Average Error: 33.8 → 6.7
Time: 7.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.93152454634661985 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -5.09754230413862019 \cdot 10^{-301}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.77031934398909708 \cdot 10^{52}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -7.93152454634661985 \cdot 10^{153}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -5.09754230413862019 \cdot 10^{-301}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.77031934398909708 \cdot 10^{52}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r213971 = b;
        double r213972 = -r213971;
        double r213973 = r213971 * r213971;
        double r213974 = 4.0;
        double r213975 = a;
        double r213976 = r213974 * r213975;
        double r213977 = c;
        double r213978 = r213976 * r213977;
        double r213979 = r213973 - r213978;
        double r213980 = sqrt(r213979);
        double r213981 = r213972 + r213980;
        double r213982 = 2.0;
        double r213983 = r213982 * r213975;
        double r213984 = r213981 / r213983;
        return r213984;
}
↓
double f(double a, double b, double c) {
        double r213985 = b;
        double r213986 = -7.93152454634662e+153;
        bool r213987 = r213985 <= r213986;
        double r213988 = 1.0;
        double r213989 = c;
        double r213990 = r213989 / r213985;
        double r213991 = a;
        double r213992 = r213985 / r213991;
        double r213993 = r213990 - r213992;
        double r213994 = r213988 * r213993;
        double r213995 = -5.09754230413862e-301;
        bool r213996 = r213985 <= r213995;
        double r213997 = -r213985;
        double r213998 = r213985 * r213985;
        double r213999 = 4.0;
        double r214000 = r213999 * r213991;
        double r214001 = r214000 * r213989;
        double r214002 = r213998 - r214001;
        double r214003 = sqrt(r214002);
        double r214004 = r213997 + r214003;
        double r214005 = 2.0;
        double r214006 = r214005 * r213991;
        double r214007 = r214004 / r214006;
        double r214008 = 1.770319343989097e+52;
        bool r214009 = r213985 <= r214008;
        double r214010 = 1.0;
        double r214011 = r214010 / r214005;
        double r214012 = r213999 * r213989;
        double r214013 = r213997 - r214003;
        double r214014 = r214012 / r214013;
        double r214015 = r214011 * r214014;
        double r214016 = -1.0;
        double r214017 = r214016 * r213990;
        double r214018 = r214009 ? r214015 : r214017;
        double r214019 = r213996 ? r214007 : r214018;
        double r214020 = r213987 ? r213994 : r214019;
        return r214020;
}

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

Original33.8
Target21.2
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.93152454634662e+153

    1. Initial program 63.8

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

    2. Taylor expanded around -inf 1.9

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

    3. Simplified1.9

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

    if -7.93152454634662e+153 < b < -5.09754230413862e-301

    1. Initial program 8.5

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

    if -5.09754230413862e-301 < b < 1.770319343989097e+52

    1. Initial program 29.0

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

    3. Applied flip-+29.0

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

    4. Simplified17.3

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

    6. Applied *-un-lft-identity17.3

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

    7. Applied *-un-lft-identity17.3

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

    8. Applied times-frac17.3

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

    9. Applied times-frac17.3

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

    10. Simplified17.3

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

    11. Simplified17.0

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\frac{\frac{4 \cdot \left(a \cdot c\right) + 0}{a}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\]

    12. Taylor expanded around 0 9.7

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

    if 1.770319343989097e+52 < b

    1. Initial program 57.0

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

    2. Taylor expanded around inf 3.7

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.93152454634661985 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -5.09754230413862019 \cdot 10^{-301}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.77031934398909708 \cdot 10^{52}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))