Average Error: 33.7 → 10.7
Time: 4.5m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -7.363255598823911 \cdot 10^{-15}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\
\;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}\\

\mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\
\;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r14044119 = b;
        double r14044120 = -r14044119;
        double r14044121 = r14044119 * r14044119;
        double r14044122 = 4.0;
        double r14044123 = a;
        double r14044124 = c;
        double r14044125 = r14044123 * r14044124;
        double r14044126 = r14044122 * r14044125;
        double r14044127 = r14044121 - r14044126;
        double r14044128 = sqrt(r14044127);
        double r14044129 = r14044120 - r14044128;
        double r14044130 = 2.0;
        double r14044131 = r14044130 * r14044123;
        double r14044132 = r14044129 / r14044131;
        return r14044132;
}
↓
double f(double a, double b, double c) {
        double r14044133 = b;
        double r14044134 = -7.363255598823911e-15;
        bool r14044135 = r14044133 <= r14044134;
        double r14044136 = c;
        double r14044137 = r14044136 / r14044133;
        double r14044138 = -r14044137;
        double r14044139 = -6.936587154412951e-28;
        bool r14044140 = r14044133 <= r14044139;
        double r14044141 = -r14044133;
        double r14044142 = 2.0;
        double r14044143 = r14044141 / r14044142;
        double r14044144 = a;
        double r14044145 = r14044143 / r14044144;
        double r14044146 = 1.0;
        double r14044147 = r14044146 / r14044144;
        double r14044148 = -4.0;
        double r14044149 = r14044144 * r14044148;
        double r14044150 = r14044133 * r14044133;
        double r14044151 = fma(r14044136, r14044149, r14044150);
        double r14044152 = sqrt(r14044151);
        double r14044153 = r14044152 / r14044142;
        double r14044154 = r14044147 * r14044153;
        double r14044155 = r14044145 - r14044154;
        double r14044156 = -2.3344326820285623e-123;
        bool r14044157 = r14044133 <= r14044156;
        double r14044158 = 1.6691257204922504e+85;
        bool r14044159 = r14044133 <= r14044158;
        double r14044160 = r14044144 / r14044153;
        double r14044161 = r14044146 / r14044160;
        double r14044162 = r14044145 - r14044161;
        double r14044163 = r14044133 / r14044144;
        double r14044164 = r14044137 - r14044163;
        double r14044165 = r14044159 ? r14044162 : r14044164;
        double r14044166 = r14044157 ? r14044138 : r14044165;
        double r14044167 = r14044140 ? r14044155 : r14044166;
        double r14044168 = r14044135 ? r14044138 : r14044167;
        return r14044168;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.7
Target21.0
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.363255598823911e-15 or -6.936587154412951e-28 < b < -2.3344326820285623e-123

    1. Initial program 50.9

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

    2. Simplified50.9

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

    3. Taylor expanded around -inf 10.6

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

    4. Simplified10.6

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

    if -7.363255598823911e-15 < b < -6.936587154412951e-28

    1. Initial program 35.8

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

    2. Simplified35.8

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

    4. Applied div-sub35.8

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

    5. Applied div-sub35.8

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

    7. Applied div-inv35.9

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

    if -2.3344326820285623e-123 < b < 1.6691257204922504e+85

    1. Initial program 12.6

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

    2. Simplified12.7

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

    4. Applied div-sub12.7

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

    5. Applied div-sub12.7

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

    7. Applied clear-num12.8

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

    if 1.6691257204922504e+85 < b

    1. Initial program 42.9

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

    2. Simplified42.9

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

    3. Taylor expanded around inf 3.7

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.363255598823911 \cdot 10^{-15}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -6.936587154412951 \cdot 10^{-28}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{a} \cdot \frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}\\ \mathbf{elif}\;b \le -2.3344326820285623 \cdot 10^{-123}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.6691257204922504 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{-b}{2}}{a} - \frac{1}{\frac{a}{\frac{\sqrt{\mathsf{fma}\left(c, a \cdot -4, b \cdot b\right)}}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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)))