Average Error: 33.8 → 9.3
Time: 18.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -0.03099989563658142946445117615894560003653:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.992812285264992608677553115821113751089 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot 4\right) \cdot a}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}}{2}}{a}\\ \mathbf{elif}\;b \le 63580190853209333432320:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]
\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 -0.03099989563658142946445117615894560003653:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.992812285264992608677553115821113751089 \cdot 10^{-289}:\\
\;\;\;\;\frac{\frac{\frac{\left(c \cdot 4\right) \cdot a}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}}{2}}{a}\\

\mathbf{elif}\;b \le 63580190853209333432320:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{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 r3559117 = b;
        double r3559118 = -r3559117;
        double r3559119 = r3559117 * r3559117;
        double r3559120 = 4.0;
        double r3559121 = a;
        double r3559122 = c;
        double r3559123 = r3559121 * r3559122;
        double r3559124 = r3559120 * r3559123;
        double r3559125 = r3559119 - r3559124;
        double r3559126 = sqrt(r3559125);
        double r3559127 = r3559118 - r3559126;
        double r3559128 = 2.0;
        double r3559129 = r3559128 * r3559121;
        double r3559130 = r3559127 / r3559129;
        return r3559130;
}
↓
double f(double a, double b, double c) {
        double r3559131 = b;
        double r3559132 = -0.03099989563658143;
        bool r3559133 = r3559131 <= r3559132;
        double r3559134 = -1.0;
        double r3559135 = c;
        double r3559136 = r3559135 / r3559131;
        double r3559137 = r3559134 * r3559136;
        double r3559138 = 5.992812285264993e-289;
        bool r3559139 = r3559131 <= r3559138;
        double r3559140 = 4.0;
        double r3559141 = r3559135 * r3559140;
        double r3559142 = a;
        double r3559143 = r3559141 * r3559142;
        double r3559144 = r3559131 * r3559131;
        double r3559145 = r3559144 - r3559143;
        double r3559146 = sqrt(r3559145);
        double r3559147 = r3559146 - r3559131;
        double r3559148 = r3559143 / r3559147;
        double r3559149 = 2.0;
        double r3559150 = r3559148 / r3559149;
        double r3559151 = r3559150 / r3559142;
        double r3559152 = 6.358019085320933e+22;
        bool r3559153 = r3559131 <= r3559152;
        double r3559154 = -r3559131;
        double r3559155 = r3559142 * r3559135;
        double r3559156 = r3559140 * r3559155;
        double r3559157 = r3559144 - r3559156;
        double r3559158 = sqrt(r3559157);
        double r3559159 = r3559154 - r3559158;
        double r3559160 = r3559142 * r3559149;
        double r3559161 = r3559159 / r3559160;
        double r3559162 = r3559131 / r3559142;
        double r3559163 = r3559136 - r3559162;
        double r3559164 = 1.0;
        double r3559165 = r3559163 * r3559164;
        double r3559166 = r3559153 ? r3559161 : r3559165;
        double r3559167 = r3559139 ? r3559151 : r3559166;
        double r3559168 = r3559133 ? r3559137 : r3559167;
        return r3559168;
}

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

Original33.8
Target20.8
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -0.03099989563658143

    1. Initial program 55.6

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

    2. Taylor expanded around -inf 6.4

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

    if -0.03099989563658143 < b < 5.992812285264993e-289

    1. Initial program 24.4

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

    3. Applied flip--24.4

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

    4. Simplified16.6

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

    5. Simplified16.6

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

    7. Applied associate-/r*16.6

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

    8. Simplified16.7

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

    if 5.992812285264993e-289 < b < 6.358019085320933e+22

    1. Initial program 10.0

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

    if 6.358019085320933e+22 < b

    1. Initial program 33.1

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

    2. Taylor expanded around inf 6.1

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

    3. Simplified6.1

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -0.03099989563658142946445117615894560003653:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.992812285264992608677553115821113751089 \cdot 10^{-289}:\\ \;\;\;\;\frac{\frac{\frac{\left(c \cdot 4\right) \cdot a}{\sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a} - b}}{2}}{a}\\ \mathbf{elif}\;b \le 63580190853209333432320:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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)))