Average Error: 32.8 → 6.4
Time: 2.0m
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.1214768270116103 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\left(a \cdot \frac{c}{b} - b\right) \cdot 2}\\ \mathbf{elif}\;b \le 1.199441090208904 \cdot 10^{-250}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 3.3389954009657566 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]
\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 -1.1214768270116103 \cdot 10^{+154}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\left(a \cdot \frac{c}{b} - b\right) \cdot 2}\\

\mathbf{elif}\;b \le 1.199441090208904 \cdot 10^{-250}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\

\mathbf{elif}\;b \le 3.3389954009657566 \cdot 10^{+124}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r18785638 = b;
        double r18785639 = -r18785638;
        double r18785640 = r18785638 * r18785638;
        double r18785641 = 4.0;
        double r18785642 = a;
        double r18785643 = c;
        double r18785644 = r18785642 * r18785643;
        double r18785645 = r18785641 * r18785644;
        double r18785646 = r18785640 - r18785645;
        double r18785647 = sqrt(r18785646);
        double r18785648 = r18785639 - r18785647;
        double r18785649 = 2.0;
        double r18785650 = r18785649 * r18785642;
        double r18785651 = r18785648 / r18785650;
        return r18785651;
}
↓
double f(double a, double b, double c) {
        double r18785652 = b;
        double r18785653 = -1.1214768270116103e+154;
        bool r18785654 = r18785652 <= r18785653;
        double r18785655 = 0.5;
        double r18785656 = 4.0;
        double r18785657 = c;
        double r18785658 = r18785656 * r18785657;
        double r18785659 = a;
        double r18785660 = r18785657 / r18785652;
        double r18785661 = r18785659 * r18785660;
        double r18785662 = r18785661 - r18785652;
        double r18785663 = 2.0;
        double r18785664 = r18785662 * r18785663;
        double r18785665 = r18785658 / r18785664;
        double r18785666 = r18785655 * r18785665;
        double r18785667 = 1.199441090208904e-250;
        bool r18785668 = r18785652 <= r18785667;
        double r18785669 = r18785652 * r18785652;
        double r18785670 = r18785659 * r18785657;
        double r18785671 = r18785656 * r18785670;
        double r18785672 = r18785669 - r18785671;
        double r18785673 = sqrt(r18785672);
        double r18785674 = r18785673 - r18785652;
        double r18785675 = r18785658 / r18785674;
        double r18785676 = r18785655 * r18785675;
        double r18785677 = 3.3389954009657566e+124;
        bool r18785678 = r18785652 <= r18785677;
        double r18785679 = -r18785652;
        double r18785680 = r18785679 - r18785673;
        double r18785681 = r18785663 * r18785659;
        double r18785682 = r18785680 / r18785681;
        double r18785683 = r18785652 / r18785659;
        double r18785684 = -r18785683;
        double r18785685 = r18785678 ? r18785682 : r18785684;
        double r18785686 = r18785668 ? r18785676 : r18785685;
        double r18785687 = r18785654 ? r18785666 : r18785686;
        return r18785687;
}

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
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1214768270116103e+154

    1. Initial program 62.9

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

    3. Applied flip--62.9

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

    4. Simplified37.6

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

    6. Applied *-un-lft-identity37.6

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

    7. Applied times-frac37.6

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

    8. Simplified37.6

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

    9. Simplified37.5

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

    10. Taylor expanded around -inf 7.1

      \[\leadsto \frac{1}{2} \cdot \frac{4 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\]

    11. Simplified1.5

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

    if -1.1214768270116103e+154 < b < 1.199441090208904e-250

    1. Initial program 32.2

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

    3. Applied flip--32.3

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

    4. Simplified16.1

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

    6. Applied *-un-lft-identity16.1

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

    7. Applied times-frac16.2

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

    8. Simplified16.2

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

    9. Simplified8.4

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

    if 1.199441090208904e-250 < b < 3.3389954009657566e+124

    1. Initial program 7.8

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

    if 3.3389954009657566e+124 < b

    1. Initial program 50.5

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

    3. Applied flip--61.9

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

    4. Simplified62.0

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

    5. Taylor expanded around 0 3.5

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

    6. Simplified3.5

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1214768270116103 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\left(a \cdot \frac{c}{b} - b\right) \cdot 2}\\ \mathbf{elif}\;b \le 1.199441090208904 \cdot 10^{-250}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{4 \cdot c}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}\\ \mathbf{elif}\;b \le 3.3389954009657566 \cdot 10^{+124}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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)))