Average Error: 34.5 → 6.9
Time: 8.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.7757959561449348 \cdot 10^{129}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.70088188146619881 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5}{c}}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\ \mathbf{elif}\;b \le 3.2649111998892948 \cdot 10^{111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -2.7757959561449348 \cdot 10^{129}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.70088188146619881 \cdot 10^{-222}:\\
\;\;\;\;\frac{1}{\frac{\frac{0.5}{c}}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\

\mathbf{elif}\;b \le 3.2649111998892948 \cdot 10^{111}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r71185 = b;
        double r71186 = -r71185;
        double r71187 = r71185 * r71185;
        double r71188 = 4.0;
        double r71189 = a;
        double r71190 = c;
        double r71191 = r71189 * r71190;
        double r71192 = r71188 * r71191;
        double r71193 = r71187 - r71192;
        double r71194 = sqrt(r71193);
        double r71195 = r71186 - r71194;
        double r71196 = 2.0;
        double r71197 = r71196 * r71189;
        double r71198 = r71195 / r71197;
        return r71198;
}
↓
double f(double a, double b, double c) {
        double r71199 = b;
        double r71200 = -2.775795956144935e+129;
        bool r71201 = r71199 <= r71200;
        double r71202 = -1.0;
        double r71203 = c;
        double r71204 = r71203 / r71199;
        double r71205 = r71202 * r71204;
        double r71206 = 9.700881881466199e-222;
        bool r71207 = r71199 <= r71206;
        double r71208 = 1.0;
        double r71209 = 0.5;
        double r71210 = r71209 / r71203;
        double r71211 = r71199 * r71199;
        double r71212 = 4.0;
        double r71213 = a;
        double r71214 = r71213 * r71203;
        double r71215 = r71212 * r71214;
        double r71216 = r71211 - r71215;
        double r71217 = sqrt(r71216);
        double r71218 = r71217 - r71199;
        double r71219 = r71208 / r71218;
        double r71220 = r71210 / r71219;
        double r71221 = r71208 / r71220;
        double r71222 = 3.264911199889295e+111;
        bool r71223 = r71199 <= r71222;
        double r71224 = -r71199;
        double r71225 = r71224 - r71217;
        double r71226 = 2.0;
        double r71227 = r71226 * r71213;
        double r71228 = r71225 / r71227;
        double r71229 = r71199 / r71213;
        double r71230 = r71202 * r71229;
        double r71231 = r71223 ? r71228 : r71230;
        double r71232 = r71207 ? r71221 : r71231;
        double r71233 = r71201 ? r71205 : r71232;
        return r71233;
}

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
Target20.6
Herbie6.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.775795956144935e+129

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 2.2

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

    if -2.775795956144935e+129 < b < 9.700881881466199e-222

    1. Initial program 31.8

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

    3. Applied flip--31.9

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

    4. Simplified15.9

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

    5. Simplified15.9

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

    7. Applied clear-num16.1

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

    8. Simplified16.1

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\frac{4 \cdot \left(a \cdot c\right) - 0}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}}\]
    9. Using strategy rm

    10. Applied div-inv16.2

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

    11. Applied associate-/r*15.3

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

    12. Simplified15.3

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

    13. Taylor expanded around 0 9.6

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{0.5}{c}}}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]

    if 9.700881881466199e-222 < b < 3.264911199889295e+111

    1. Initial program 8.1

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

    if 3.264911199889295e+111 < b

    1. Initial program 49.7

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

    3. Applied flip--63.3

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

    4. Simplified62.3

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

    5. Simplified62.3

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

    6. Taylor expanded around 0 3.6

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.7757959561449348 \cdot 10^{129}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.70088188146619881 \cdot 10^{-222}:\\ \;\;\;\;\frac{1}{\frac{\frac{0.5}{c}}{\frac{1}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\\ \mathbf{elif}\;b \le 3.2649111998892948 \cdot 10^{111}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

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

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