Average Error: 32.9 → 12.3
Time: 2.3m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.06600713448898 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{-a}}}{a}\\ \mathbf{elif}\;b \le -1.8674816151448643 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}\right)}}{a}\\ \mathbf{elif}\;b \le 5.436017879840864 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]
\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 -7.06600713448898 \cdot 10^{+148}:\\
\;\;\;\;\frac{\frac{c}{\frac{b}{-a}}}{a}\\

\mathbf{elif}\;b \le -1.8674816151448643 \cdot 10^{-121}:\\
\;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}\right)}}{a}\\

\mathbf{elif}\;b \le 5.436017879840864 \cdot 10^{+134}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r6521047 = b;
        double r6521048 = -r6521047;
        double r6521049 = r6521047 * r6521047;
        double r6521050 = 4.0;
        double r6521051 = a;
        double r6521052 = c;
        double r6521053 = r6521051 * r6521052;
        double r6521054 = r6521050 * r6521053;
        double r6521055 = r6521049 - r6521054;
        double r6521056 = sqrt(r6521055);
        double r6521057 = r6521048 - r6521056;
        double r6521058 = 2.0;
        double r6521059 = r6521058 * r6521051;
        double r6521060 = r6521057 / r6521059;
        return r6521060;
}
↓
double f(double a, double b, double c) {
        double r6521061 = b;
        double r6521062 = -7.06600713448898e+148;
        bool r6521063 = r6521061 <= r6521062;
        double r6521064 = c;
        double r6521065 = a;
        double r6521066 = -r6521065;
        double r6521067 = r6521061 / r6521066;
        double r6521068 = r6521064 / r6521067;
        double r6521069 = r6521068 / r6521065;
        double r6521070 = -1.8674816151448643e-121;
        bool r6521071 = r6521061 <= r6521070;
        double r6521072 = 4.0;
        double r6521073 = r6521072 * r6521065;
        double r6521074 = r6521073 * r6521064;
        double r6521075 = 2.0;
        double r6521076 = -r6521061;
        double r6521077 = r6521064 * r6521065;
        double r6521078 = -4.0;
        double r6521079 = r6521061 * r6521061;
        double r6521080 = fma(r6521077, r6521078, r6521079);
        double r6521081 = sqrt(r6521080);
        double r6521082 = r6521076 + r6521081;
        double r6521083 = r6521075 * r6521082;
        double r6521084 = r6521074 / r6521083;
        double r6521085 = r6521084 / r6521065;
        double r6521086 = 5.436017879840864e+134;
        bool r6521087 = r6521061 <= r6521086;
        double r6521088 = r6521076 - r6521081;
        double r6521089 = r6521088 / r6521075;
        double r6521090 = r6521089 / r6521065;
        double r6521091 = r6521061 / r6521065;
        double r6521092 = -r6521091;
        double r6521093 = r6521087 ? r6521090 : r6521092;
        double r6521094 = r6521071 ? r6521085 : r6521093;
        double r6521095 = r6521063 ? r6521069 : r6521094;
        return r6521095;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target20.6
Herbie12.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -7.06600713448898e+148

    1. Initial program 62.4

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

    2. Simplified62.4

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

    3. Taylor expanded around 0 62.4

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

    4. Simplified62.4

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

    5. Taylor expanded around -inf 15.6

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

    6. Simplified17.3

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

    if -7.06600713448898e+148 < b < -1.8674816151448643e-121

    1. Initial program 41.6

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

    2. Simplified41.6

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

    4. Applied flip--41.7

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

    5. Applied associate-/l/41.7

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

    6. Simplified15.7

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

    if -1.8674816151448643e-121 < b < 5.436017879840864e+134

    1. Initial program 11.0

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

    2. Simplified11.0

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

    3. Taylor expanded around 0 11.0

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

    4. Simplified11.0

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

    if 5.436017879840864e+134 < b

    1. Initial program 53.5

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

    2. Simplified53.5

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

    3. Taylor expanded around 0 53.5

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

    4. Simplified53.5

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

    6. Applied add-sqr-sqrt53.7

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

    7. Applied associate-/r*53.6

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

    8. Taylor expanded around inf 4.8

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

    9. Simplified3.7

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

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.06600713448898 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{c}{\frac{b}{-a}}}{a}\\ \mathbf{elif}\;b \le -1.8674816151448643 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\left(4 \cdot a\right) \cdot c}{2 \cdot \left(\left(-b\right) + \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}\right)}}{a}\\ \mathbf{elif}\;b \le 5.436017879840864 \cdot 10^{+134}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 +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)))