Average Error: 34.2 → 8.3
Time: 15.6s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.450829996567047685966692456342790556879 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.829508521752968200470883584177554571598 \cdot 10^{-234}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{2}\\ \mathbf{elif}\;b \le 2.080207944760409970231511154919136216465 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{4 \cdot a}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]
\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 -3.450829996567047685966692456342790556879 \cdot 10^{138}:\\
\;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\

\mathbf{elif}\;b \le 2.829508521752968200470883584177554571598 \cdot 10^{-234}:\\
\;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{2}\\

\mathbf{elif}\;b \le 2.080207944760409970231511154919136216465 \cdot 10^{-46}:\\
\;\;\;\;\frac{\frac{4 \cdot a}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{c}{b} \cdot -1\\

\end{array}
double f(double a, double b, double c) {
        double r147714 = b;
        double r147715 = -r147714;
        double r147716 = r147714 * r147714;
        double r147717 = 4.0;
        double r147718 = a;
        double r147719 = r147717 * r147718;
        double r147720 = c;
        double r147721 = r147719 * r147720;
        double r147722 = r147716 - r147721;
        double r147723 = sqrt(r147722);
        double r147724 = r147715 + r147723;
        double r147725 = 2.0;
        double r147726 = r147725 * r147718;
        double r147727 = r147724 / r147726;
        return r147727;
}
↓
double f(double a, double b, double c) {
        double r147728 = b;
        double r147729 = -3.450829996567048e+138;
        bool r147730 = r147728 <= r147729;
        double r147731 = c;
        double r147732 = r147731 / r147728;
        double r147733 = a;
        double r147734 = r147728 / r147733;
        double r147735 = r147732 - r147734;
        double r147736 = 1.0;
        double r147737 = r147735 * r147736;
        double r147738 = 2.829508521752968e-234;
        bool r147739 = r147728 <= r147738;
        double r147740 = r147728 * r147728;
        double r147741 = 4.0;
        double r147742 = r147741 * r147733;
        double r147743 = r147742 * r147731;
        double r147744 = r147740 - r147743;
        double r147745 = sqrt(r147744);
        double r147746 = -r147728;
        double r147747 = r147745 + r147746;
        double r147748 = 1.0;
        double r147749 = r147748 / r147733;
        double r147750 = 2.0;
        double r147751 = r147749 / r147750;
        double r147752 = r147747 * r147751;
        double r147753 = 2.08020794476041e-46;
        bool r147754 = r147728 <= r147753;
        double r147755 = r147742 / r147750;
        double r147756 = r147731 * r147733;
        double r147757 = r147756 * r147741;
        double r147758 = r147740 - r147757;
        double r147759 = sqrt(r147758);
        double r147760 = r147746 - r147759;
        double r147761 = r147731 / r147760;
        double r147762 = r147755 * r147761;
        double r147763 = r147762 / r147733;
        double r147764 = -1.0;
        double r147765 = r147732 * r147764;
        double r147766 = r147754 ? r147763 : r147765;
        double r147767 = r147739 ? r147752 : r147766;
        double r147768 = r147730 ? r147737 : r147767;
        return r147768;
}

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

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

Derivation

  1. Split input into 4 regimes

  2. if b < -3.450829996567048e+138

    1. Initial program 58.5

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -3.450829996567048e+138 < b < 2.829508521752968e-234

    1. Initial program 8.7

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

    3. Applied div-inv8.9

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

    4. Simplified8.9

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

    if 2.829508521752968e-234 < b < 2.08020794476041e-46

    1. Initial program 25.7

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

    3. Applied flip-+25.7

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

    4. Simplified19.0

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

    6. Applied associate-/r*19.0

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

    7. Simplified14.7

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

    if 2.08020794476041e-46 < b

    1. Initial program 54.2

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

    2. Taylor expanded around inf 7.6

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

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.450829996567047685966692456342790556879 \cdot 10^{138}:\\ \;\;\;\;\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 1\\ \mathbf{elif}\;b \le 2.829508521752968200470883584177554571598 \cdot 10^{-234}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)\right) \cdot \frac{\frac{1}{a}}{2}\\ \mathbf{elif}\;b \le 2.080207944760409970231511154919136216465 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{4 \cdot a}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -1\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "The quadratic formula (r1)"

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

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