Average Error: 32.8 → 8.9
Time: 35.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.877342284320474 \cdot 10^{+38}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -4.726535681060057 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 2}{a}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \mathbf{elif}\;b \le 9.19242293018462 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -9.877342284320474 \cdot 10^{+38}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le -4.726535681060057 \cdot 10^{-132}:\\
\;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 2}{a}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\

\mathbf{elif}\;b \le 9.19242293018462 \cdot 10^{+63}:\\
\;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r6060500 = b;
        double r6060501 = -r6060500;
        double r6060502 = r6060500 * r6060500;
        double r6060503 = 4.0;
        double r6060504 = a;
        double r6060505 = c;
        double r6060506 = r6060504 * r6060505;
        double r6060507 = r6060503 * r6060506;
        double r6060508 = r6060502 - r6060507;
        double r6060509 = sqrt(r6060508);
        double r6060510 = r6060501 - r6060509;
        double r6060511 = 2.0;
        double r6060512 = r6060511 * r6060504;
        double r6060513 = r6060510 / r6060512;
        return r6060513;
}
↓
double f(double a, double b, double c) {
        double r6060514 = b;
        double r6060515 = -9.877342284320474e+38;
        bool r6060516 = r6060514 <= r6060515;
        double r6060517 = c;
        double r6060518 = r6060517 / r6060514;
        double r6060519 = -r6060518;
        double r6060520 = -4.726535681060057e-132;
        bool r6060521 = r6060514 <= r6060520;
        double r6060522 = a;
        double r6060523 = r6060522 * r6060517;
        double r6060524 = 2.0;
        double r6060525 = r6060523 * r6060524;
        double r6060526 = r6060525 / r6060522;
        double r6060527 = r6060514 * r6060514;
        double r6060528 = 4.0;
        double r6060529 = r6060523 * r6060528;
        double r6060530 = r6060527 - r6060529;
        double r6060531 = sqrt(r6060530);
        double r6060532 = r6060531 - r6060514;
        double r6060533 = r6060526 / r6060532;
        double r6060534 = 9.19242293018462e+63;
        bool r6060535 = r6060514 <= r6060534;
        double r6060536 = 1.0;
        double r6060537 = r6060522 * r6060524;
        double r6060538 = -r6060514;
        double r6060539 = r6060538 - r6060531;
        double r6060540 = r6060537 / r6060539;
        double r6060541 = r6060536 / r6060540;
        double r6060542 = r6060514 / r6060522;
        double r6060543 = r6060518 - r6060542;
        double r6060544 = r6060535 ? r6060541 : r6060543;
        double r6060545 = r6060521 ? r6060533 : r6060544;
        double r6060546 = r6060516 ? r6060519 : r6060545;
        return r6060546;
}

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

Original32.8
Target20.1
Herbie8.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.877342284320474e+38

    1. Initial program 55.4

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

    2. Taylor expanded around -inf 4.5

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

    3. Simplified4.5

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

    if -9.877342284320474e+38 < b < -4.726535681060057e-132

    1. Initial program 37.3

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

    3. Applied flip--37.4

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

    4. Simplified15.8

      \[\leadsto \frac{\frac{\color{blue}{0 + \left(a \cdot c\right) \cdot 4}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}{2 \cdot a}\]

    5. Simplified15.8

      \[\leadsto \frac{\frac{0 + \left(a \cdot c\right) \cdot 4}{\color{blue}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}}{2 \cdot a}\]
    6. Using strategy rm

    7. Applied div-inv15.9

      \[\leadsto \color{blue}{\frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b} \cdot \frac{1}{2 \cdot a}}\]

    8. Simplified15.8

      \[\leadsto \frac{0 + \left(a \cdot c\right) \cdot 4}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b} \cdot \color{blue}{\frac{\frac{1}{2}}{a}}\]
    9. Using strategy rm

    10. Applied associate-*l/15.5

      \[\leadsto \color{blue}{\frac{\left(0 + \left(a \cdot c\right) \cdot 4\right) \cdot \frac{\frac{1}{2}}{a}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}}\]

    11. Simplified15.4

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

    if -4.726535681060057e-132 < b < 9.19242293018462e+63

    1. Initial program 11.8

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

    3. Applied clear-num11.9

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

    if 9.19242293018462e+63 < b

    1. Initial program 38.2

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

    2. Taylor expanded around inf 4.6

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.877342284320474 \cdot 10^{+38}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -4.726535681060057 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot 2}{a}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b}\\ \mathbf{elif}\;b \le 9.19242293018462 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))