Average Error: 33.1 → 11.0
Time: 36.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.7503548021140933 \cdot 10^{-65}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -5.61762387795767 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{-\mathsf{fma}\left(b, b \cdot b, \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}, \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le -4.884190020998732 \cdot 10^{-159}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 7.377921431051488 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\]
\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 -2.7503548021140933 \cdot 10^{-65}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -5.61762387795767 \cdot 10^{-100}:\\
\;\;\;\;\frac{\frac{-\mathsf{fma}\left(b, b \cdot b, \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}, \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b, b \cdot b\right)}}{2 \cdot a}\\

\mathbf{elif}\;b \le -4.884190020998732 \cdot 10^{-159}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 7.377921431051488 \cdot 10^{+75}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\

\end{array}
double f(double a, double b, double c) {
        double r2316141 = b;
        double r2316142 = -r2316141;
        double r2316143 = r2316141 * r2316141;
        double r2316144 = 4.0;
        double r2316145 = a;
        double r2316146 = c;
        double r2316147 = r2316145 * r2316146;
        double r2316148 = r2316144 * r2316147;
        double r2316149 = r2316143 - r2316148;
        double r2316150 = sqrt(r2316149);
        double r2316151 = r2316142 - r2316150;
        double r2316152 = 2.0;
        double r2316153 = r2316152 * r2316145;
        double r2316154 = r2316151 / r2316153;
        return r2316154;
}
↓
double f(double a, double b, double c) {
        double r2316155 = b;
        double r2316156 = -2.7503548021140933e-65;
        bool r2316157 = r2316155 <= r2316156;
        double r2316158 = c;
        double r2316159 = r2316158 / r2316155;
        double r2316160 = -r2316159;
        double r2316161 = -5.61762387795767e-100;
        bool r2316162 = r2316155 <= r2316161;
        double r2316163 = r2316155 * r2316155;
        double r2316164 = a;
        double r2316165 = -4.0;
        double r2316166 = r2316164 * r2316165;
        double r2316167 = fma(r2316166, r2316158, r2316163);
        double r2316168 = sqrt(r2316167);
        double r2316169 = r2316168 * r2316167;
        double r2316170 = fma(r2316155, r2316163, r2316169);
        double r2316171 = -r2316170;
        double r2316172 = r2316168 - r2316155;
        double r2316173 = fma(r2316168, r2316172, r2316163);
        double r2316174 = r2316171 / r2316173;
        double r2316175 = 2.0;
        double r2316176 = r2316175 * r2316164;
        double r2316177 = r2316174 / r2316176;
        double r2316178 = -4.884190020998732e-159;
        bool r2316179 = r2316155 <= r2316178;
        double r2316180 = 7.377921431051488e+75;
        bool r2316181 = r2316155 <= r2316180;
        double r2316182 = -r2316155;
        double r2316183 = r2316182 - r2316168;
        double r2316184 = r2316183 / r2316176;
        double r2316185 = r2316182 - r2316155;
        double r2316186 = r2316185 / r2316176;
        double r2316187 = r2316181 ? r2316184 : r2316186;
        double r2316188 = r2316179 ? r2316160 : r2316187;
        double r2316189 = r2316162 ? r2316177 : r2316188;
        double r2316190 = r2316157 ? r2316160 : r2316189;
        return r2316190;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.1
Target20.8
Herbie11.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.7503548021140933e-65 or -5.61762387795767e-100 < b < -4.884190020998732e-159

    1. Initial program 49.6

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

    2. Taylor expanded around inf 49.6

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

    3. Simplified49.7

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

    4. Taylor expanded around -inf 12.1

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

    5. Simplified12.1

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

    if -2.7503548021140933e-65 < b < -5.61762387795767e-100

    1. Initial program 26.8

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

    3. Applied flip3--34.0

      \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{3} - {\left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}^{3}}{\left(-b\right) \cdot \left(-b\right) + \left(\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} + \left(-b\right) \cdot \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}}}{2 \cdot a}\]

    4. Simplified34.1

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

    5. Simplified34.1

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

    if -4.884190020998732e-159 < b < 7.377921431051488e+75

    1. Initial program 11.2

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

    2. Taylor expanded around inf 11.2

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

    3. Simplified11.2

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

    if 7.377921431051488e+75 < b

    1. Initial program 39.3

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

    2. Taylor expanded around inf 39.3

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

    3. Simplified39.3

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

    4. Taylor expanded around 0 4.5

      \[\leadsto \frac{\left(-b\right) - \color{blue}{b}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.7503548021140933 \cdot 10^{-65}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -5.61762387795767 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{-\mathsf{fma}\left(b, b \cdot b, \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} \cdot \mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)\right)}{\mathsf{fma}\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}, \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{elif}\;b \le -4.884190020998732 \cdot 10^{-159}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 7.377921431051488 \cdot 10^{+75}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))