Average Error: 34.5 → 6.5
Time: 11.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.631041364691662867016708422572785521166 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 1.292564275165585001505551266323067045785 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2}{\frac{4}{\frac{1}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.631041364691662867016708422572785521166 \cdot 10^{-308}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\

\mathbf{elif}\;b \le 1.292564275165585001505551266323067045785 \cdot 10^{140}:\\
\;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2}{\frac{4}{\frac{1}{c}}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r120169 = b;
        double r120170 = -r120169;
        double r120171 = r120169 * r120169;
        double r120172 = 4.0;
        double r120173 = a;
        double r120174 = r120172 * r120173;
        double r120175 = c;
        double r120176 = r120174 * r120175;
        double r120177 = r120171 - r120176;
        double r120178 = sqrt(r120177);
        double r120179 = r120170 + r120178;
        double r120180 = 2.0;
        double r120181 = r120180 * r120173;
        double r120182 = r120179 / r120181;
        return r120182;
}
↓
double f(double a, double b, double c) {
        double r120183 = b;
        double r120184 = -1.8572382657132166e+109;
        bool r120185 = r120183 <= r120184;
        double r120186 = 1.0;
        double r120187 = c;
        double r120188 = r120187 / r120183;
        double r120189 = a;
        double r120190 = r120183 / r120189;
        double r120191 = r120188 - r120190;
        double r120192 = r120186 * r120191;
        double r120193 = 4.631041364691663e-308;
        bool r120194 = r120183 <= r120193;
        double r120195 = 1.0;
        double r120196 = 2.0;
        double r120197 = r120196 * r120189;
        double r120198 = r120183 * r120183;
        double r120199 = 4.0;
        double r120200 = r120199 * r120189;
        double r120201 = r120200 * r120187;
        double r120202 = r120198 - r120201;
        double r120203 = sqrt(r120202);
        double r120204 = r120203 - r120183;
        double r120205 = r120197 / r120204;
        double r120206 = r120195 / r120205;
        double r120207 = 1.292564275165585e+140;
        bool r120208 = r120183 <= r120207;
        double r120209 = -r120183;
        double r120210 = r120209 - r120203;
        double r120211 = r120195 / r120210;
        double r120212 = r120195 / r120187;
        double r120213 = r120199 / r120212;
        double r120214 = r120196 / r120213;
        double r120215 = r120211 / r120214;
        double r120216 = -1.0;
        double r120217 = r120216 * r120188;
        double r120218 = r120208 ? r120215 : r120217;
        double r120219 = r120194 ? r120206 : r120218;
        double r120220 = r120185 ? r120192 : r120219;
        return r120220;
}

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

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8572382657132166e+109

    1. Initial program 50.1

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

    2. Taylor expanded around -inf 3.6

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

    3. Simplified3.6

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

    if -1.8572382657132166e+109 < b < 4.631041364691663e-308

    1. Initial program 9.0

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

    3. Applied clear-num9.2

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

    4. Simplified9.2

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

    if 4.631041364691663e-308 < b < 1.292564275165585e+140

    1. Initial program 34.6

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

    3. Applied flip-+34.7

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

    4. Simplified15.8

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

    6. Applied clear-num16.0

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

    7. Simplified16.0

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

    9. Applied div-inv16.3

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

    10. Applied *-un-lft-identity16.3

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

    11. Applied times-frac16.2

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

    12. Applied associate-/l*15.1

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

    13. Simplified14.7

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

    15. Applied associate-/l*14.7

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

    16. Simplified8.2

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

    if 1.292564275165585e+140 < b

    1. Initial program 62.6

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

    2. Taylor expanded around inf 1.4

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

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.857238265713216596268581045781308602833 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.631041364691662867016708422572785521166 \cdot 10^{-308}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}}\\ \mathbf{elif}\;b \le 1.292564275165585001505551266323067045785 \cdot 10^{140}:\\ \;\;\;\;\frac{\frac{1}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{\frac{2}{\frac{4}{\frac{1}{c}}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(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)))