Average Error: 34.4 → 10.6
Time: 21.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{c}{\frac{b}{a}}, 2, b \cdot -2\right)}{a}}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4\right)\right) \cdot a\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{c}{\frac{b}{a}}, 2, b \cdot -2\right)}{a}}{2}\\

\mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4\right)\right) \cdot a\right)} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r4203183 = b;
        double r4203184 = -r4203183;
        double r4203185 = r4203183 * r4203183;
        double r4203186 = 4.0;
        double r4203187 = a;
        double r4203188 = c;
        double r4203189 = r4203187 * r4203188;
        double r4203190 = r4203186 * r4203189;
        double r4203191 = r4203185 - r4203190;
        double r4203192 = sqrt(r4203191);
        double r4203193 = r4203184 + r4203192;
        double r4203194 = 2.0;
        double r4203195 = r4203194 * r4203187;
        double r4203196 = r4203193 / r4203195;
        return r4203196;
}
↓
double f(double a, double b, double c) {
        double r4203197 = b;
        double r4203198 = -2.221067196710922e+149;
        bool r4203199 = r4203197 <= r4203198;
        double r4203200 = c;
        double r4203201 = a;
        double r4203202 = r4203197 / r4203201;
        double r4203203 = r4203200 / r4203202;
        double r4203204 = 2.0;
        double r4203205 = -2.0;
        double r4203206 = r4203197 * r4203205;
        double r4203207 = fma(r4203203, r4203204, r4203206);
        double r4203208 = r4203207 / r4203201;
        double r4203209 = r4203208 / r4203204;
        double r4203210 = 2.8983489306952693e-35;
        bool r4203211 = r4203197 <= r4203210;
        double r4203212 = 4.0;
        double r4203213 = -r4203212;
        double r4203214 = r4203200 * r4203213;
        double r4203215 = r4203214 * r4203201;
        double r4203216 = fma(r4203197, r4203197, r4203215);
        double r4203217 = sqrt(r4203216);
        double r4203218 = r4203217 - r4203197;
        double r4203219 = r4203218 / r4203201;
        double r4203220 = r4203219 / r4203204;
        double r4203221 = -2.0;
        double r4203222 = r4203200 / r4203197;
        double r4203223 = r4203221 * r4203222;
        double r4203224 = r4203223 / r4203204;
        double r4203225 = r4203211 ? r4203220 : r4203224;
        double r4203226 = r4203199 ? r4203209 : r4203225;
        return r4203226;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Target21.5
Herbie10.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.221067196710922e+149

    1. Initial program 62.3

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

    2. Simplified62.3

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

    4. Applied fma-neg62.3

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

    5. Taylor expanded around -inf 11.0

      \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{a}}{2}\]

    6. Simplified2.9

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

    if -2.221067196710922e+149 < b < 2.8983489306952693e-35

    1. Initial program 14.6

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

    2. Simplified14.6

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

    4. Applied fma-neg14.6

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

    if 2.8983489306952693e-35 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.221067196710922123169723133116561516447 \cdot 10^{149}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\frac{c}{\frac{b}{a}}, 2, b \cdot -2\right)}{a}}{2}\\ \mathbf{elif}\;b \le 2.898348930695269343280527497904161468201 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot \left(-4\right)\right) \cdot a\right)} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))