Average Error: 34.5 → 9.6
Time: 37.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.8071664004115455 \cdot 10^{+149}:\\ \;\;\;\;\frac{-2 \cdot b}{2.0 \cdot a}\\ \mathbf{elif}\;b \le 1.1804820682342164 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4.0 \cdot \left(c \cdot a\right)} - b}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1.0 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -5.8071664004115455 \cdot 10^{+149}:\\
\;\;\;\;\frac{-2 \cdot b}{2.0 \cdot a}\\

\mathbf{elif}\;b \le 1.1804820682342164 \cdot 10^{-93}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - 4.0 \cdot \left(c \cdot a\right)} - b}{2.0 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;-1.0 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r3200258 = b;
        double r3200259 = -r3200258;
        double r3200260 = r3200258 * r3200258;
        double r3200261 = 4.0;
        double r3200262 = a;
        double r3200263 = c;
        double r3200264 = r3200262 * r3200263;
        double r3200265 = r3200261 * r3200264;
        double r3200266 = r3200260 - r3200265;
        double r3200267 = sqrt(r3200266);
        double r3200268 = r3200259 + r3200267;
        double r3200269 = 2.0;
        double r3200270 = r3200269 * r3200262;
        double r3200271 = r3200268 / r3200270;
        return r3200271;
}
↓
double f(double a, double b, double c) {
        double r3200272 = b;
        double r3200273 = -5.8071664004115455e+149;
        bool r3200274 = r3200272 <= r3200273;
        double r3200275 = -2.0;
        double r3200276 = r3200275 * r3200272;
        double r3200277 = 2.0;
        double r3200278 = a;
        double r3200279 = r3200277 * r3200278;
        double r3200280 = r3200276 / r3200279;
        double r3200281 = 1.1804820682342164e-93;
        bool r3200282 = r3200272 <= r3200281;
        double r3200283 = r3200272 * r3200272;
        double r3200284 = 4.0;
        double r3200285 = c;
        double r3200286 = r3200285 * r3200278;
        double r3200287 = r3200284 * r3200286;
        double r3200288 = r3200283 - r3200287;
        double r3200289 = sqrt(r3200288);
        double r3200290 = r3200289 - r3200272;
        double r3200291 = r3200290 / r3200279;
        double r3200292 = -1.0;
        double r3200293 = r3200285 / r3200272;
        double r3200294 = r3200292 * r3200293;
        double r3200295 = r3200282 ? r3200291 : r3200294;
        double r3200296 = r3200274 ? r3200280 : r3200295;
        return r3200296;
}

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

Original34.5
Target20.9
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.8071664004115455e+149

    1. Initial program 62.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv62.8

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2.0 \cdot a}}\]
    4. Using strategy rm

    5. Applied associate-*r/62.8

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot 1}{2.0 \cdot a}}\]

    6. Simplified62.8

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} - b}}{2.0 \cdot a}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt62.8

      \[\leadsto \frac{\sqrt{\color{blue}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0}}} - b}{2.0 \cdot a}\]

    9. Applied sqrt-prod62.8

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0}} \cdot \sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0}}} - b}{2.0 \cdot a}\]

    10. Applied fma-neg62.8

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0}}, \sqrt{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0}}, -b\right)}}{2.0 \cdot a}\]

    11. Taylor expanded around -inf 3.0

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2.0 \cdot a}\]

    12. Simplified3.0

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2.0 \cdot a}\]

    if -5.8071664004115455e+149 < b < 1.1804820682342164e-93

    1. Initial program 11.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]
    2. Using strategy rm

    3. Applied div-inv11.7

      \[\leadsto \color{blue}{\left(\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2.0 \cdot a}}\]
    4. Using strategy rm

    5. Applied associate-*r/11.6

      \[\leadsto \color{blue}{\frac{\left(\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}\right) \cdot 1}{2.0 \cdot a}}\]

    6. Simplified11.6

      \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4.0} - b}}{2.0 \cdot a}\]

    if 1.1804820682342164e-93 < b

    1. Initial program 52.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4.0 \cdot \left(a \cdot c\right)}}{2.0 \cdot a}\]

    2. Taylor expanded around inf 9.1

      \[\leadsto \color{blue}{-1.0 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.8071664004115455 \cdot 10^{+149}:\\ \;\;\;\;\frac{-2 \cdot b}{2.0 \cdot a}\\ \mathbf{elif}\;b \le 1.1804820682342164 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 4.0 \cdot \left(c \cdot a\right)} - b}{2.0 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1.0 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))