Average Error: 34.0 → 9.9
Time: 14.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.683005404599734610349454603271447966358 \cdot 10^{80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.982634002611897496887233296015886335875 \cdot 10^{-268}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(4, -a \cdot c, b \cdot b\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2098867031.934578418731689453125:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.683005404599734610349454603271447966358 \cdot 10^{80}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -6.982634002611897496887233296015886335875 \cdot 10^{-268}:\\
\;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(4, -a \cdot c, b \cdot b\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le 2098867031.934578418731689453125:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r61142 = b;
        double r61143 = -r61142;
        double r61144 = r61142 * r61142;
        double r61145 = 4.0;
        double r61146 = a;
        double r61147 = c;
        double r61148 = r61146 * r61147;
        double r61149 = r61145 * r61148;
        double r61150 = r61144 - r61149;
        double r61151 = sqrt(r61150);
        double r61152 = r61143 - r61151;
        double r61153 = 2.0;
        double r61154 = r61153 * r61146;
        double r61155 = r61152 / r61154;
        return r61155;
}
↓
double f(double a, double b, double c) {
        double r61156 = b;
        double r61157 = -1.6830054045997346e+80;
        bool r61158 = r61156 <= r61157;
        double r61159 = -1.0;
        double r61160 = c;
        double r61161 = r61160 / r61156;
        double r61162 = r61159 * r61161;
        double r61163 = -6.982634002611897e-268;
        bool r61164 = r61156 <= r61163;
        double r61165 = 4.0;
        double r61166 = a;
        double r61167 = r61166 * r61160;
        double r61168 = r61165 * r61167;
        double r61169 = -r61167;
        double r61170 = r61156 * r61156;
        double r61171 = fma(r61165, r61169, r61170);
        double r61172 = sqrt(r61171);
        double r61173 = r61172 - r61156;
        double r61174 = r61168 / r61173;
        double r61175 = 2.0;
        double r61176 = r61175 * r61166;
        double r61177 = r61174 / r61176;
        double r61178 = 2098867031.9345784;
        bool r61179 = r61156 <= r61178;
        double r61180 = -r61156;
        double r61181 = r61180 / r61176;
        double r61182 = 2.0;
        double r61183 = pow(r61156, r61182);
        double r61184 = r61183 - r61168;
        double r61185 = sqrt(r61184);
        double r61186 = r61185 / r61176;
        double r61187 = r61181 - r61186;
        double r61188 = 1.0;
        double r61189 = r61156 / r61166;
        double r61190 = r61161 - r61189;
        double r61191 = r61188 * r61190;
        double r61192 = r61179 ? r61187 : r61191;
        double r61193 = r61164 ? r61177 : r61192;
        double r61194 = r61158 ? r61162 : r61193;
        return r61194;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.0
Target20.9
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6830054045997346e+80

    1. Initial program 58.0

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

    2. Taylor expanded around -inf 2.8

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

    if -1.6830054045997346e+80 < b < -6.982634002611897e-268

    1. Initial program 33.7

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

    3. Applied div-inv33.8

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]
    4. Using strategy rm

    5. Applied associate-*r/33.7

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

    6. Simplified33.7

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

    8. Applied flip--33.8

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

    9. Simplified17.2

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\left(a \cdot c\right) \cdot 4\right)}}}{2 \cdot a}\]

    10. Simplified17.2

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\sqrt{\mathsf{fma}\left(4, -a \cdot c, b \cdot b\right)} - b}}}{2 \cdot a}\]

    if -6.982634002611897e-268 < b < 2098867031.9345784

    1. Initial program 11.8

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

    3. Applied div-inv11.9

      \[\leadsto \color{blue}{\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}}\]
    4. Using strategy rm

    5. Applied associate-*r/11.8

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

    6. Simplified11.8

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

    8. Applied div-sub11.8

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

    9. Simplified11.8

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

    if 2098867031.9345784 < b

    1. Initial program 32.3

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

    2. Taylor expanded around inf 7.0

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

    3. Simplified7.0

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.683005404599734610349454603271447966358 \cdot 10^{80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -6.982634002611897496887233296015886335875 \cdot 10^{-268}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(4, -a \cdot c, b \cdot b\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le 2098867031.934578418731689453125:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))