Average Error: 34.1 → 6.5
Time: 5.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7431685240570133 \cdot 10^{102}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.0417939395900796 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.37351117144741807 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \left(\frac{4}{1} \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.7431685240570133 \cdot 10^{102}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.0417939395900796 \cdot 10^{-259}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 9.37351117144741807 \cdot 10^{103}:\\
\;\;\;\;\frac{1}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \left(\frac{4}{1} \cdot c\right)\\

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

\end{array}
double f(double a, double b, double c) {
        double r118066 = b;
        double r118067 = -r118066;
        double r118068 = r118066 * r118066;
        double r118069 = 4.0;
        double r118070 = a;
        double r118071 = r118069 * r118070;
        double r118072 = c;
        double r118073 = r118071 * r118072;
        double r118074 = r118068 - r118073;
        double r118075 = sqrt(r118074);
        double r118076 = r118067 + r118075;
        double r118077 = 2.0;
        double r118078 = r118077 * r118070;
        double r118079 = r118076 / r118078;
        return r118079;
}
↓
double f(double a, double b, double c) {
        double r118080 = b;
        double r118081 = -1.7431685240570133e+102;
        bool r118082 = r118080 <= r118081;
        double r118083 = 1.0;
        double r118084 = c;
        double r118085 = r118084 / r118080;
        double r118086 = a;
        double r118087 = r118080 / r118086;
        double r118088 = r118085 - r118087;
        double r118089 = r118083 * r118088;
        double r118090 = 1.0417939395900796e-259;
        bool r118091 = r118080 <= r118090;
        double r118092 = -r118080;
        double r118093 = r118080 * r118080;
        double r118094 = 4.0;
        double r118095 = r118094 * r118086;
        double r118096 = r118095 * r118084;
        double r118097 = r118093 - r118096;
        double r118098 = sqrt(r118097);
        double r118099 = r118092 + r118098;
        double r118100 = 2.0;
        double r118101 = r118100 * r118086;
        double r118102 = r118099 / r118101;
        double r118103 = 9.373511171447418e+103;
        bool r118104 = r118080 <= r118103;
        double r118105 = 1.0;
        double r118106 = r118092 - r118098;
        double r118107 = r118100 * r118106;
        double r118108 = r118105 / r118107;
        double r118109 = r118094 / r118105;
        double r118110 = r118109 * r118084;
        double r118111 = r118108 * r118110;
        double r118112 = -1.0;
        double r118113 = r118112 * r118085;
        double r118114 = r118104 ? r118111 : r118113;
        double r118115 = r118091 ? r118102 : r118114;
        double r118116 = r118082 ? r118089 : r118115;
        return r118116;
}

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

Original34.1
Target20.5
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.7431685240570133e+102

    1. Initial program 47.5

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

    2. Taylor expanded around -inf 3.1

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

    3. Simplified3.1

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

    if -1.7431685240570133e+102 < b < 1.0417939395900796e-259

    1. Initial program 9.7

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

    if 1.0417939395900796e-259 < b < 9.373511171447418e+103

    1. Initial program 34.9

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

    3. Applied flip-+35.0

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

    4. Simplified17.0

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity17.0

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

    7. Applied *-un-lft-identity17.0

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

    8. Applied times-frac17.0

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

    9. Applied associate-/l*17.3

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

    10. Simplified16.1

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

    12. Applied associate-/l*16.1

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

    13. Simplified8.3

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

    15. Applied associate-*l/8.2

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

    16. Applied associate-/r/7.8

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

    17. Simplified7.8

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

    if 9.373511171447418e+103 < b

    1. Initial program 59.9

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

    2. Taylor expanded around inf 2.3

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7431685240570133 \cdot 10^{102}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.0417939395900796 \cdot 10^{-259}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.37351117144741807 \cdot 10^{103}:\\ \;\;\;\;\frac{1}{2 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)} \cdot \left(\frac{4}{1} \cdot c\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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)))