Average Error: 34.3 → 8.0
Time: 16.4s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9661478263987.111328125:\\ \;\;\;\;\frac{-1 \cdot c}{b}\\ \mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\frac{4 \cdot a}{2} \cdot \frac{c}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}\right)\\ \mathbf{elif}\;b \le 6.033691444141405046034068616119572110107 \cdot 10^{84}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -9661478263987.111328125:\\
\;\;\;\;\frac{-1 \cdot c}{b}\\

\mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\
\;\;\;\;\frac{1}{a} \cdot \left(\frac{4 \cdot a}{2} \cdot \frac{c}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}\right)\\

\mathbf{elif}\;b \le 6.033691444141405046034068616119572110107 \cdot 10^{84}:\\
\;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r74669 = b;
        double r74670 = -r74669;
        double r74671 = r74669 * r74669;
        double r74672 = 4.0;
        double r74673 = a;
        double r74674 = c;
        double r74675 = r74673 * r74674;
        double r74676 = r74672 * r74675;
        double r74677 = r74671 - r74676;
        double r74678 = sqrt(r74677);
        double r74679 = r74670 - r74678;
        double r74680 = 2.0;
        double r74681 = r74680 * r74673;
        double r74682 = r74679 / r74681;
        return r74682;
}
↓
double f(double a, double b, double c) {
        double r74683 = b;
        double r74684 = -9661478263987.111;
        bool r74685 = r74683 <= r74684;
        double r74686 = -1.0;
        double r74687 = c;
        double r74688 = r74686 * r74687;
        double r74689 = r74688 / r74683;
        double r74690 = 8.958852798091287e-209;
        bool r74691 = r74683 <= r74690;
        double r74692 = 1.0;
        double r74693 = a;
        double r74694 = r74692 / r74693;
        double r74695 = 4.0;
        double r74696 = r74695 * r74693;
        double r74697 = 2.0;
        double r74698 = r74696 / r74697;
        double r74699 = r74683 * r74683;
        double r74700 = r74687 * r74695;
        double r74701 = r74693 * r74700;
        double r74702 = r74699 - r74701;
        double r74703 = sqrt(r74702);
        double r74704 = r74703 - r74683;
        double r74705 = r74687 / r74704;
        double r74706 = r74698 * r74705;
        double r74707 = r74694 * r74706;
        double r74708 = 6.033691444141405e+84;
        bool r74709 = r74683 <= r74708;
        double r74710 = -r74683;
        double r74711 = r74697 * r74693;
        double r74712 = r74710 / r74711;
        double r74713 = r74687 * r74693;
        double r74714 = r74695 * r74713;
        double r74715 = r74699 - r74714;
        double r74716 = sqrt(r74715);
        double r74717 = r74716 / r74711;
        double r74718 = r74712 - r74717;
        double r74719 = 1.0;
        double r74720 = r74687 / r74683;
        double r74721 = r74683 / r74693;
        double r74722 = r74720 - r74721;
        double r74723 = r74719 * r74722;
        double r74724 = r74709 ? r74718 : r74723;
        double r74725 = r74691 ? r74707 : r74724;
        double r74726 = r74685 ? r74689 : r74725;
        return r74726;
}

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

Original34.3
Target21.1
Herbie8.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9661478263987.111

    1. Initial program 56.4

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

    2. Taylor expanded around -inf 5.0

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

    3. Simplified5.0

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

    if -9661478263987.111 < b < 8.958852798091287e-209

    1. Initial program 25.5

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

    3. Applied flip--25.7

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

    4. Simplified17.5

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

    5. Simplified17.5

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

    7. Applied associate-/r*17.5

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

    8. Simplified14.8

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

    10. Applied div-inv15.0

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

    if 8.958852798091287e-209 < b < 6.033691444141405e+84

    1. Initial program 7.0

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

    3. Applied div-sub7.0

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

    4. Simplified7.0

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

    5. Simplified7.0

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

    if 6.033691444141405e+84 < b

    1. Initial program 44.0

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

    2. Taylor expanded around inf 3.5

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

    3. Simplified3.5

      \[\leadsto \color{blue}{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9661478263987.111328125:\\ \;\;\;\;\frac{-1 \cdot c}{b}\\ \mathbf{elif}\;b \le 8.958852798091287000832395933283492118861 \cdot 10^{-209}:\\ \;\;\;\;\frac{1}{a} \cdot \left(\frac{4 \cdot a}{2} \cdot \frac{c}{\sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)} - b}\right)\\ \mathbf{elif}\;b \le 6.033691444141405046034068616119572110107 \cdot 10^{84}:\\ \;\;\;\;\frac{-b}{2 \cdot a} - \frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0.0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))