Average Error: 33.6 → 9.8
Time: 18.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.396811349079212 \cdot 10^{+61}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3659668388152999 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -3.396811349079212 \cdot 10^{+61}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.3659668388152999 \cdot 10^{-67}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2670060 = b;
        double r2670061 = -r2670060;
        double r2670062 = r2670060 * r2670060;
        double r2670063 = 4.0;
        double r2670064 = a;
        double r2670065 = c;
        double r2670066 = r2670064 * r2670065;
        double r2670067 = r2670063 * r2670066;
        double r2670068 = r2670062 - r2670067;
        double r2670069 = sqrt(r2670068);
        double r2670070 = r2670061 + r2670069;
        double r2670071 = 2.0;
        double r2670072 = r2670071 * r2670064;
        double r2670073 = r2670070 / r2670072;
        return r2670073;
}
↓
double f(double a, double b, double c) {
        double r2670074 = b;
        double r2670075 = -3.396811349079212e+61;
        bool r2670076 = r2670074 <= r2670075;
        double r2670077 = c;
        double r2670078 = r2670077 / r2670074;
        double r2670079 = a;
        double r2670080 = r2670074 / r2670079;
        double r2670081 = r2670078 - r2670080;
        double r2670082 = 1.3659668388152999e-67;
        bool r2670083 = r2670074 <= r2670082;
        double r2670084 = r2670074 * r2670074;
        double r2670085 = 4.0;
        double r2670086 = r2670079 * r2670085;
        double r2670087 = r2670086 * r2670077;
        double r2670088 = r2670084 - r2670087;
        double r2670089 = sqrt(r2670088);
        double r2670090 = r2670089 - r2670074;
        double r2670091 = 2.0;
        double r2670092 = r2670079 * r2670091;
        double r2670093 = r2670090 / r2670092;
        double r2670094 = -r2670078;
        double r2670095 = r2670083 ? r2670093 : r2670094;
        double r2670096 = r2670076 ? r2670081 : r2670095;
        return r2670096;
}

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

Original33.6
Target20.8
Herbie9.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.396811349079212e+61

    1. Initial program 37.6

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

    2. Simplified37.6

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

    3. Taylor expanded around -inf 4.3

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

    if -3.396811349079212e+61 < b < 1.3659668388152999e-67

    1. Initial program 13.9

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

    2. Simplified13.9

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

    if 1.3659668388152999e-67 < b

    1. Initial program 53.0

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

    2. Simplified53.0

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

    4. Applied div-inv53.0

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

    5. Simplified53.0

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

    6. Taylor expanded around inf 8.1

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

    7. Simplified8.1

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

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.396811349079212 \cdot 10^{+61}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3659668388152999 \cdot 10^{-67}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(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)))