Average Error: 33.3 → 10.4
Time: 34.6s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.179137486378021 \cdot 10^{-24}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -4.179137486378021 \cdot 10^{-24}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\
\;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\end{array}
double f(double a, double b, double c) {
        double r2391207 = b;
        double r2391208 = -r2391207;
        double r2391209 = r2391207 * r2391207;
        double r2391210 = 4.0;
        double r2391211 = a;
        double r2391212 = c;
        double r2391213 = r2391211 * r2391212;
        double r2391214 = r2391210 * r2391213;
        double r2391215 = r2391209 - r2391214;
        double r2391216 = sqrt(r2391215);
        double r2391217 = r2391208 - r2391216;
        double r2391218 = 2.0;
        double r2391219 = r2391218 * r2391211;
        double r2391220 = r2391217 / r2391219;
        return r2391220;
}
↓
double f(double a, double b, double c) {
        double r2391221 = b;
        double r2391222 = -4.179137486378021e-24;
        bool r2391223 = r2391221 <= r2391222;
        double r2391224 = c;
        double r2391225 = r2391224 / r2391221;
        double r2391226 = -r2391225;
        double r2391227 = 2.3648644896474148e+52;
        bool r2391228 = r2391221 <= r2391227;
        double r2391229 = -r2391221;
        double r2391230 = a;
        double r2391231 = r2391224 * r2391230;
        double r2391232 = -4.0;
        double r2391233 = r2391221 * r2391221;
        double r2391234 = fma(r2391231, r2391232, r2391233);
        double r2391235 = sqrt(r2391234);
        double r2391236 = r2391229 - r2391235;
        double r2391237 = 0.5;
        double r2391238 = r2391236 * r2391237;
        double r2391239 = r2391238 / r2391230;
        double r2391240 = r2391221 / r2391230;
        double r2391241 = r2391225 - r2391240;
        double r2391242 = r2391228 ? r2391239 : r2391241;
        double r2391243 = r2391223 ? r2391226 : r2391242;
        return r2391243;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.7
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 < -4.179137486378021e-24

    1. Initial program 54.6

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

    3. Applied div-inv54.6

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

    4. Simplified54.6

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

    5. Taylor expanded around -inf 7.1

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

    6. Simplified7.1

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

    if -4.179137486378021e-24 < b < 2.3648644896474148e+52

    1. Initial program 15.1

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

    3. Applied div-inv15.3

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

    4. Simplified15.3

      \[\leadsto \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]
    5. Using strategy rm

    6. Applied associate-*r/15.1

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

    7. Simplified15.1

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

    if 2.3648644896474148e+52 < b

    1. Initial program 36.7

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

    3. Applied div-inv36.9

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

    4. Simplified36.9

      \[\leadsto \left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]
    5. Using strategy rm

    6. Applied associate-*r/36.7

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

    7. Simplified36.7

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

    8. Taylor expanded around inf 5.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.179137486378021 \cdot 10^{-24}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.3648644896474148 \cdot 10^{+52}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +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)))