Average Error: 33.3 → 10.4
Time: 17.7s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{a}}{2}\\ \end{array}\]
\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 -5.961198324014865 \cdot 10^{-88}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{a}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1409209 = b;
        double r1409210 = -r1409209;
        double r1409211 = r1409209 * r1409209;
        double r1409212 = 4.0;
        double r1409213 = a;
        double r1409214 = c;
        double r1409215 = r1409213 * r1409214;
        double r1409216 = r1409212 * r1409215;
        double r1409217 = r1409211 - r1409216;
        double r1409218 = sqrt(r1409217);
        double r1409219 = r1409210 - r1409218;
        double r1409220 = 2.0;
        double r1409221 = r1409220 * r1409213;
        double r1409222 = r1409219 / r1409221;
        return r1409222;
}
↓
double f(double a, double b, double c) {
        double r1409223 = b;
        double r1409224 = -5.961198324014865e-88;
        bool r1409225 = r1409223 <= r1409224;
        double r1409226 = c;
        double r1409227 = -r1409226;
        double r1409228 = r1409227 / r1409223;
        double r1409229 = 6.384705165981893e+101;
        bool r1409230 = r1409223 <= r1409229;
        double r1409231 = -4.0;
        double r1409232 = a;
        double r1409233 = r1409226 * r1409232;
        double r1409234 = r1409223 * r1409223;
        double r1409235 = fma(r1409231, r1409233, r1409234);
        double r1409236 = sqrt(r1409235);
        double r1409237 = r1409236 + r1409223;
        double r1409238 = -r1409232;
        double r1409239 = r1409237 / r1409238;
        double r1409240 = 2.0;
        double r1409241 = r1409239 / r1409240;
        double r1409242 = r1409223 / r1409226;
        double r1409243 = r1409232 / r1409242;
        double r1409244 = r1409243 - r1409223;
        double r1409245 = r1409240 * r1409244;
        double r1409246 = r1409245 / r1409232;
        double r1409247 = r1409246 / r1409240;
        double r1409248 = r1409230 ? r1409241 : r1409247;
        double r1409249 = r1409225 ? r1409228 : r1409248;
        return r1409249;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.2
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.961198324014865e-88

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified9.8

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

    if -5.961198324014865e-88 < b < 6.384705165981893e+101

    1. Initial program 13.1

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

    2. Simplified13.1

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

    4. Applied frac-2neg13.1

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

    5. Simplified13.2

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

    if 6.384705165981893e+101 < b

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around inf 9.5

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

    4. Simplified3.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(-4, c \cdot a, b \cdot b\right)} + b}{-a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 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)))