Average Error: 33.6 → 10.4
Time: 21.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -2.1144981103869975 \cdot 10^{+131}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\
\;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{2}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3783990 = b;
        double r3783991 = -r3783990;
        double r3783992 = r3783990 * r3783990;
        double r3783993 = 4.0;
        double r3783994 = a;
        double r3783995 = r3783993 * r3783994;
        double r3783996 = c;
        double r3783997 = r3783995 * r3783996;
        double r3783998 = r3783992 - r3783997;
        double r3783999 = sqrt(r3783998);
        double r3784000 = r3783991 + r3783999;
        double r3784001 = 2.0;
        double r3784002 = r3784001 * r3783994;
        double r3784003 = r3784000 / r3784002;
        return r3784003;
}
↓
double f(double a, double b, double c) {
        double r3784004 = b;
        double r3784005 = -2.1144981103869975e+131;
        bool r3784006 = r3784004 <= r3784005;
        double r3784007 = c;
        double r3784008 = r3784007 / r3784004;
        double r3784009 = a;
        double r3784010 = r3784004 / r3784009;
        double r3784011 = r3784008 - r3784010;
        double r3784012 = 4.5810084990875205e-68;
        bool r3784013 = r3784004 <= r3784012;
        double r3784014 = 1.0;
        double r3784015 = r3784004 * r3784004;
        double r3784016 = -4.0;
        double r3784017 = r3784016 * r3784009;
        double r3784018 = r3784007 * r3784017;
        double r3784019 = r3784015 + r3784018;
        double r3784020 = sqrt(r3784019);
        double r3784021 = r3784020 - r3784004;
        double r3784022 = 2.0;
        double r3784023 = r3784021 / r3784022;
        double r3784024 = r3784009 / r3784023;
        double r3784025 = r3784014 / r3784024;
        double r3784026 = -r3784008;
        double r3784027 = r3784013 ? r3784025 : r3784026;
        double r3784028 = r3784006 ? r3784011 : r3784027;
        return r3784028;
}

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

Original33.6
Target21.0
Herbie10.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.1144981103869975e+131

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 2.6

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

    if -2.1144981103869975e+131 < b < 4.5810084990875205e-68

    1. Initial program 13.3

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

    3. Applied div-inv13.5

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

    4. Simplified13.5

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

    6. Applied associate-*r/13.3

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

    7. Simplified13.3

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

    9. Applied clear-num13.4

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

    if 4.5810084990875205e-68 < b

    1. Initial program 52.0

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

    2. Taylor expanded around inf 9.3

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

    3. Simplified9.3

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

  4. Final simplification10.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1144981103869975 \cdot 10^{+131}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 4.5810084990875205 \cdot 10^{-68}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\sqrt{b \cdot b + c \cdot \left(-4 \cdot a\right)} - b}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))