Average Error: 32.9 → 12.3
Time: 2.3m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.06600713448898 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{-a}}}{a}\\ \mathbf{elif}\;b \le -1.8674816151448643 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}\right)}}{a}\\ \mathbf{elif}\;b \le 5.436017879840864 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]
\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 -7.06600713448898 \cdot 10^{+148}:\\
\;\;\;\;\frac{\frac{c}{\frac{b}{-a}}}{a}\\

\mathbf{elif}\;b \le -1.8674816151448643 \cdot 10^{-121}:\\
\;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}\right)}}{a}\\

\mathbf{elif}\;b \le 5.436017879840864 \cdot 10^{+134}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r7230798 = b;
        double r7230799 = -r7230798;
        double r7230800 = r7230798 * r7230798;
        double r7230801 = 4.0;
        double r7230802 = a;
        double r7230803 = c;
        double r7230804 = r7230802 * r7230803;
        double r7230805 = r7230801 * r7230804;
        double r7230806 = r7230800 - r7230805;
        double r7230807 = sqrt(r7230806);
        double r7230808 = r7230799 - r7230807;
        double r7230809 = 2.0;
        double r7230810 = r7230809 * r7230802;
        double r7230811 = r7230808 / r7230810;
        return r7230811;
}
↓
double f(double a, double b, double c) {
        double r7230812 = b;
        double r7230813 = -7.06600713448898e+148;
        bool r7230814 = r7230812 <= r7230813;
        double r7230815 = c;
        double r7230816 = a;
        double r7230817 = -r7230816;
        double r7230818 = r7230812 / r7230817;
        double r7230819 = r7230815 / r7230818;
        double r7230820 = r7230819 / r7230816;
        double r7230821 = -1.8674816151448643e-121;
        bool r7230822 = r7230812 <= r7230821;
        double r7230823 = 4.0;
        double r7230824 = r7230823 * r7230816;
        double r7230825 = r7230824 * r7230815;
        double r7230826 = 2.0;
        double r7230827 = -r7230812;
        double r7230828 = r7230815 * r7230816;
        double r7230829 = -4.0;
        double r7230830 = r7230812 * r7230812;
        double r7230831 = fma(r7230828, r7230829, r7230830);
        double r7230832 = sqrt(r7230831);
        double r7230833 = r7230827 + r7230832;
        double r7230834 = r7230826 * r7230833;
        double r7230835 = r7230825 / r7230834;
        double r7230836 = r7230835 / r7230816;
        double r7230837 = 5.436017879840864e+134;
        bool r7230838 = r7230812 <= r7230837;
        double r7230839 = r7230827 - r7230832;
        double r7230840 = r7230839 / r7230826;
        double r7230841 = r7230840 / r7230816;
        double r7230842 = r7230812 / r7230816;
        double r7230843 = -r7230842;
        double r7230844 = r7230838 ? r7230841 : r7230843;
        double r7230845 = r7230822 ? r7230836 : r7230844;
        double r7230846 = r7230814 ? r7230820 : r7230845;
        return r7230846;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target20.6
Herbie12.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.06600713448898e+148

    1. Initial program 62.4

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

    2. Simplified62.4

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

    3. Taylor expanded around -inf 62.4

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

    4. Simplified62.4

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

    5. Taylor expanded around -inf 15.6

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

    6. Simplified17.3

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

    if -7.06600713448898e+148 < b < -1.8674816151448643e-121

    1. Initial program 41.6

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

    2. Simplified41.6

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

    4. Applied flip--41.7

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

    5. Applied associate-/l/41.7

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

    6. Simplified15.7

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

    if -1.8674816151448643e-121 < b < 5.436017879840864e+134

    1. Initial program 11.0

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

    2. Simplified11.0

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

    3. Taylor expanded around -inf 11.0

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

    4. Simplified11.0

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

    if 5.436017879840864e+134 < b

    1. Initial program 53.5

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

    2. Simplified53.5

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

    3. Taylor expanded around -inf 53.5

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

    4. Simplified53.5

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

    6. Applied add-sqr-sqrt53.7

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

    7. Applied associate-/r*53.6

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

    8. Taylor expanded around inf 4.8

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

    9. Simplified3.7

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.06600713448898 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{-a}}}{a}\\ \mathbf{elif}\;b \le -1.8674816151448643 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}\right)}}{a}\\ \mathbf{elif}\;b \le 5.436017879840864 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +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)))