Average Error: 34.1 → 9.9
Time: 15.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.39564812417811377078958072800119881067 \cdot 10^{-52}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.369507469709798282050760971696230368519 \cdot 10^{103}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4\right) \cdot a, c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -4.39564812417811377078958072800119881067 \cdot 10^{-52}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 2.369507469709798282050760971696230368519 \cdot 10^{103}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4\right) \cdot a, c, b \cdot b\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r58107 = b;
        double r58108 = -r58107;
        double r58109 = r58107 * r58107;
        double r58110 = 4.0;
        double r58111 = a;
        double r58112 = c;
        double r58113 = r58111 * r58112;
        double r58114 = r58110 * r58113;
        double r58115 = r58109 - r58114;
        double r58116 = sqrt(r58115);
        double r58117 = r58108 - r58116;
        double r58118 = 2.0;
        double r58119 = r58118 * r58111;
        double r58120 = r58117 / r58119;
        return r58120;
}
↓
double f(double a, double b, double c) {
        double r58121 = b;
        double r58122 = -4.395648124178114e-52;
        bool r58123 = r58121 <= r58122;
        double r58124 = -1.0;
        double r58125 = c;
        double r58126 = r58125 / r58121;
        double r58127 = r58124 * r58126;
        double r58128 = 2.3695074697097983e+103;
        bool r58129 = r58121 <= r58128;
        double r58130 = -r58121;
        double r58131 = 4.0;
        double r58132 = -r58131;
        double r58133 = a;
        double r58134 = r58132 * r58133;
        double r58135 = r58121 * r58121;
        double r58136 = fma(r58134, r58125, r58135);
        double r58137 = sqrt(r58136);
        double r58138 = r58130 - r58137;
        double r58139 = 2.0;
        double r58140 = r58139 * r58133;
        double r58141 = r58138 / r58140;
        double r58142 = r58121 / r58133;
        double r58143 = r58124 * r58142;
        double r58144 = r58129 ? r58141 : r58143;
        double r58145 = r58123 ? r58127 : r58144;
        return r58145;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.1
Target20.7
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.395648124178114e-52

    1. Initial program 53.9

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

    2. Taylor expanded around -inf 7.7

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

    if -4.395648124178114e-52 < b < 2.3695074697097983e+103

    1. Initial program 13.8

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

    2. Taylor expanded around 0 13.8

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

    3. Simplified13.8

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

    if 2.3695074697097983e+103 < b

    1. Initial program 47.8

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

    2. Taylor expanded around 0 47.8

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

    3. Simplified47.8

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

    5. Applied clear-num47.9

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

    6. Taylor expanded around 0 3.3

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.39564812417811377078958072800119881067 \cdot 10^{-52}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 2.369507469709798282050760971696230368519 \cdot 10^{103}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(-4\right) \cdot a, c, b \cdot b\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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)))