Average Error: 33.2 → 9.9
Time: 18.9s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]
\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 -1.8774910265390396 \cdot 10^{-73}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\
\;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, 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 r1490794 = b;
        double r1490795 = -r1490794;
        double r1490796 = r1490794 * r1490794;
        double r1490797 = 4.0;
        double r1490798 = a;
        double r1490799 = c;
        double r1490800 = r1490798 * r1490799;
        double r1490801 = r1490797 * r1490800;
        double r1490802 = r1490796 - r1490801;
        double r1490803 = sqrt(r1490802);
        double r1490804 = r1490795 - r1490803;
        double r1490805 = 2.0;
        double r1490806 = r1490805 * r1490798;
        double r1490807 = r1490804 / r1490806;
        return r1490807;
}
↓
double f(double a, double b, double c) {
        double r1490808 = b;
        double r1490809 = -1.8774910265390396e-73;
        bool r1490810 = r1490808 <= r1490809;
        double r1490811 = -2.0;
        double r1490812 = c;
        double r1490813 = r1490812 / r1490808;
        double r1490814 = r1490811 * r1490813;
        double r1490815 = 2.0;
        double r1490816 = r1490814 / r1490815;
        double r1490817 = 2.5703497435733685e+102;
        bool r1490818 = r1490808 <= r1490817;
        double r1490819 = 1.0;
        double r1490820 = -r1490808;
        double r1490821 = a;
        double r1490822 = -4.0;
        double r1490823 = r1490821 * r1490822;
        double r1490824 = r1490808 * r1490808;
        double r1490825 = fma(r1490823, r1490812, r1490824);
        double r1490826 = sqrt(r1490825);
        double r1490827 = r1490820 - r1490826;
        double r1490828 = r1490827 / r1490821;
        double r1490829 = r1490819 / r1490828;
        double r1490830 = r1490819 / r1490829;
        double r1490831 = r1490830 / r1490815;
        double r1490832 = r1490808 / r1490821;
        double r1490833 = r1490813 - r1490832;
        double r1490834 = r1490833 * r1490815;
        double r1490835 = r1490834 / r1490815;
        double r1490836 = r1490818 ? r1490831 : r1490835;
        double r1490837 = r1490810 ? r1490816 : r1490836;
        return r1490837;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

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

Derivation

  1. Split input into 3 regimes

  2. if b < -1.8774910265390396e-73

    1. Initial program 52.5

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

    2. Simplified52.5

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

    3. Taylor expanded around -inf 8.6

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

    if -1.8774910265390396e-73 < b < 2.5703497435733685e+102

    1. Initial program 13.1

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

    2. Simplified13.1

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

    4. Applied clear-num13.2

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

    6. Applied clear-num13.2

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

    if 2.5703497435733685e+102 < b

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around inf 3.0

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

    4. Simplified3.0

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8774910265390396 \cdot 10^{-73}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 2.5703497435733685 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)}}{a}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +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)))