Average Error: 33.4 → 9.9
Time: 25.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3353129 = b;
        double r3353130 = -r3353129;
        double r3353131 = r3353129 * r3353129;
        double r3353132 = 4.0;
        double r3353133 = a;
        double r3353134 = r3353132 * r3353133;
        double r3353135 = c;
        double r3353136 = r3353134 * r3353135;
        double r3353137 = r3353131 - r3353136;
        double r3353138 = sqrt(r3353137);
        double r3353139 = r3353130 + r3353138;
        double r3353140 = 2.0;
        double r3353141 = r3353140 * r3353133;
        double r3353142 = r3353139 / r3353141;
        return r3353142;
}
↓
double f(double a, double b, double c) {
        double r3353143 = b;
        double r3353144 = -1.0027271082217074e+110;
        bool r3353145 = r3353143 <= r3353144;
        double r3353146 = c;
        double r3353147 = r3353146 / r3353143;
        double r3353148 = a;
        double r3353149 = r3353143 / r3353148;
        double r3353150 = r3353147 - r3353149;
        double r3353151 = 2.0;
        double r3353152 = r3353150 * r3353151;
        double r3353153 = r3353152 / r3353151;
        double r3353154 = 2.326372645943808e-74;
        bool r3353155 = r3353143 <= r3353154;
        double r3353156 = -4.0;
        double r3353157 = r3353148 * r3353156;
        double r3353158 = r3353143 * r3353143;
        double r3353159 = fma(r3353146, r3353157, r3353158);
        double r3353160 = sqrt(r3353159);
        double r3353161 = r3353160 - r3353143;
        double r3353162 = r3353161 / r3353148;
        double r3353163 = r3353162 / r3353151;
        double r3353164 = -2.0;
        double r3353165 = r3353164 * r3353147;
        double r3353166 = r3353165 / r3353151;
        double r3353167 = r3353155 ? r3353163 : r3353166;
        double r3353168 = r3353145 ? r3353153 : r3353167;
        return r3353168;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.4
Target20.3
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0027271082217074e+110

    1. Initial program 46.7

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

    2. Simplified46.7

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

    3. Taylor expanded around -inf 3.6

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

    4. Simplified3.6

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

    if -1.0027271082217074e+110 < b < 2.326372645943808e-74

    1. Initial program 12.8

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

    2. Simplified12.8

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

    3. Taylor expanded around 0 12.8

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

    4. Simplified12.8

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

    if 2.326372645943808e-74 < b

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around inf 8.8

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0027271082217074 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.326372645943808 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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

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