Average Error: 34.3 → 8.0
Time: 17.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9661478263987.111328125:\\ \;\;\;\;\frac{-1 \cdot c}{b}\\ \mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\frac{4 \cdot a}{2} \cdot \frac{c}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}\right)\\ \mathbf{elif}\;b \le 6.033691444141405046034068616119572110107 \cdot 10^{84}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -9661478263987.111328125:\\
\;\;\;\;\frac{-1 \cdot c}{b}\\

\mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\frac{4 \cdot a}{2} \cdot \frac{c}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}\right)\\

\mathbf{elif}\;b \le 6.033691444141405046034068616119572110107 \cdot 10^{84}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\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 r63137 = b;
        double r63138 = -r63137;
        double r63139 = r63137 * r63137;
        double r63140 = 4.0;
        double r63141 = a;
        double r63142 = c;
        double r63143 = r63141 * r63142;
        double r63144 = r63140 * r63143;
        double r63145 = r63139 - r63144;
        double r63146 = sqrt(r63145);
        double r63147 = r63138 - r63146;
        double r63148 = 2.0;
        double r63149 = r63148 * r63141;
        double r63150 = r63147 / r63149;
        return r63150;
}
↓
double f(double a, double b, double c) {
        double r63151 = b;
        double r63152 = -9661478263987.111;
        bool r63153 = r63151 <= r63152;
        double r63154 = -1.0;
        double r63155 = c;
        double r63156 = r63154 * r63155;
        double r63157 = r63156 / r63151;
        double r63158 = 8.958852798091287e-209;
        bool r63159 = r63151 <= r63158;
        double r63160 = 1.0;
        double r63161 = a;
        double r63162 = r63160 / r63161;
        double r63163 = 4.0;
        double r63164 = r63163 * r63161;
        double r63165 = 2.0;
        double r63166 = r63164 / r63165;
        double r63167 = r63151 * r63151;
        double r63168 = r63155 * r63163;
        double r63169 = r63161 * r63168;
        double r63170 = r63167 - r63169;
        double r63171 = sqrt(r63170);
        double r63172 = r63171 - r63151;
        double r63173 = r63155 / r63172;
        double r63174 = r63166 * r63173;
        double r63175 = r63162 * r63174;
        double r63176 = 6.033691444141405e+84;
        bool r63177 = r63151 <= r63176;
        double r63178 = -r63151;
        double r63179 = r63165 * r63161;
        double r63180 = r63178 / r63179;
        double r63181 = r63155 * r63161;
        double r63182 = r63163 * r63181;
        double r63183 = r63167 - r63182;
        double r63184 = sqrt(r63183);
        double r63185 = r63184 / r63179;
        double r63186 = r63180 - r63185;
        double r63187 = 1.0;
        double r63188 = r63155 / r63151;
        double r63189 = r63151 / r63161;
        double r63190 = r63188 - r63189;
        double r63191 = r63187 * r63190;
        double r63192 = r63177 ? r63186 : r63191;
        double r63193 = r63159 ? r63175 : r63192;
        double r63194 = r63153 ? r63157 : r63193;
        return r63194;
}

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

Original34.3
Target21.1
Herbie8.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9661478263987.111

    1. Initial program 56.4

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

    2. Taylor expanded around -inf 5.0

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

    3. Simplified5.0

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

    if -9661478263987.111 < b < 8.958852798091287e-209

    1. Initial program 25.5

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

    3. Applied flip--25.7

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

    4. Simplified17.5

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

    5. Simplified17.5

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

    7. Applied associate-/r*17.5

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

    8. Simplified14.8

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

    10. Applied div-inv15.0

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

    if 8.958852798091287e-209 < b < 6.033691444141405e+84

    1. Initial program 7.0

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

    3. Applied div-sub7.0

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

    4. Simplified7.0

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

    5. Simplified7.0

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

    if 6.033691444141405e+84 < b

    1. Initial program 44.0

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

    2. Taylor expanded around inf 3.5

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

    3. Simplified3.5

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9661478263987.111328125:\\ \;\;\;\;\frac{-1 \cdot c}{b}\\ \mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\frac{4 \cdot a}{2} \cdot \frac{c}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}\right)\\ \mathbf{elif}\;b \le 6.033691444141405046034068616119572110107 \cdot 10^{84}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))