Average Error: 34.2 → 10.0
Time: 5.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -4.01157973271056712 \cdot 10^{-81}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r93957 = b;
        double r93958 = -r93957;
        double r93959 = r93957 * r93957;
        double r93960 = 4.0;
        double r93961 = a;
        double r93962 = c;
        double r93963 = r93961 * r93962;
        double r93964 = r93960 * r93963;
        double r93965 = r93959 - r93964;
        double r93966 = sqrt(r93965);
        double r93967 = r93958 - r93966;
        double r93968 = 2.0;
        double r93969 = r93968 * r93961;
        double r93970 = r93967 / r93969;
        return r93970;
}
↓
double f(double a, double b, double c) {
        double r93971 = b;
        double r93972 = -4.011579732710567e-81;
        bool r93973 = r93971 <= r93972;
        double r93974 = -1.0;
        double r93975 = c;
        double r93976 = r93975 / r93971;
        double r93977 = r93974 * r93976;
        double r93978 = 1.3176462918432122e+99;
        bool r93979 = r93971 <= r93978;
        double r93980 = -r93971;
        double r93981 = 2.0;
        double r93982 = a;
        double r93983 = r93981 * r93982;
        double r93984 = r93980 / r93983;
        double r93985 = r93971 * r93971;
        double r93986 = 4.0;
        double r93987 = r93982 * r93975;
        double r93988 = r93986 * r93987;
        double r93989 = r93985 - r93988;
        double r93990 = sqrt(r93989);
        double r93991 = r93990 / r93983;
        double r93992 = r93984 - r93991;
        double r93993 = 1.0;
        double r93994 = r93971 / r93982;
        double r93995 = r93976 - r93994;
        double r93996 = r93993 * r93995;
        double r93997 = r93979 ? r93992 : r93996;
        double r93998 = r93973 ? r93977 : r93997;
        return r93998;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.2
Target21.5
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.011579732710567e-81

    1. Initial program 52.8

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

    2. Taylor expanded around -inf 9.4

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

    if -4.011579732710567e-81 < b < 1.3176462918432122e+99

    1. Initial program 12.9

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

    3. Applied div-sub12.9

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

    if 1.3176462918432122e+99 < b

    1. Initial program 46.8

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.01157973271056712 \cdot 10^{-81}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 1.3176462918432122 \cdot 10^{99}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))