Average Error: 34.1 → 9.9
Time: 16.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.39564812417811377078958072800119881067 \cdot 10^{-52}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.369507469709798282050760971696230368519 \cdot 10^{103}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4\right) \cdot a, c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -4.39564812417811377078958072800119881067 \cdot 10^{-52}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.369507469709798282050760971696230368519 \cdot 10^{103}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4\right) \cdot a, c, b \cdot b\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r110696 = b;
        double r110697 = -r110696;
        double r110698 = r110696 * r110696;
        double r110699 = 4.0;
        double r110700 = a;
        double r110701 = c;
        double r110702 = r110700 * r110701;
        double r110703 = r110699 * r110702;
        double r110704 = r110698 - r110703;
        double r110705 = sqrt(r110704);
        double r110706 = r110697 - r110705;
        double r110707 = 2.0;
        double r110708 = r110707 * r110700;
        double r110709 = r110706 / r110708;
        return r110709;
}
↓
double f(double a, double b, double c) {
        double r110710 = b;
        double r110711 = -4.395648124178114e-52;
        bool r110712 = r110710 <= r110711;
        double r110713 = -1.0;
        double r110714 = c;
        double r110715 = r110714 / r110710;
        double r110716 = r110713 * r110715;
        double r110717 = 2.3695074697097983e+103;
        bool r110718 = r110710 <= r110717;
        double r110719 = -r110710;
        double r110720 = 4.0;
        double r110721 = -r110720;
        double r110722 = a;
        double r110723 = r110721 * r110722;
        double r110724 = r110710 * r110710;
        double r110725 = fma(r110723, r110714, r110724);
        double r110726 = sqrt(r110725);
        double r110727 = r110719 - r110726;
        double r110728 = 2.0;
        double r110729 = r110728 * r110722;
        double r110730 = r110727 / r110729;
        double r110731 = r110710 / r110722;
        double r110732 = r110713 * r110731;
        double r110733 = r110718 ? r110730 : r110732;
        double r110734 = r110712 ? r110716 : r110733;
        return r110734;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.1
Target20.7
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 3 regimes

  2. if b < -4.395648124178114e-52

    1. Initial program 53.9

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

    2. Taylor expanded around -inf 7.7

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

    if -4.395648124178114e-52 < b < 2.3695074697097983e+103

    1. Initial program 13.8

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

    2. Taylor expanded around 0 13.8

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

    3. Simplified13.8

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

    if 2.3695074697097983e+103 < b

    1. Initial program 47.8

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

    2. Taylor expanded around 0 47.8

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

    3. Simplified47.8

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

    5. Applied clear-num47.9

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

    6. Taylor expanded around 0 3.3

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.39564812417811377078958072800119881067 \cdot 10^{-52}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.369507469709798282050760971696230368519 \cdot 10^{103}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4\right) \cdot a, c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :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)))