Average Error: 34.1 → 10.0
Time: 16.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r70137 = b;
        double r70138 = -r70137;
        double r70139 = r70137 * r70137;
        double r70140 = 4.0;
        double r70141 = a;
        double r70142 = c;
        double r70143 = r70141 * r70142;
        double r70144 = r70140 * r70143;
        double r70145 = r70139 - r70144;
        double r70146 = sqrt(r70145);
        double r70147 = r70138 + r70146;
        double r70148 = 2.0;
        double r70149 = r70148 * r70141;
        double r70150 = r70147 / r70149;
        return r70150;
}
↓
double f(double a, double b, double c) {
        double r70151 = b;
        double r70152 = -1.119811545308531e+143;
        bool r70153 = r70151 <= r70152;
        double r70154 = 1.0;
        double r70155 = c;
        double r70156 = r70155 / r70151;
        double r70157 = a;
        double r70158 = r70151 / r70157;
        double r70159 = r70156 - r70158;
        double r70160 = r70154 * r70159;
        double r70161 = 4.718890261991469e-106;
        bool r70162 = r70151 <= r70161;
        double r70163 = r70151 * r70151;
        double r70164 = r70157 * r70155;
        double r70165 = 4.0;
        double r70166 = r70164 * r70165;
        double r70167 = r70163 - r70166;
        double r70168 = sqrt(r70167);
        double r70169 = r70168 - r70151;
        double r70170 = 2.0;
        double r70171 = r70170 * r70157;
        double r70172 = r70169 / r70171;
        double r70173 = -1.0;
        double r70174 = r70173 * r70156;
        double r70175 = r70162 ? r70172 : r70174;
        double r70176 = r70153 ? r70160 : r70175;
        return r70176;
}

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

Original34.1
Target21.0
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.119811545308531e+143

    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

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

    3. Taylor expanded around -inf 2.4

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

    4. Simplified2.4

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

    if -1.119811545308531e+143 < b < 4.718890261991469e-106

    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

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

    4. Applied div-inv11.2

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

    6. Applied associate-*r/11.1

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

    7. Simplified11.1

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

    if 4.718890261991469e-106 < b

    1. Initial program 52.4

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

    2. Simplified52.4

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

    3. Taylor expanded around inf 10.9

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.11981154530853106611761327467786604265 \cdot 10^{143}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 4.718890261991468628346768591871377778707 \cdot 10^{-106}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))