Average Error: 33.3 → 10.0
Time: 28.4s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -4.82289647433212 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.289226058156428 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -4.82289647433212 \cdot 10^{+153}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 3.289226058156428 \cdot 10^{-70}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a} - \frac{b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r2290075 = b;
        double r2290076 = -r2290075;
        double r2290077 = r2290075 * r2290075;
        double r2290078 = 4.0;
        double r2290079 = a;
        double r2290080 = c;
        double r2290081 = r2290079 * r2290080;
        double r2290082 = r2290078 * r2290081;
        double r2290083 = r2290077 - r2290082;
        double r2290084 = sqrt(r2290083);
        double r2290085 = r2290076 + r2290084;
        double r2290086 = 2.0;
        double r2290087 = r2290086 * r2290079;
        double r2290088 = r2290085 / r2290087;
        return r2290088;
}
↓
double f(double a, double b, double c) {
        double r2290089 = b;
        double r2290090 = -4.82289647433212e+153;
        bool r2290091 = r2290089 <= r2290090;
        double r2290092 = c;
        double r2290093 = r2290092 / r2290089;
        double r2290094 = a;
        double r2290095 = r2290089 / r2290094;
        double r2290096 = r2290093 - r2290095;
        double r2290097 = 2.0;
        double r2290098 = r2290096 * r2290097;
        double r2290099 = r2290098 / r2290097;
        double r2290100 = 3.289226058156428e-70;
        bool r2290101 = r2290089 <= r2290100;
        double r2290102 = r2290089 * r2290089;
        double r2290103 = 4.0;
        double r2290104 = r2290103 * r2290094;
        double r2290105 = r2290092 * r2290104;
        double r2290106 = r2290102 - r2290105;
        double r2290107 = sqrt(r2290106);
        double r2290108 = r2290107 / r2290094;
        double r2290109 = r2290108 - r2290095;
        double r2290110 = r2290109 / r2290097;
        double r2290111 = -2.0;
        double r2290112 = r2290111 * r2290093;
        double r2290113 = r2290112 / r2290097;
        double r2290114 = r2290101 ? r2290110 : r2290113;
        double r2290115 = r2290091 ? r2290099 : r2290114;
        return r2290115;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original33.3
Target20.7
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -4.82289647433212e+153

    1. Initial program 60.9

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

    2. Simplified60.9

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

    4. Applied div-inv60.9

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

    6. Applied associate-*r/60.9

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

    7. Simplified60.9

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

    8. Taylor expanded around -inf 2.3

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

    9. Simplified2.3

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

    if -4.82289647433212e+153 < b < 3.289226058156428e-70

    1. Initial program 12.4

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

    2. Simplified12.4

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

    4. Applied div-inv12.6

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

    6. Applied associate-*r/12.4

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

    7. Simplified12.4

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

    9. Applied div-sub12.4

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

    if 3.289226058156428e-70 < b

    1. Initial program 52.2

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

    2. Simplified52.2

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

    4. Applied div-inv52.2

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

    6. Applied associate-*r/52.2

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

    7. Simplified52.2

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

    8. Taylor expanded around inf 9.1

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.82289647433212 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 3.289226058156428 \cdot 10^{-70}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}{a} - \frac{b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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)))