Average Error: 33.1 → 6.9
Time: 22.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6441900461248674 \cdot 10^{+79}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.5207465453202035 \cdot 10^{-263}:\\ \;\;\;\;\frac{\left(4 \cdot c\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{2}\\ \mathbf{elif}\;b \le 3.846543337744466 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{-a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\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 -1.6441900461248674 \cdot 10^{+79}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 1.5207465453202035 \cdot 10^{-263}:\\
\;\;\;\;\frac{\left(4 \cdot c\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{2}\\

\mathbf{elif}\;b \le 3.846543337744466 \cdot 10^{+48}:\\
\;\;\;\;\frac{\frac{b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{-a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3277141 = b;
        double r3277142 = -r3277141;
        double r3277143 = r3277141 * r3277141;
        double r3277144 = 4.0;
        double r3277145 = a;
        double r3277146 = c;
        double r3277147 = r3277145 * r3277146;
        double r3277148 = r3277144 * r3277147;
        double r3277149 = r3277143 - r3277148;
        double r3277150 = sqrt(r3277149);
        double r3277151 = r3277142 - r3277150;
        double r3277152 = 2.0;
        double r3277153 = r3277152 * r3277145;
        double r3277154 = r3277151 / r3277153;
        return r3277154;
}
↓
double f(double a, double b, double c) {
        double r3277155 = b;
        double r3277156 = -1.6441900461248674e+79;
        bool r3277157 = r3277155 <= r3277156;
        double r3277158 = -2.0;
        double r3277159 = c;
        double r3277160 = r3277159 / r3277155;
        double r3277161 = r3277158 * r3277160;
        double r3277162 = 2.0;
        double r3277163 = r3277161 / r3277162;
        double r3277164 = 1.5207465453202035e-263;
        bool r3277165 = r3277155 <= r3277164;
        double r3277166 = 4.0;
        double r3277167 = r3277166 * r3277159;
        double r3277168 = 1.0;
        double r3277169 = -4.0;
        double r3277170 = r3277169 * r3277159;
        double r3277171 = a;
        double r3277172 = r3277155 * r3277155;
        double r3277173 = fma(r3277170, r3277171, r3277172);
        double r3277174 = sqrt(r3277173);
        double r3277175 = r3277174 - r3277155;
        double r3277176 = r3277168 / r3277175;
        double r3277177 = r3277167 * r3277176;
        double r3277178 = r3277177 / r3277162;
        double r3277179 = 3.846543337744466e+48;
        bool r3277180 = r3277155 <= r3277179;
        double r3277181 = r3277171 * r3277159;
        double r3277182 = fma(r3277169, r3277181, r3277172);
        double r3277183 = sqrt(r3277182);
        double r3277184 = r3277155 + r3277183;
        double r3277185 = -r3277171;
        double r3277186 = r3277184 / r3277185;
        double r3277187 = r3277186 / r3277162;
        double r3277188 = r3277155 / r3277171;
        double r3277189 = r3277160 - r3277188;
        double r3277190 = r3277189 * r3277162;
        double r3277191 = r3277190 / r3277162;
        double r3277192 = r3277180 ? r3277187 : r3277191;
        double r3277193 = r3277165 ? r3277178 : r3277192;
        double r3277194 = r3277157 ? r3277163 : r3277193;
        return r3277194;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.1
Target20.3
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6441900461248674e+79

    1. Initial program 57.2

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

    2. Simplified57.2

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

    3. Taylor expanded around -inf 3.4

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]

    if -1.6441900461248674e+79 < b < 1.5207465453202035e-263

    1. Initial program 29.6

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

    2. Simplified29.5

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

    4. Applied frac-2neg29.5

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

    5. Simplified29.6

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

    7. Applied clear-num29.6

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

    9. Applied flip-+29.7

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

    10. Applied associate-/r/29.8

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

    11. Applied add-cube-cbrt29.8

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{-a}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b} \cdot \left(\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b\right)}}{2}\]

    12. Applied times-frac29.8

      \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{-a}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} \cdot \sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b \cdot b}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}}}{2}\]

    13. Simplified15.8

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{-a}} \cdot \frac{\sqrt[3]{1}}{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} - b}}{2}\]

    14. Simplified15.8

      \[\leadsto \frac{\frac{\mathsf{fma}\left(a \cdot c, -4, 0\right)}{-a} \cdot \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}}{2}\]

    15. Taylor expanded around 0 9.2

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

    if 1.5207465453202035e-263 < b < 3.846543337744466e+48

    1. Initial program 9.4

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

    2. Simplified9.4

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

    4. Applied frac-2neg9.4

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

    5. Simplified9.4

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

    if 3.846543337744466e+48 < b

    1. Initial program 35.6

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

    2. Simplified35.6

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

    3. Taylor expanded around inf 5.1

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified5.1

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6441900461248674 \cdot 10^{+79}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.5207465453202035 \cdot 10^{-263}:\\ \;\;\;\;\frac{\left(4 \cdot c\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(-4 \cdot c, a, b \cdot b\right)} - b}}{2}\\ \mathbf{elif}\;b \le 3.846543337744466 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{b + \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{-a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +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)))