Average Error: 33.3 → 10.3
Time: 22.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\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 -3.263941314600607 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\
\;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)} - b\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r1055032 = b;
        double r1055033 = -r1055032;
        double r1055034 = r1055032 * r1055032;
        double r1055035 = 4.0;
        double r1055036 = a;
        double r1055037 = c;
        double r1055038 = r1055036 * r1055037;
        double r1055039 = r1055035 * r1055038;
        double r1055040 = r1055034 - r1055039;
        double r1055041 = sqrt(r1055040);
        double r1055042 = r1055033 + r1055041;
        double r1055043 = 2.0;
        double r1055044 = r1055043 * r1055036;
        double r1055045 = r1055042 / r1055044;
        return r1055045;
}
↓
double f(double a, double b, double c) {
        double r1055046 = b;
        double r1055047 = -3.263941314600607e+152;
        bool r1055048 = r1055046 <= r1055047;
        double r1055049 = c;
        double r1055050 = r1055049 / r1055046;
        double r1055051 = a;
        double r1055052 = r1055046 / r1055051;
        double r1055053 = r1055050 - r1055052;
        double r1055054 = 1.8378252714625124e-19;
        bool r1055055 = r1055046 <= r1055054;
        double r1055056 = r1055049 * r1055051;
        double r1055057 = -4.0;
        double r1055058 = r1055056 * r1055057;
        double r1055059 = fma(r1055046, r1055046, r1055058);
        double r1055060 = sqrt(r1055059);
        double r1055061 = r1055060 - r1055046;
        double r1055062 = 0.5;
        double r1055063 = r1055061 * r1055062;
        double r1055064 = r1055063 / r1055051;
        double r1055065 = -r1055049;
        double r1055066 = r1055065 / r1055046;
        double r1055067 = r1055055 ? r1055064 : r1055066;
        double r1055068 = r1055048 ? r1055053 : r1055067;
        return r1055068;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.3
Target20.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.263941314600607e+152

    1. Initial program 60.1

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

    2. Simplified60.1

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

    4. Applied *-un-lft-identity60.1

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

    5. Applied div-inv60.1

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

    6. Applied times-frac60.1

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

    7. Simplified60.1

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

    8. Simplified60.1

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

    10. Applied associate-*r/60.1

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

    11. Taylor expanded around -inf 2.3

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

    if -3.263941314600607e+152 < b < 1.8378252714625124e-19

    1. Initial program 14.2

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

    2. Simplified14.2

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

    4. Applied *-un-lft-identity14.2

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

    5. Applied div-inv14.2

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

    6. Applied times-frac14.3

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

    7. Simplified14.3

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

    8. Simplified14.3

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

    10. Applied associate-*r/14.2

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

    if 1.8378252714625124e-19 < b

    1. Initial program 54.4

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

    2. Simplified54.4

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

    3. Taylor expanded around inf 7.0

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

    4. Simplified7.0

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.263941314600607 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.8378252714625124 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(b, b, \left(\left(c \cdot a\right) \cdot -4\right)\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))