Average Error: 33.6 → 10.4
Time: 20.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{1}{2}}}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{1}{\frac{\frac{a}{\frac{1}{2}}}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3010430 = b;
        double r3010431 = -r3010430;
        double r3010432 = r3010430 * r3010430;
        double r3010433 = 4.0;
        double r3010434 = a;
        double r3010435 = r3010433 * r3010434;
        double r3010436 = c;
        double r3010437 = r3010435 * r3010436;
        double r3010438 = r3010432 - r3010437;
        double r3010439 = sqrt(r3010438);
        double r3010440 = r3010431 + r3010439;
        double r3010441 = 2.0;
        double r3010442 = r3010441 * r3010434;
        double r3010443 = r3010440 / r3010442;
        return r3010443;
}
↓
double f(double a, double b, double c) {
        double r3010444 = b;
        double r3010445 = -2.1144981103869975e+131;
        bool r3010446 = r3010444 <= r3010445;
        double r3010447 = c;
        double r3010448 = r3010447 / r3010444;
        double r3010449 = a;
        double r3010450 = r3010444 / r3010449;
        double r3010451 = r3010448 - r3010450;
        double r3010452 = 4.5810084990875205e-68;
        bool r3010453 = r3010444 <= r3010452;
        double r3010454 = 1.0;
        double r3010455 = 0.5;
        double r3010456 = r3010449 / r3010455;
        double r3010457 = -4.0;
        double r3010458 = r3010449 * r3010457;
        double r3010459 = r3010458 * r3010447;
        double r3010460 = r3010444 * r3010444;
        double r3010461 = r3010459 + r3010460;
        double r3010462 = sqrt(r3010461);
        double r3010463 = r3010462 - r3010444;
        double r3010464 = r3010456 / r3010463;
        double r3010465 = r3010454 / r3010464;
        double r3010466 = -r3010448;
        double r3010467 = r3010453 ? r3010465 : r3010466;
        double r3010468 = r3010446 ? r3010451 : r3010467;
        return r3010468;
}

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

Derivation

  1. Split input into 3 regimes

  2. if b < -2.1144981103869975e+131

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 2.6

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

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

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

    3. Applied div-inv13.5

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

    4. Simplified13.5

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

    6. Applied associate-*r/13.3

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

    7. Simplified13.3

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

    9. Applied div-inv13.3

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

    10. Applied associate-/l*13.3

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

    11. Simplified13.3

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

    13. Applied clear-num13.4

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

    if 4.5810084990875205e-68 < b

    1. Initial program 52.0

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{\frac{1}{2}}}{\sqrt{\left(a \cdot -4\right) \cdot c + b \cdot b} - b}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :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)))