Average Error: 32.9 → 28.6
Time: 1.6m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 7.844448680425584 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{a}{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
\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 7.844448680425584 \cdot 10^{+101}:\\
\;\;\;\;\frac{\frac{1}{2}}{\frac{a}{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}}\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}
double f(double a, double b, double c) {
        double r3909972 = b;
        double r3909973 = -r3909972;
        double r3909974 = r3909972 * r3909972;
        double r3909975 = 4.0;
        double r3909976 = a;
        double r3909977 = c;
        double r3909978 = r3909976 * r3909977;
        double r3909979 = r3909975 * r3909978;
        double r3909980 = r3909974 - r3909979;
        double r3909981 = sqrt(r3909980);
        double r3909982 = r3909973 + r3909981;
        double r3909983 = 2.0;
        double r3909984 = r3909983 * r3909976;
        double r3909985 = r3909982 / r3909984;
        return r3909985;
}
↓
double f(double a, double b, double c) {
        double r3909986 = b;
        double r3909987 = 7.844448680425584e+101;
        bool r3909988 = r3909986 <= r3909987;
        double r3909989 = 0.5;
        double r3909990 = a;
        double r3909991 = c;
        double r3909992 = r3909991 * r3909990;
        double r3909993 = -4.0;
        double r3909994 = r3909986 * r3909986;
        double r3909995 = fma(r3909992, r3909993, r3909994);
        double r3909996 = sqrt(r3909995);
        double r3909997 = r3909996 - r3909986;
        double r3909998 = r3909990 / r3909997;
        double r3909999 = r3909989 / r3909998;
        double r3910000 = 0.0;
        double r3910001 = r3909988 ? r3909999 : r3910000;
        return r3910001;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original32.9
Target20.2
Herbie28.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if b < 7.844448680425584e+101

    1. Initial program 25.3

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

    2. Simplified25.3

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

    4. Applied div-sub25.3

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

    5. Applied div-sub25.4

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

    7. Applied sub-div25.3

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

    9. Applied div-inv25.3

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

    10. Applied div-inv25.3

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

    11. Applied distribute-rgt-out--25.3

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

    12. Applied associate-/l*25.3

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

    if 7.844448680425584e+101 < b

    1. Initial program 59.0

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

    2. Simplified59.0

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

    4. Applied div-sub59.0

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

    5. Applied div-sub59.5

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

    6. Taylor expanded around 0 39.9

      \[\leadsto \color{blue}{0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification28.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 7.844448680425584 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{1}{2}}{\frac{a}{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))