Average Error: 33.6 → 6.3
Time: 43.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.4105949946140778 \cdot 10^{+148}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.7020775143638545 \cdot 10^{-308}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 4.2298575609145854 \cdot 10^{+96}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
\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 -2.4105949946140778 \cdot 10^{+148}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le -1.7020775143638545 \cdot 10^{-308}:\\
\;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}\\

\mathbf{elif}\;b \le 4.2298575609145854 \cdot 10^{+96}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r4889039 = b;
        double r4889040 = -r4889039;
        double r4889041 = r4889039 * r4889039;
        double r4889042 = 4.0;
        double r4889043 = a;
        double r4889044 = c;
        double r4889045 = r4889043 * r4889044;
        double r4889046 = r4889042 * r4889045;
        double r4889047 = r4889041 - r4889046;
        double r4889048 = sqrt(r4889047);
        double r4889049 = r4889040 - r4889048;
        double r4889050 = 2.0;
        double r4889051 = r4889050 * r4889043;
        double r4889052 = r4889049 / r4889051;
        return r4889052;
}
↓
double f(double a, double b, double c) {
        double r4889053 = b;
        double r4889054 = -2.4105949946140778e+148;
        bool r4889055 = r4889053 <= r4889054;
        double r4889056 = c;
        double r4889057 = -r4889056;
        double r4889058 = r4889057 / r4889053;
        double r4889059 = -1.7020775143638545e-308;
        bool r4889060 = r4889053 <= r4889059;
        double r4889061 = 2.0;
        double r4889062 = r4889061 * r4889056;
        double r4889063 = r4889053 * r4889053;
        double r4889064 = 4.0;
        double r4889065 = a;
        double r4889066 = r4889056 * r4889065;
        double r4889067 = r4889064 * r4889066;
        double r4889068 = r4889063 - r4889067;
        double r4889069 = sqrt(r4889068);
        double r4889070 = -r4889053;
        double r4889071 = r4889069 + r4889070;
        double r4889072 = r4889062 / r4889071;
        double r4889073 = 4.2298575609145854e+96;
        bool r4889074 = r4889053 <= r4889073;
        double r4889075 = r4889070 - r4889069;
        double r4889076 = r4889065 * r4889061;
        double r4889077 = r4889075 / r4889076;
        double r4889078 = r4889070 / r4889065;
        double r4889079 = r4889074 ? r4889077 : r4889078;
        double r4889080 = r4889060 ? r4889072 : r4889079;
        double r4889081 = r4889055 ? r4889058 : r4889080;
        return r4889081;
}

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

Original33.6
Target20.7
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -2.4105949946140778e+148

    1. Initial program 62.4

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

    3. Applied div-inv62.4

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

    4. Taylor expanded around -inf 1.3

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

    5. Simplified1.3

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

    if -2.4105949946140778e+148 < b < -1.7020775143638545e-308

    1. Initial program 35.1

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

    3. Applied div-inv35.2

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

    5. Applied flip--35.3

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

    6. Applied associate-*l/35.3

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

    7. Simplified14.5

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

    8. Taylor expanded around inf 8.1

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

    if -1.7020775143638545e-308 < b < 4.2298575609145854e+96

    1. Initial program 8.3

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

    3. Applied div-inv8.5

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

    5. Applied un-div-inv8.3

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

    if 4.2298575609145854e+96 < b

    1. Initial program 43.5

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

    3. Applied div-inv43.6

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

    5. Applied flip--61.5

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

    6. Applied associate-*l/61.5

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

    7. Simplified61.6

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

    8. Taylor expanded around inf 61.5

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

    9. Taylor expanded around 0 4.1

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

    10. Simplified4.1

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

  4. Final simplification6.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.4105949946140778 \cdot 10^{+148}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.7020775143638545 \cdot 10^{-308}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}\\ \mathbf{elif}\;b \le 4.2298575609145854 \cdot 10^{+96}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))