Average Error: 33.2 → 7.0
Time: 1.2m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.7966305506212728 \cdot 10^{+65}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -2.436990347475487 \cdot 10^{-257}:\\ \;\;\;\;c \cdot \frac{-2}{b - \sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*}}\\ \mathbf{elif}\;b \le 2.598286182153128 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*} + b\right) \cdot \frac{-1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]
double f(double a, double b, double c) {
        double r6076285 = b;
        double r6076286 = -r6076285;
        double r6076287 = r6076285 * r6076285;
        double r6076288 = 4.0;
        double r6076289 = a;
        double r6076290 = c;
        double r6076291 = r6076289 * r6076290;
        double r6076292 = r6076288 * r6076291;
        double r6076293 = r6076287 - r6076292;
        double r6076294 = sqrt(r6076293);
        double r6076295 = r6076286 - r6076294;
        double r6076296 = 2.0;
        double r6076297 = r6076296 * r6076289;
        double r6076298 = r6076295 / r6076297;
        return r6076298;
}
↓
double f(double a, double b, double c) {
        double r6076299 = b;
        double r6076300 = -1.7966305506212728e+65;
        bool r6076301 = r6076299 <= r6076300;
        double r6076302 = c;
        double r6076303 = r6076302 / r6076299;
        double r6076304 = -r6076303;
        double r6076305 = -2.436990347475487e-257;
        bool r6076306 = r6076299 <= r6076305;
        double r6076307 = -2.0;
        double r6076308 = a;
        double r6076309 = r6076308 * r6076302;
        double r6076310 = -4.0;
        double r6076311 = r6076309 * r6076310;
        double r6076312 = fma(r6076299, r6076299, r6076311);
        double r6076313 = sqrt(r6076312);
        double r6076314 = r6076299 - r6076313;
        double r6076315 = r6076307 / r6076314;
        double r6076316 = r6076302 * r6076315;
        double r6076317 = 2.598286182153128e+84;
        bool r6076318 = r6076299 <= r6076317;
        double r6076319 = r6076302 * r6076310;
        double r6076320 = r6076299 * r6076299;
        double r6076321 = fma(r6076308, r6076319, r6076320);
        double r6076322 = sqrt(r6076321);
        double r6076323 = r6076322 + r6076299;
        double r6076324 = -0.5;
        double r6076325 = r6076323 * r6076324;
        double r6076326 = r6076325 / r6076308;
        double r6076327 = -r6076299;
        double r6076328 = r6076327 / r6076308;
        double r6076329 = r6076318 ? r6076326 : r6076328;
        double r6076330 = r6076306 ? r6076316 : r6076329;
        double r6076331 = r6076301 ? r6076304 : r6076330;
        return r6076331;
}


\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 -1.7966305506212728 \cdot 10^{+65}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -2.436990347475487 \cdot 10^{-257}:\\
\;\;\;\;c \cdot \frac{-2}{b - \sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*}}\\

\mathbf{elif}\;b \le 2.598286182153128 \cdot 10^{+84}:\\
\;\;\;\;\frac{\left(\sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*} + b\right) \cdot \frac{-1}{2}}{a}\\

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

\end{array}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.2
Target20.1
Herbie7.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.7966305506212728e+65

    1. Initial program 57.3

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

    2. Simplified57.3

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

    3. Taylor expanded around -inf 3.8

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

    4. Simplified3.8

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

    if -1.7966305506212728e+65 < b < -2.436990347475487e-257

    1. Initial program 31.7

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

    2. Simplified31.6

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

    4. Applied *-un-lft-identity31.6

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

    5. Applied div-inv31.6

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

    6. Applied times-frac31.7

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

    7. Simplified31.7

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

    8. Simplified31.7

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

    10. Applied flip-+31.8

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

    11. Applied distribute-neg-frac31.8

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

    12. Applied frac-times36.6

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

    13. Simplified21.3

      \[\leadsto \frac{\color{blue}{\left(0 - a \cdot \left(c \cdot -4\right)\right) \cdot \frac{-1}{2}}}{\left(b - \sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*}\right) \cdot a}\]
    14. Using strategy rm

    15. Applied sub0-neg21.3

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

    16. Applied distribute-lft-neg-out21.3

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

    17. Applied distribute-frac-neg21.3

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

    18. Simplified8.3

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

    if -2.436990347475487e-257 < b < 2.598286182153128e+84

    1. Initial program 10.0

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

    2. Simplified10.0

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

    4. Applied *-un-lft-identity10.0

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

    5. Applied div-inv10.0

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

    6. Applied times-frac10.1

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

    7. Simplified10.1

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

    8. Simplified10.1

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

    10. Applied associate-*r/10.0

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

    if 2.598286182153128e+84 < b

    1. Initial program 40.7

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

    2. Simplified40.7

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

    4. Applied *-un-lft-identity40.7

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

    5. Applied div-inv40.7

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

    6. Applied times-frac40.8

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

    7. Simplified40.7

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

    8. Simplified40.7

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

    10. Applied flip-+60.9

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

    11. Applied distribute-neg-frac60.9

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

    12. Applied frac-times61.2

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

    13. Simplified61.4

      \[\leadsto \frac{\color{blue}{\left(0 - a \cdot \left(c \cdot -4\right)\right) \cdot \frac{-1}{2}}}{\left(b - \sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*}\right) \cdot a}\]
    14. Using strategy rm

    15. Applied sub0-neg61.4

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

    16. Applied distribute-lft-neg-out61.4

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

    17. Applied distribute-frac-neg61.4

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

    18. Simplified61.1

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

    19. Taylor expanded around 0 4.5

      \[\leadsto -\color{blue}{\frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification7.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.7966305506212728 \cdot 10^{+65}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -2.436990347475487 \cdot 10^{-257}:\\ \;\;\;\;c \cdot \frac{-2}{b - \sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*}}\\ \mathbf{elif}\;b \le 2.598286182153128 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*} + b\right) \cdot \frac{-1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 +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)))