Average Error: 34.6 → 8.7
Time: 18.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1150955755735961567232:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1150955755735961567232:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\
\;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r89152 = b;
        double r89153 = -r89152;
        double r89154 = r89152 * r89152;
        double r89155 = 4.0;
        double r89156 = a;
        double r89157 = c;
        double r89158 = r89156 * r89157;
        double r89159 = r89155 * r89158;
        double r89160 = r89154 - r89159;
        double r89161 = sqrt(r89160);
        double r89162 = r89153 - r89161;
        double r89163 = 2.0;
        double r89164 = r89163 * r89156;
        double r89165 = r89162 / r89164;
        return r89165;
}
↓
double f(double a, double b, double c) {
        double r89166 = b;
        double r89167 = -1.1509557557359616e+21;
        bool r89168 = r89166 <= r89167;
        double r89169 = -1.0;
        double r89170 = c;
        double r89171 = r89170 / r89166;
        double r89172 = r89169 * r89171;
        double r89173 = -3.1153949179978696e-213;
        bool r89174 = r89166 <= r89173;
        double r89175 = 4.0;
        double r89176 = a;
        double r89177 = r89176 * r89170;
        double r89178 = r89175 * r89177;
        double r89179 = -r89178;
        double r89180 = fma(r89166, r89166, r89179);
        double r89181 = sqrt(r89180);
        double r89182 = r89181 - r89166;
        double r89183 = r89178 / r89182;
        double r89184 = 1.0;
        double r89185 = 2.0;
        double r89186 = r89185 * r89176;
        double r89187 = r89184 / r89186;
        double r89188 = r89183 * r89187;
        double r89189 = 1.974261024048121e+145;
        bool r89190 = r89166 <= r89189;
        double r89191 = -r89166;
        double r89192 = r89166 * r89166;
        double r89193 = r89192 - r89178;
        double r89194 = sqrt(r89193);
        double r89195 = r89191 - r89194;
        double r89196 = r89195 / r89186;
        double r89197 = 1.0;
        double r89198 = r89166 / r89176;
        double r89199 = r89171 - r89198;
        double r89200 = r89197 * r89199;
        double r89201 = r89190 ? r89196 : r89200;
        double r89202 = r89174 ? r89188 : r89201;
        double r89203 = r89168 ? r89172 : r89202;
        return r89203;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.6
Target20.9
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1509557557359616e+21

    1. Initial program 56.3

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

    2. Taylor expanded around -inf 4.5

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

    if -1.1509557557359616e+21 < b < -3.1153949179978696e-213

    1. Initial program 31.5

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

    3. Applied flip--31.5

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

    4. Simplified17.7

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

    5. Simplified17.7

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

    7. Applied div-inv17.8

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

    if -3.1153949179978696e-213 < b < 1.974261024048121e+145

    1. Initial program 9.9

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

    if 1.974261024048121e+145 < b

    1. Initial program 60.1

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

    2. Taylor expanded around inf 2.3

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

    3. Simplified2.3

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1150955755735961567232:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -3.11539491799786956147131222652382589094 \cdot 10^{-213}:\\ \;\;\;\;\frac{4 \cdot \left(a \cdot c\right)}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(a \cdot c\right)\right)} - b} \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.974261024048120880950549217298529943371 \cdot 10^{145}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +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)))