Average Error: 34.1 → 10.0
Time: 40.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0240531044330778 \cdot 10^{-88}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.0064301601964595 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2\right)\\ \end{array}\]
\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 -1.0240531044330778 \cdot 10^{-88}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 1.0064301601964595 \cdot 10^{+151}:\\
\;\;\;\;\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{\frac{1}{2}}{a} \cdot \left(\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2\right)\\

\end{array}
double f(double a, double b, double c) {
        double r3326225 = b;
        double r3326226 = -r3326225;
        double r3326227 = r3326225 * r3326225;
        double r3326228 = 4.0;
        double r3326229 = a;
        double r3326230 = c;
        double r3326231 = r3326229 * r3326230;
        double r3326232 = r3326228 * r3326231;
        double r3326233 = r3326227 - r3326232;
        double r3326234 = sqrt(r3326233);
        double r3326235 = r3326226 - r3326234;
        double r3326236 = 2.0;
        double r3326237 = r3326236 * r3326229;
        double r3326238 = r3326235 / r3326237;
        return r3326238;
}
↓
double f(double a, double b, double c) {
        double r3326239 = b;
        double r3326240 = -1.0240531044330778e-88;
        bool r3326241 = r3326239 <= r3326240;
        double r3326242 = c;
        double r3326243 = r3326242 / r3326239;
        double r3326244 = -r3326243;
        double r3326245 = 1.0064301601964595e+151;
        bool r3326246 = r3326239 <= r3326245;
        double r3326247 = -r3326239;
        double r3326248 = a;
        double r3326249 = r3326242 * r3326248;
        double r3326250 = -4.0;
        double r3326251 = r3326239 * r3326239;
        double r3326252 = fma(r3326249, r3326250, r3326251);
        double r3326253 = sqrt(r3326252);
        double r3326254 = r3326247 - r3326253;
        double r3326255 = 0.5;
        double r3326256 = r3326254 * r3326255;
        double r3326257 = r3326256 / r3326248;
        double r3326258 = r3326255 / r3326248;
        double r3326259 = r3326239 / r3326242;
        double r3326260 = r3326248 / r3326259;
        double r3326261 = r3326260 - r3326239;
        double r3326262 = 2.0;
        double r3326263 = r3326261 * r3326262;
        double r3326264 = r3326258 * r3326263;
        double r3326265 = r3326246 ? r3326257 : r3326264;
        double r3326266 = r3326241 ? r3326244 : r3326265;
        return r3326266;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.1
Target21.0
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0240531044330778e-88

    1. Initial program 52.3

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

    3. Applied div-inv52.3

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

    4. Simplified52.3

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

    6. Applied associate-*r/52.3

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

    7. Simplified52.3

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

    8. Taylor expanded around -inf 9.2

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

    9. Simplified9.2

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

    if -1.0240531044330778e-88 < b < 1.0064301601964595e+151

    1. Initial program 12.3

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

    3. Applied div-inv12.4

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

    4. Simplified12.4

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

    6. Applied associate-*r/12.3

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

    7. Simplified12.3

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

    if 1.0064301601964595e+151 < b

    1. Initial program 59.7

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

    3. Applied div-inv59.7

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

    4. Simplified59.7

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

    5. Taylor expanded around inf 12.4

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

    6. Simplified3.0

      \[\leadsto \color{blue}{\left(\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2\right)} \cdot \frac{\frac{1}{2}}{a}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0240531044330778 \cdot 10^{-88}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 1.0064301601964595 \cdot 10^{+151}:\\ \;\;\;\;\frac{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot a, -4, b \cdot b\right)}\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 2\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 +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)))