Average Error: 32.9 → 10.3
Time: 20.5s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\
\;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2164940 = b;
        double r2164941 = -r2164940;
        double r2164942 = r2164940 * r2164940;
        double r2164943 = 4.0;
        double r2164944 = a;
        double r2164945 = r2164943 * r2164944;
        double r2164946 = c;
        double r2164947 = r2164945 * r2164946;
        double r2164948 = r2164942 - r2164947;
        double r2164949 = sqrt(r2164948);
        double r2164950 = r2164941 + r2164949;
        double r2164951 = 2.0;
        double r2164952 = r2164951 * r2164944;
        double r2164953 = r2164950 / r2164952;
        return r2164953;
}
↓
double f(double a, double b, double c) {
        double r2164954 = b;
        double r2164955 = -9.088000531423294e+152;
        bool r2164956 = r2164954 <= r2164955;
        double r2164957 = c;
        double r2164958 = r2164957 / r2164954;
        double r2164959 = a;
        double r2164960 = r2164954 / r2164959;
        double r2164961 = r2164958 - r2164960;
        double r2164962 = 9.354082991670835e-125;
        bool r2164963 = r2164954 <= r2164962;
        double r2164964 = -4.0;
        double r2164965 = r2164959 * r2164964;
        double r2164966 = r2164954 * r2164954;
        double r2164967 = fma(r2164965, r2164957, r2164966);
        double r2164968 = sqrt(r2164967);
        double r2164969 = r2164968 - r2164954;
        double r2164970 = 0.5;
        double r2164971 = r2164969 * r2164970;
        double r2164972 = r2164971 / r2164959;
        double r2164973 = -r2164957;
        double r2164974 = r2164973 / r2164954;
        double r2164975 = r2164963 ? r2164972 : r2164974;
        double r2164976 = r2164956 ? r2164961 : r2164975;
        return r2164976;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target20.3
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.088000531423294e+152

    1. Initial program 60.4

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

    2. Simplified60.4

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

    3. Taylor expanded around -inf 1.5

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

    if -9.088000531423294e+152 < b < 9.354082991670835e-125

    1. Initial program 10.9

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

    2. Simplified10.9

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

    4. Applied *-un-lft-identity10.9

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

    5. Applied div-inv11.1

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

    6. Applied times-frac11.1

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

    7. Simplified11.1

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

    8. Simplified11.1

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

    10. Applied associate-*r/10.9

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

    if 9.354082991670835e-125 < b

    1. Initial program 49.8

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

    2. Simplified49.8

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

    3. Taylor expanded around inf 11.9

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

    4. Simplified11.9

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.088000531423294 \cdot 10^{+152}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.354082991670835 \cdot 10^{-125}:\\ \;\;\;\;\frac{\left(\sqrt{\mathsf{fma}\left(a \cdot -4, c, b \cdot b\right)} - b\right) \cdot \frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))