Average Error: 34.5 → 6.4
Time: 5.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.889080831912834239838349081155498349678 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.433188992089221261295095985980317261031 \cdot 10^{-271}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{elif}\;b \le 1.668280145404843890899097968396793485605 \cdot 10^{139}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4}}}{2} \cdot c\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.889080831912834239838349081155498349678 \cdot 10^{153}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 5.433188992089221261295095985980317261031 \cdot 10^{-271}:\\
\;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

\mathbf{elif}\;b \le 1.668280145404843890899097968396793485605 \cdot 10^{139}:\\
\;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4}}}{2} \cdot c\\

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

\end{array}
double f(double a, double b, double c) {
        double r99061 = b;
        double r99062 = -r99061;
        double r99063 = r99061 * r99061;
        double r99064 = 4.0;
        double r99065 = a;
        double r99066 = c;
        double r99067 = r99065 * r99066;
        double r99068 = r99064 * r99067;
        double r99069 = r99063 - r99068;
        double r99070 = sqrt(r99069);
        double r99071 = r99062 + r99070;
        double r99072 = 2.0;
        double r99073 = r99072 * r99065;
        double r99074 = r99071 / r99073;
        return r99074;
}
↓
double f(double a, double b, double c) {
        double r99075 = b;
        double r99076 = -8.889080831912834e+153;
        bool r99077 = r99075 <= r99076;
        double r99078 = 1.0;
        double r99079 = c;
        double r99080 = r99079 / r99075;
        double r99081 = a;
        double r99082 = r99075 / r99081;
        double r99083 = r99080 - r99082;
        double r99084 = r99078 * r99083;
        double r99085 = 5.433188992089221e-271;
        bool r99086 = r99075 <= r99085;
        double r99087 = -r99075;
        double r99088 = r99075 * r99075;
        double r99089 = 4.0;
        double r99090 = r99081 * r99079;
        double r99091 = r99089 * r99090;
        double r99092 = r99088 - r99091;
        double r99093 = sqrt(r99092);
        double r99094 = r99087 + r99093;
        double r99095 = 2.0;
        double r99096 = r99094 / r99095;
        double r99097 = r99096 / r99081;
        double r99098 = 1.668280145404844e+139;
        bool r99099 = r99075 <= r99098;
        double r99100 = 1.0;
        double r99101 = r99087 - r99093;
        double r99102 = r99101 / r99089;
        double r99103 = r99100 / r99102;
        double r99104 = r99103 / r99095;
        double r99105 = r99104 * r99079;
        double r99106 = -1.0;
        double r99107 = r99106 * r99080;
        double r99108 = r99099 ? r99105 : r99107;
        double r99109 = r99086 ? r99097 : r99108;
        double r99110 = r99077 ? r99084 : r99109;
        return r99110;
}

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

Original34.5
Target20.8
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.889080831912834e+153

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 2.8

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

    3. Simplified2.8

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

    if -8.889080831912834e+153 < b < 5.433188992089221e-271

    1. Initial program 9.1

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

    3. Applied associate-/r*9.1

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

    if 5.433188992089221e-271 < b < 1.668280145404844e+139

    1. Initial program 35.7

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

    3. Applied flip-+35.7

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

    4. Simplified15.7

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

    6. Applied clear-num15.9

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

    7. Simplified15.9

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

    9. Applied div-inv16.3

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

    10. Applied add-sqr-sqrt16.3

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

    11. Applied times-frac16.1

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

    12. Applied times-frac15.0

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

    13. Simplified15.0

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

    14. Simplified14.5

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

    15. Taylor expanded around 0 7.5

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

    if 1.668280145404844e+139 < b

    1. Initial program 62.5

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

    2. Taylor expanded around inf 1.5

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.889080831912834239838349081155498349678 \cdot 10^{153}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 5.433188992089221261295095985980317261031 \cdot 10^{-271}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{elif}\;b \le 1.668280145404843890899097968396793485605 \cdot 10^{139}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{4}}}{2} \cdot c\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :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)))