Average Error: 34.1 → 9.4
Time: 18.5s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.913936217078672453781998496151874189163 \cdot 10^{60}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.876852397051050336397941792630169519921 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -1.124423417365483683607573317023703677331 \cdot 10^{-68}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.282387544777820560455992664410231115991 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]
\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 -9.913936217078672453781998496151874189163 \cdot 10^{60}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -2.876852397051050336397941792630169519921 \cdot 10^{-50}:\\
\;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}{2 \cdot a}\\

\mathbf{elif}\;b \le -1.124423417365483683607573317023703677331 \cdot 10^{-68}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 9.282387544777820560455992664410231115991 \cdot 10^{92}:\\
\;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r61699 = b;
        double r61700 = -r61699;
        double r61701 = r61699 * r61699;
        double r61702 = 4.0;
        double r61703 = a;
        double r61704 = c;
        double r61705 = r61703 * r61704;
        double r61706 = r61702 * r61705;
        double r61707 = r61701 - r61706;
        double r61708 = sqrt(r61707);
        double r61709 = r61700 - r61708;
        double r61710 = 2.0;
        double r61711 = r61710 * r61703;
        double r61712 = r61709 / r61711;
        return r61712;
}
↓
double f(double a, double b, double c) {
        double r61713 = b;
        double r61714 = -9.913936217078672e+60;
        bool r61715 = r61713 <= r61714;
        double r61716 = -1.0;
        double r61717 = c;
        double r61718 = r61717 / r61713;
        double r61719 = r61716 * r61718;
        double r61720 = -2.8768523970510503e-50;
        bool r61721 = r61713 <= r61720;
        double r61722 = 4.0;
        double r61723 = a;
        double r61724 = r61722 * r61723;
        double r61725 = r61717 * r61724;
        double r61726 = r61713 * r61713;
        double r61727 = r61724 * r61717;
        double r61728 = -r61727;
        double r61729 = r61726 + r61728;
        double r61730 = sqrt(r61729);
        double r61731 = r61730 - r61713;
        double r61732 = r61725 / r61731;
        double r61733 = 2.0;
        double r61734 = r61733 * r61723;
        double r61735 = r61732 / r61734;
        double r61736 = -1.1244234173654837e-68;
        bool r61737 = r61713 <= r61736;
        double r61738 = 9.28238754477782e+92;
        bool r61739 = r61713 <= r61738;
        double r61740 = -r61713;
        double r61741 = r61723 * r61717;
        double r61742 = r61722 * r61741;
        double r61743 = r61726 - r61742;
        double r61744 = sqrt(r61743);
        double r61745 = r61740 - r61744;
        double r61746 = r61745 / r61733;
        double r61747 = r61746 / r61723;
        double r61748 = -2.0;
        double r61749 = r61748 * r61713;
        double r61750 = r61749 / r61734;
        double r61751 = r61739 ? r61747 : r61750;
        double r61752 = r61737 ? r61719 : r61751;
        double r61753 = r61721 ? r61735 : r61752;
        double r61754 = r61715 ? r61719 : r61753;
        return r61754;
}

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

Original34.1
Target20.8
Herbie9.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -9.913936217078672e+60 or -2.8768523970510503e-50 < b < -1.1244234173654837e-68

    1. Initial program 56.8

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

    2. Taylor expanded around -inf 5.0

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

    if -9.913936217078672e+60 < b < -2.8768523970510503e-50

    1. Initial program 43.8

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

    3. Applied flip--43.9

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

    4. Simplified13.5

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

    5. Simplified13.5

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

    7. Applied sub-neg13.5

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

    8. Simplified13.5

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

    if -1.1244234173654837e-68 < b < 9.28238754477782e+92

    1. Initial program 13.4

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

    3. Applied associate-/r*13.4

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

    if 9.28238754477782e+92 < b

    1. Initial program 45.0

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

    3. Applied flip--62.8

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

    4. Simplified61.9

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

    5. Simplified61.9

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

    6. Taylor expanded around 0 3.6

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.913936217078672453781998496151874189163 \cdot 10^{60}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -2.876852397051050336397941792630169519921 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\sqrt{b \cdot b + \left(-\left(4 \cdot a\right) \cdot c\right)} - b}}{2 \cdot a}\\ \mathbf{elif}\;b \le -1.124423417365483683607573317023703677331 \cdot 10^{-68}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 9.282387544777820560455992664410231115991 \cdot 10^{92}:\\ \;\;\;\;\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot b}{2 \cdot a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019347 
(FPCore (a b c)
  :name "The quadratic formula (r2)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))