Average Error: 33.5 → 10.5
Time: 28.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -227369802444031.66:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} \cdot \frac{1}{a} + b \cdot \frac{-1}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -227369802444031.66:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} \cdot \frac{1}{a} + b \cdot \frac{-1}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r4179654 = b;
        double r4179655 = -r4179654;
        double r4179656 = r4179654 * r4179654;
        double r4179657 = 4.0;
        double r4179658 = a;
        double r4179659 = r4179657 * r4179658;
        double r4179660 = c;
        double r4179661 = r4179659 * r4179660;
        double r4179662 = r4179656 - r4179661;
        double r4179663 = sqrt(r4179662);
        double r4179664 = r4179655 + r4179663;
        double r4179665 = 2.0;
        double r4179666 = r4179665 * r4179658;
        double r4179667 = r4179664 / r4179666;
        return r4179667;
}
↓
double f(double a, double b, double c) {
        double r4179668 = b;
        double r4179669 = -227369802444031.66;
        bool r4179670 = r4179668 <= r4179669;
        double r4179671 = c;
        double r4179672 = r4179671 / r4179668;
        double r4179673 = a;
        double r4179674 = r4179668 / r4179673;
        double r4179675 = r4179672 - r4179674;
        double r4179676 = 2.0;
        double r4179677 = r4179675 * r4179676;
        double r4179678 = r4179677 / r4179676;
        double r4179679 = 2.0617732603635578e-61;
        bool r4179680 = r4179668 <= r4179679;
        double r4179681 = r4179668 * r4179668;
        double r4179682 = r4179673 * r4179671;
        double r4179683 = 4.0;
        double r4179684 = r4179682 * r4179683;
        double r4179685 = r4179681 - r4179684;
        double r4179686 = sqrt(r4179685);
        double r4179687 = 1.0;
        double r4179688 = r4179687 / r4179673;
        double r4179689 = r4179686 * r4179688;
        double r4179690 = -1.0;
        double r4179691 = r4179690 / r4179673;
        double r4179692 = r4179668 * r4179691;
        double r4179693 = r4179689 + r4179692;
        double r4179694 = r4179693 / r4179676;
        double r4179695 = -2.0;
        double r4179696 = r4179695 * r4179672;
        double r4179697 = r4179696 / r4179676;
        double r4179698 = r4179680 ? r4179694 : r4179697;
        double r4179699 = r4179670 ? r4179678 : r4179698;
        return r4179699;
}

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

Original33.5
Target20.6
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -227369802444031.66

    1. Initial program 32.9

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

    2. Simplified32.9

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

    3. Taylor expanded around -inf 6.8

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

    4. Simplified6.8

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

    if -227369802444031.66 < b < 2.0617732603635578e-61

    1. Initial program 14.9

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

    2. Simplified15.0

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

    4. Applied div-inv15.0

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

    6. Applied *-commutative15.0

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

    8. Applied sub-neg15.0

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

    9. Applied distribute-lft-in15.0

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

    if 2.0617732603635578e-61 < b

    1. Initial program 52.8

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

    2. Simplified52.8

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

    3. Taylor expanded around inf 8.3

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -227369802444031.66:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 2.0617732603635578 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} \cdot \frac{1}{a} + b \cdot \frac{-1}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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)))