Average Error: 33.0 → 10.5
Time: 35.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -6.615151909502748 \cdot 10^{-87}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.5387363548079373 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\sqrt{2} \cdot \frac{a}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]
\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 -6.615151909502748 \cdot 10^{-87}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 3.5387363548079373 \cdot 10^{+99}:\\
\;\;\;\;\frac{1}{\sqrt{2} \cdot \frac{a}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2539238 = b;
        double r2539239 = -r2539238;
        double r2539240 = r2539238 * r2539238;
        double r2539241 = 4.0;
        double r2539242 = a;
        double r2539243 = c;
        double r2539244 = r2539242 * r2539243;
        double r2539245 = r2539241 * r2539244;
        double r2539246 = r2539240 - r2539245;
        double r2539247 = sqrt(r2539246);
        double r2539248 = r2539239 - r2539247;
        double r2539249 = 2.0;
        double r2539250 = r2539249 * r2539242;
        double r2539251 = r2539248 / r2539250;
        return r2539251;
}
↓
double f(double a, double b, double c) {
        double r2539252 = b;
        double r2539253 = -6.615151909502748e-87;
        bool r2539254 = r2539252 <= r2539253;
        double r2539255 = c;
        double r2539256 = r2539255 / r2539252;
        double r2539257 = -r2539256;
        double r2539258 = 3.5387363548079373e+99;
        bool r2539259 = r2539252 <= r2539258;
        double r2539260 = 1.0;
        double r2539261 = 2.0;
        double r2539262 = sqrt(r2539261);
        double r2539263 = a;
        double r2539264 = -r2539252;
        double r2539265 = -4.0;
        double r2539266 = r2539263 * r2539265;
        double r2539267 = r2539252 * r2539252;
        double r2539268 = fma(r2539266, r2539255, r2539267);
        double r2539269 = sqrt(r2539268);
        double r2539270 = r2539264 - r2539269;
        double r2539271 = sqrt(r2539262);
        double r2539272 = r2539270 / r2539271;
        double r2539273 = r2539272 / r2539271;
        double r2539274 = r2539263 / r2539273;
        double r2539275 = r2539262 * r2539274;
        double r2539276 = r2539260 / r2539275;
        double r2539277 = r2539252 / r2539263;
        double r2539278 = -r2539277;
        double r2539279 = r2539259 ? r2539276 : r2539278;
        double r2539280 = r2539254 ? r2539257 : r2539279;
        return r2539280;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.0
Target20.1
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -6.615151909502748e-87

    1. Initial program 51.9

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

    2. Simplified52.0

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

    3. Taylor expanded around -inf 10.0

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

    4. Simplified10.0

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

    if -6.615151909502748e-87 < b < 3.5387363548079373e+99

    1. Initial program 12.8

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

    2. Simplified12.8

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

    4. Applied *-un-lft-identity12.8

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

    5. Applied associate-/l*12.9

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

    7. Applied add-sqr-sqrt13.6

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

    8. Applied *-un-lft-identity13.6

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

    9. Applied *-un-lft-identity13.6

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

    10. Applied distribute-rgt-neg-in13.6

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

    11. Applied distribute-lft-out--13.6

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

    12. Applied times-frac13.4

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

    13. Applied *-un-lft-identity13.4

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

    14. Applied times-frac13.3

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

    15. Simplified13.3

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

    17. Applied add-sqr-sqrt13.0

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

    18. Applied associate-/r*13.0

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

    if 3.5387363548079373e+99 < b

    1. Initial program 44.4

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

    2. Simplified44.4

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

    4. Applied *-un-lft-identity44.4

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

    5. Applied associate-/l*44.5

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

    6. Taylor expanded around 0 3.9

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

    7. Simplified3.9

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -6.615151909502748 \cdot 10^{-87}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 3.5387363548079373 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\sqrt{2} \cdot \frac{a}{\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)}}{\sqrt{\sqrt{2}}}}{\sqrt{\sqrt{2}}}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +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)))