Average Error: 32.5 → 9.9
Time: 25.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.4266250849096228 \cdot 10^{-56}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -1.4266250849096228 \cdot 10^{-56}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2621657 = b;
        double r2621658 = -r2621657;
        double r2621659 = r2621657 * r2621657;
        double r2621660 = 4.0;
        double r2621661 = a;
        double r2621662 = c;
        double r2621663 = r2621661 * r2621662;
        double r2621664 = r2621660 * r2621663;
        double r2621665 = r2621659 - r2621664;
        double r2621666 = sqrt(r2621665);
        double r2621667 = r2621658 - r2621666;
        double r2621668 = 2.0;
        double r2621669 = r2621668 * r2621661;
        double r2621670 = r2621667 / r2621669;
        return r2621670;
}
↓
double f(double a, double b, double c) {
        double r2621671 = b;
        double r2621672 = -1.4266250849096228e-56;
        bool r2621673 = r2621671 <= r2621672;
        double r2621674 = c;
        double r2621675 = r2621674 / r2621671;
        double r2621676 = -r2621675;
        double r2621677 = 2.2373425340727037e+98;
        bool r2621678 = r2621671 <= r2621677;
        double r2621679 = 1.0;
        double r2621680 = 2.0;
        double r2621681 = a;
        double r2621682 = r2621680 * r2621681;
        double r2621683 = -r2621671;
        double r2621684 = r2621671 * r2621671;
        double r2621685 = -4.0;
        double r2621686 = r2621685 * r2621674;
        double r2621687 = r2621681 * r2621686;
        double r2621688 = r2621684 + r2621687;
        double r2621689 = sqrt(r2621688);
        double r2621690 = r2621683 - r2621689;
        double r2621691 = r2621682 / r2621690;
        double r2621692 = r2621679 / r2621691;
        double r2621693 = r2621671 / r2621681;
        double r2621694 = r2621675 - r2621693;
        double r2621695 = r2621678 ? r2621692 : r2621694;
        double r2621696 = r2621673 ? r2621676 : r2621695;
        return r2621696;
}

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

Original32.5
Target20.0
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.4266250849096228e-56

    1. Initial program 52.5

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

    3. Applied sub-neg52.5

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

    4. Simplified52.5

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

    5. Taylor expanded around -inf 8.3

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

    6. Simplified8.3

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

    if -1.4266250849096228e-56 < b < 2.2373425340727037e+98

    1. Initial program 12.7

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

    3. Applied sub-neg12.7

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

    4. Simplified12.8

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

    6. Applied *-un-lft-identity12.8

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

    7. Applied associate-/l*12.9

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

    if 2.2373425340727037e+98 < b

    1. Initial program 43.2

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

    2. Taylor expanded around inf 4.9

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.4266250849096228 \cdot 10^{-56}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.2373425340727037 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :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)))