Average Error: 33.3 → 10.4
Time: 18.8s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{a}}{2}\\ \end{array}\]
\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 -5.961198324014865 \cdot 10^{-88}:\\
\;\;\;\;\frac{-c}{b}\\

\mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{a}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r1455954 = b;
        double r1455955 = -r1455954;
        double r1455956 = r1455954 * r1455954;
        double r1455957 = 4.0;
        double r1455958 = a;
        double r1455959 = c;
        double r1455960 = r1455958 * r1455959;
        double r1455961 = r1455957 * r1455960;
        double r1455962 = r1455956 - r1455961;
        double r1455963 = sqrt(r1455962);
        double r1455964 = r1455955 - r1455963;
        double r1455965 = 2.0;
        double r1455966 = r1455965 * r1455958;
        double r1455967 = r1455964 / r1455966;
        return r1455967;
}
↓
double f(double a, double b, double c) {
        double r1455968 = b;
        double r1455969 = -5.961198324014865e-88;
        bool r1455970 = r1455968 <= r1455969;
        double r1455971 = c;
        double r1455972 = -r1455971;
        double r1455973 = r1455972 / r1455968;
        double r1455974 = 6.384705165981893e+101;
        bool r1455975 = r1455968 <= r1455974;
        double r1455976 = -r1455968;
        double r1455977 = -4.0;
        double r1455978 = a;
        double r1455979 = r1455978 * r1455971;
        double r1455980 = r1455968 * r1455968;
        double r1455981 = fma(r1455977, r1455979, r1455980);
        double r1455982 = sqrt(r1455981);
        double r1455983 = r1455976 - r1455982;
        double r1455984 = r1455983 / r1455978;
        double r1455985 = 2.0;
        double r1455986 = r1455984 / r1455985;
        double r1455987 = r1455968 / r1455971;
        double r1455988 = r1455978 / r1455987;
        double r1455989 = r1455988 - r1455968;
        double r1455990 = r1455985 * r1455989;
        double r1455991 = r1455990 / r1455978;
        double r1455992 = r1455991 / r1455985;
        double r1455993 = r1455975 ? r1455986 : r1455992;
        double r1455994 = r1455970 ? r1455973 : r1455993;
        return r1455994;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

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

Derivation

  1. Split input into 3 regimes

  2. if b < -5.961198324014865e-88

    1. Initial program 51.4

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

    2. Taylor expanded around -inf 9.8

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

    3. Simplified9.8

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

    if -5.961198324014865e-88 < b < 6.384705165981893e+101

    1. Initial program 13.1

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

    2. Simplified13.1

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

    4. Applied *-un-lft-identity13.1

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

    5. Applied associate-/r*13.1

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

    6. Simplified13.2

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

    if 6.384705165981893e+101 < b

    1. Initial program 43.9

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

    2. Simplified43.9

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

    3. Taylor expanded around inf 9.5

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

    4. Simplified3.8

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.961198324014865 \cdot 10^{-88}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 6.384705165981893 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(-4, a \cdot c, b \cdot b\right)}}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \left(\frac{a}{\frac{b}{c}} - b\right)}{a}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))