Average Error: 34.2 → 9.8
Time: 1.1m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.446364120144488225689247638815792820209 \cdot 10^{-54}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.060593179890297485125936384251379457639 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -9.446364120144488225689247638815792820209 \cdot 10^{-54}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 6.060593179890297485125936384251379457639 \cdot 10^{143}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\end{array}
double f(double a, double b, double c) {
        double r3297138 = b;
        double r3297139 = -r3297138;
        double r3297140 = r3297138 * r3297138;
        double r3297141 = 4.0;
        double r3297142 = a;
        double r3297143 = c;
        double r3297144 = r3297142 * r3297143;
        double r3297145 = r3297141 * r3297144;
        double r3297146 = r3297140 - r3297145;
        double r3297147 = sqrt(r3297146);
        double r3297148 = r3297139 - r3297147;
        double r3297149 = 2.0;
        double r3297150 = r3297149 * r3297142;
        double r3297151 = r3297148 / r3297150;
        return r3297151;
}
↓
double f(double a, double b, double c) {
        double r3297152 = b;
        double r3297153 = -9.446364120144488e-54;
        bool r3297154 = r3297152 <= r3297153;
        double r3297155 = -1.0;
        double r3297156 = c;
        double r3297157 = r3297156 / r3297152;
        double r3297158 = r3297155 * r3297157;
        double r3297159 = 6.0605931798902975e+143;
        bool r3297160 = r3297152 <= r3297159;
        double r3297161 = -r3297152;
        double r3297162 = r3297152 * r3297152;
        double r3297163 = 4.0;
        double r3297164 = a;
        double r3297165 = r3297163 * r3297164;
        double r3297166 = r3297165 * r3297156;
        double r3297167 = r3297162 - r3297166;
        double r3297168 = sqrt(r3297167);
        double r3297169 = r3297161 - r3297168;
        double r3297170 = 2.0;
        double r3297171 = r3297164 * r3297170;
        double r3297172 = r3297169 / r3297171;
        double r3297173 = r3297152 / r3297164;
        double r3297174 = r3297157 - r3297173;
        double r3297175 = 1.0;
        double r3297176 = r3297174 * r3297175;
        double r3297177 = r3297160 ? r3297172 : r3297176;
        double r3297178 = r3297154 ? r3297158 : r3297177;
        return r3297178;
}

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

Original34.2
Target21.1
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.446364120144488e-54

    1. Initial program 54.2

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

    2. Taylor expanded around -inf 7.7

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

    if -9.446364120144488e-54 < b < 6.0605931798902975e+143

    1. Initial program 13.1

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

    2. Taylor expanded around 0 13.1

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

    3. Simplified13.1

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

    if 6.0605931798902975e+143 < b

    1. Initial program 59.9

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

    2. Taylor expanded around inf 2.1

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

    3. Simplified2.1

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.446364120144488225689247638815792820209 \cdot 10^{-54}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 6.060593179890297485125936384251379457639 \cdot 10^{143}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))