double r92716 = fma(r92713, r92714, r92715);
        double r92717 = sqrt(r92716);
        double r92718 = r92717 + r92695;
        double r92719 = r92711 * r92718;
        double r92720 = r92695 / r92702;
        double r92721 = r92700 / r92695;
        double r92722 = r92720 - r92721;
        double r92723 = -r92722;
        double r92724 = r92723 * r92699;
        double r92725 = r92708 ? r92719 : r92724;
        double r92726 = r92697 ? r92706 : r92725;
        return r92726;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.2
Target21.2
Herbie11.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2471576748785859e-136

    1. Initial program 50.5

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

    2. Simplified50.5

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

    4. Applied clear-num50.5

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

    5. Simplified50.5

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

    6. Taylor expanded around -inf 13.3

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

    7. Simplified13.3

      \[\leadsto -\frac{1}{\color{blue}{\left(\frac{b}{c} - \frac{a}{b}\right) \cdot 1}}\]

    if -1.2471576748785859e-136 < b < 9.027398388687083e+77

    1. Initial program 12.0

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

    2. Simplified12.0

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

    4. Applied div-inv12.2

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

    5. Simplified12.2

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

    if 9.027398388687083e+77 < b

    1. Initial program 42.5

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

    2. Simplified42.5

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

    4. Applied div-inv42.6

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

    5. Simplified42.6

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

    6. Taylor expanded around inf 5.0

      \[\leadsto -\color{blue}{\left(1 \cdot \frac{b}{a} - 1 \cdot \frac{c}{b}\right)}\]

    7. Simplified5.0

      \[\leadsto -\color{blue}{1 \cdot \left(\frac{b}{a} - \frac{c}{b}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.247157674878585888858757389039773391247 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{1 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \le 9.027398388687083073747117877445020640893 \cdot 10^{77}:\\ \;\;\;\;\frac{-1}{a \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(-a \cdot c, 4, b \cdot b\right)} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\left(\frac{b}{a} - \frac{c}{b}\right)\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))