Average Error: 33.8 → 10.4
Time: 21.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, \frac{c}{b} \cdot 2\right)}{2}\\ \mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\
\;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, \frac{c}{b} \cdot 2\right)}{2}\\

\mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3950066 = b;
        double r3950067 = -r3950066;
        double r3950068 = r3950066 * r3950066;
        double r3950069 = 4.0;
        double r3950070 = a;
        double r3950071 = c;
        double r3950072 = r3950070 * r3950071;
        double r3950073 = r3950069 * r3950072;
        double r3950074 = r3950068 - r3950073;
        double r3950075 = sqrt(r3950074);
        double r3950076 = r3950067 + r3950075;
        double r3950077 = 2.0;
        double r3950078 = r3950077 * r3950070;
        double r3950079 = r3950076 / r3950078;
        return r3950079;
}
↓
double f(double a, double b, double c) {
        double r3950080 = b;
        double r3950081 = -8.213216247196925e+129;
        bool r3950082 = r3950080 <= r3950081;
        double r3950083 = a;
        double r3950084 = r3950080 / r3950083;
        double r3950085 = -2.0;
        double r3950086 = c;
        double r3950087 = r3950086 / r3950080;
        double r3950088 = 2.0;
        double r3950089 = r3950087 * r3950088;
        double r3950090 = fma(r3950084, r3950085, r3950089);
        double r3950091 = r3950090 / r3950088;
        double r3950092 = 6.088267304256603e-81;
        bool r3950093 = r3950080 <= r3950092;
        double r3950094 = 1.0;
        double r3950095 = r3950080 * r3950080;
        double r3950096 = 4.0;
        double r3950097 = r3950086 * r3950096;
        double r3950098 = r3950083 * r3950097;
        double r3950099 = r3950095 - r3950098;
        double r3950100 = sqrt(r3950099);
        double r3950101 = r3950100 - r3950080;
        double r3950102 = r3950083 / r3950101;
        double r3950103 = r3950094 / r3950102;
        double r3950104 = r3950103 / r3950088;
        double r3950105 = -2.0;
        double r3950106 = r3950105 * r3950087;
        double r3950107 = r3950106 / r3950088;
        double r3950108 = r3950093 ? r3950104 : r3950107;
        double r3950109 = r3950082 ? r3950091 : r3950108;
        return r3950109;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.8
Target20.6
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.213216247196925e+129

    1. Initial program 53.9

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

    2. Simplified53.9

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

    3. Taylor expanded around -inf 2.5

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified2.5

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

    if -8.213216247196925e+129 < b < 6.088267304256603e-81

    1. Initial program 12.3

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

    2. Simplified12.4

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

    4. Applied clear-num12.5

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

    if 6.088267304256603e-81 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    3. Taylor expanded around inf 10.5

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.213216247196925388401125773743990732555 \cdot 10^{129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{b}{a}, -2, \frac{c}{b} \cdot 2\right)}{2}\\ \mathbf{elif}\;b \le 6.088267304256603437292930310963869002155 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

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

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