Average Error: 33.1 → 9.2
Time: 3.4m
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6124744946043857 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 2.039797431776216 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\ \mathbf{elif}\;b \le 8421022438072682.0:\\ \;\;\;\;\frac{\frac{\frac{\left(a \cdot -4\right) \cdot c}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -1.6124744946043857 \cdot 10^{+143}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 2.039797431776216 \cdot 10^{-113}:\\
\;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\

\mathbf{elif}\;b \le 8421022438072682.0:\\
\;\;\;\;\frac{\frac{\frac{\left(a \cdot -4\right) \cdot c}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r11721062 = b;
        double r11721063 = -r11721062;
        double r11721064 = r11721062 * r11721062;
        double r11721065 = 4.0;
        double r11721066 = a;
        double r11721067 = c;
        double r11721068 = r11721066 * r11721067;
        double r11721069 = r11721065 * r11721068;
        double r11721070 = r11721064 - r11721069;
        double r11721071 = sqrt(r11721070);
        double r11721072 = r11721063 + r11721071;
        double r11721073 = 2.0;
        double r11721074 = r11721073 * r11721066;
        double r11721075 = r11721072 / r11721074;
        return r11721075;
}
↓
double f(double a, double b, double c) {
        double r11721076 = b;
        double r11721077 = -1.6124744946043857e+143;
        bool r11721078 = r11721076 <= r11721077;
        double r11721079 = c;
        double r11721080 = r11721079 / r11721076;
        double r11721081 = a;
        double r11721082 = r11721076 / r11721081;
        double r11721083 = r11721080 - r11721082;
        double r11721084 = 2.039797431776216e-113;
        bool r11721085 = r11721076 <= r11721084;
        double r11721086 = r11721079 * r11721081;
        double r11721087 = -4.0;
        double r11721088 = r11721076 * r11721076;
        double r11721089 = fma(r11721086, r11721087, r11721088);
        double r11721090 = sqrt(r11721089);
        double r11721091 = r11721090 - r11721076;
        double r11721092 = 2.0;
        double r11721093 = r11721091 / r11721092;
        double r11721094 = r11721093 / r11721081;
        double r11721095 = 8421022438072682.0;
        bool r11721096 = r11721076 <= r11721095;
        double r11721097 = r11721081 * r11721087;
        double r11721098 = r11721097 * r11721079;
        double r11721099 = fma(r11721097, r11721079, r11721088);
        double r11721100 = sqrt(r11721099);
        double r11721101 = r11721100 + r11721076;
        double r11721102 = r11721098 / r11721101;
        double r11721103 = r11721102 / r11721092;
        double r11721104 = r11721103 / r11721081;
        double r11721105 = -r11721080;
        double r11721106 = r11721096 ? r11721104 : r11721105;
        double r11721107 = r11721085 ? r11721094 : r11721106;
        double r11721108 = r11721078 ? r11721083 : r11721107;
        return r11721108;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original33.1
Target20.2
Herbie9.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.6124744946043857e+143

    1. Initial program 57.1

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

    2. Simplified57.1

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

    3. Taylor expanded around -inf 2.6

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

    if -1.6124744946043857e+143 < b < 2.039797431776216e-113

    1. Initial program 11.1

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

    2. Simplified11.1

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

    3. Taylor expanded around -inf 11.1

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

    4. Simplified11.1

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

    if 2.039797431776216e-113 < b < 8421022438072682.0

    1. Initial program 38.5

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

    2. Simplified38.5

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

    4. Applied flip--38.6

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

    5. Simplified18.7

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

    6. Simplified18.7

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

    if 8421022438072682.0 < b

    1. Initial program 54.6

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

    2. Simplified54.6

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

    3. Taylor expanded around inf 5.8

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

    4. Simplified5.8

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6124744946043857 \cdot 10^{+143}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 2.039797431776216 \cdot 10^{-113}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(\left(c \cdot a\right), -4, \left(b \cdot b\right)\right)} - b}{2}}{a}\\ \mathbf{elif}\;b \le 8421022438072682.0:\\ \;\;\;\;\frac{\frac{\frac{\left(a \cdot -4\right) \cdot c}{\sqrt{\mathsf{fma}\left(\left(a \cdot -4\right), c, \left(b \cdot b\right)\right)} + b}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +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)))