Average Error: 34.2 → 11.4
Time: 18.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.247157674878585888858757389039773391247 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{1 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \le 9.027398388687083073747117877445020640893 \cdot 10^{77}:\\ \;\;\;\;\frac{-1}{a \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(-a \cdot c, 4, b \cdot b\right)} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1\right) \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\\ \end{array}\]
\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 -1.247157674878585888858757389039773391247 \cdot 10^{-136}:\\
\;\;\;\;\frac{-1}{1 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\

\mathbf{elif}\;b \le 9.027398388687083073747117877445020640893 \cdot 10^{77}:\\
\;\;\;\;\frac{-1}{a \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(-a \cdot c, 4, b \cdot b\right)} + b\right)\\

\mathbf{else}:\\
\;\;\;\;\left(-1\right) \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r47038 = b;
        double r47039 = -r47038;
        double r47040 = r47038 * r47038;
        double r47041 = 4.0;
        double r47042 = a;
        double r47043 = c;
        double r47044 = r47042 * r47043;
        double r47045 = r47041 * r47044;
        double r47046 = r47040 - r47045;
        double r47047 = sqrt(r47046);
        double r47048 = r47039 - r47047;
        double r47049 = 2.0;
        double r47050 = r47049 * r47042;
        double r47051 = r47048 / r47050;
        return r47051;
}
↓
double f(double a, double b, double c) {
        double r47052 = b;
        double r47053 = -1.2471576748785859e-136;
        bool r47054 = r47052 <= r47053;
        double r47055 = -1.0;
        double r47056 = 1.0;
        double r47057 = c;
        double r47058 = r47052 / r47057;
        double r47059 = a;
        double r47060 = r47059 / r47052;
        double r47061 = r47058 - r47060;
        double r47062 = r47056 * r47061;
        double r47063 = r47055 / r47062;
        double r47064 = 9.027398388687083e+77;
        bool r47065 = r47052 <= r47064;
        double r47066 = 2.0;
        double r47067 = r47059 * r47066;
        double r47068 = r47055 / r47067;
        double r47069 = r47059 * r47057;
        double r47070 = -r47069;
        double r47071 = 4.0;
        double r47072 = r47052 * r47052;
        double r47073 = fma(r47070, r47071, r47072);
        double r47074 = sqrt(r47073);
        double r47075 = r47074 + r47052;
        double r47076 = r47068 * r47075;
        double r47077 = -r47056;
        double r47078 = r47052 / r47059;
        double r47079 = r47057 / r47052;
        double r47080 = r47078 - r47079;
        double r47081 = r47077 * r47080;
        double r47082 = r47065 ? r47076 : r47081;
        double r47083 = r47054 ? r47063 : r47082;
        return r47083;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.2
Target21.2
Herbie11.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2471576748785859e-136

    1. Initial program 50.5

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

    2. Simplified50.5

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

    4. Applied clear-num50.5

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

    5. Simplified50.5

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

    6. Taylor expanded around -inf 13.3

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

    7. Simplified13.3

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

    if -1.2471576748785859e-136 < b < 9.027398388687083e+77

    1. Initial program 12.0

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

    2. Simplified12.0

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

    4. Applied div-inv12.2

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

    5. Simplified12.2

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

    if 9.027398388687083e+77 < b

    1. Initial program 42.5

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

    2. Simplified42.5

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

    4. Applied clear-num42.6

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

    5. Simplified42.6

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

    6. Taylor expanded around inf 5.0

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

    7. Simplified5.0

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

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.247157674878585888858757389039773391247 \cdot 10^{-136}:\\ \;\;\;\;\frac{-1}{1 \cdot \left(\frac{b}{c} - \frac{a}{b}\right)}\\ \mathbf{elif}\;b \le 9.027398388687083073747117877445020640893 \cdot 10^{77}:\\ \;\;\;\;\frac{-1}{a \cdot 2} \cdot \left(\sqrt{\mathsf{fma}\left(-a \cdot c, 4, b \cdot b\right)} + b\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-1\right) \cdot \left(\frac{b}{a} - \frac{c}{b}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))