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 r24669960 = b;
        double r24669961 = -r24669960;
        double r24669962 = r24669960 * r24669960;
        double r24669963 = 4.0;
        double r24669964 = a;
        double r24669965 = c;
        double r24669966 = r24669964 * r24669965;
        double r24669967 = r24669963 * r24669966;
        double r24669968 = r24669962 - r24669967;
        double r24669969 = sqrt(r24669968);
        double r24669970 = r24669961 - r24669969;
        double r24669971 = 2.0;
        double r24669972 = r24669971 * r24669964;
        double r24669973 = r24669970 / r24669972;
        return r24669973;
}
↓
double f(double a, double b, double c) {
        double r24669974 = b;
        double r24669975 = -1.7966305506212728e+65;
        bool r24669976 = r24669974 <= r24669975;
        double r24669977 = c;
        double r24669978 = r24669977 / r24669974;
        double r24669979 = -r24669978;
        double r24669980 = -2.436990347475487e-257;
        bool r24669981 = r24669974 <= r24669980;
        double r24669982 = -2.0;
        double r24669983 = a;
        double r24669984 = r24669983 * r24669977;
        double r24669985 = -4.0;
        double r24669986 = r24669984 * r24669985;
        double r24669987 = fma(r24669974, r24669974, r24669986);
        double r24669988 = sqrt(r24669987);
        double r24669989 = r24669974 - r24669988;
        double r24669990 = r24669982 / r24669989;
        double r24669991 = r24669977 * r24669990;
        double r24669992 = 2.598286182153128e+84;
        bool r24669993 = r24669974 <= r24669992;
        double r24669994 = r24669977 * r24669985;
        double r24669995 = r24669974 * r24669974;
        double r24669996 = fma(r24669983, r24669994, r24669995);
        double r24669997 = sqrt(r24669996);
        double r24669998 = r24669997 + r24669974;
        double r24669999 = -0.5;
        double r24670000 = r24669998 * r24669999;
        double r24670001 = r24670000 / r24669983;
        double r24670002 = -r24669974;
        double r24670003 = r24670002 / r24669983;
        double r24670004 = r24669993 ? r24670001 : r24670003;
        double r24670005 = r24669981 ? r24669991 : r24670004;
        double r24670006 = r24669976 ? r24669979 : r24670005;
        return r24670006;
}


\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. Using strategy rm

    4. Applied *-un-lft-identity57.3

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

    5. Applied div-inv57.3

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

    6. Applied times-frac57.3

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

    7. Simplified57.3

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

    8. Simplified57.3

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

    9. Taylor expanded around -inf 3.8

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

    10. 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. Taylor expanded around 0 4.5

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

    15. Simplified4.5

      \[\leadsto \color{blue}{-\frac{b}{a}}\]
  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 "The quadratic formula (r2)"

  :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)))