Average Error: 32.7 → 10.0
Time: 30.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.3975762435547 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -7.3975762435547 \cdot 10^{+118}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3202390 = b;
        double r3202391 = -r3202390;
        double r3202392 = r3202390 * r3202390;
        double r3202393 = 4.0;
        double r3202394 = a;
        double r3202395 = r3202393 * r3202394;
        double r3202396 = c;
        double r3202397 = r3202395 * r3202396;
        double r3202398 = r3202392 - r3202397;
        double r3202399 = sqrt(r3202398);
        double r3202400 = r3202391 + r3202399;
        double r3202401 = 2.0;
        double r3202402 = r3202401 * r3202394;
        double r3202403 = r3202400 / r3202402;
        return r3202403;
}
↓
double f(double a, double b, double c) {
        double r3202404 = b;
        double r3202405 = -7.3975762435547e+118;
        bool r3202406 = r3202404 <= r3202405;
        double r3202407 = c;
        double r3202408 = r3202407 / r3202404;
        double r3202409 = a;
        double r3202410 = r3202404 / r3202409;
        double r3202411 = r3202408 - r3202410;
        double r3202412 = 1.3115303715225787e-131;
        bool r3202413 = r3202404 <= r3202412;
        double r3202414 = r3202404 * r3202404;
        double r3202415 = 4.0;
        double r3202416 = r3202409 * r3202415;
        double r3202417 = r3202416 * r3202407;
        double r3202418 = r3202414 - r3202417;
        double r3202419 = sqrt(r3202418);
        double r3202420 = r3202419 - r3202404;
        double r3202421 = 2.0;
        double r3202422 = r3202420 / r3202421;
        double r3202423 = r3202422 / r3202409;
        double r3202424 = -r3202408;
        double r3202425 = r3202413 ? r3202423 : r3202424;
        double r3202426 = r3202406 ? r3202411 : r3202425;
        return r3202426;
}

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

Original32.7
Target20.0
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.3975762435547e+118

    1. Initial program 49.0

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

    2. Simplified49.0

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

    3. Taylor expanded around inf 49.0

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

    4. Simplified49.0

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

    5. Taylor expanded around -inf 3.0

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

    if -7.3975762435547e+118 < b < 1.3115303715225787e-131

    1. Initial program 10.7

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

    2. Simplified10.7

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

    3. Taylor expanded around inf 10.7

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

    4. Simplified10.7

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

    if 1.3115303715225787e-131 < b

    1. Initial program 50.3

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

    2. Simplified50.3

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

    3. Taylor expanded around inf 50.3

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

    4. Simplified50.3

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

    5. Taylor expanded around inf 11.7

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

    6. Simplified11.7

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.3975762435547 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(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)))