Average Error: 34.1 → 10.5
Time: 20.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.825478720088060668779950456669858064189 \cdot 10^{107}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{b}{a} \cdot 0.5\\ \end{array}\]
\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 -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 4.825478720088060668779950456669858064189 \cdot 10^{107}:\\
\;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{b}{a} \cdot 0.5\\

\end{array}
double f(double a, double b, double c) {
        double r4406761 = b;
        double r4406762 = -r4406761;
        double r4406763 = r4406761 * r4406761;
        double r4406764 = 4.0;
        double r4406765 = a;
        double r4406766 = c;
        double r4406767 = r4406765 * r4406766;
        double r4406768 = r4406764 * r4406767;
        double r4406769 = r4406763 - r4406768;
        double r4406770 = sqrt(r4406769);
        double r4406771 = r4406762 - r4406770;
        double r4406772 = 2.0;
        double r4406773 = r4406772 * r4406765;
        double r4406774 = r4406771 / r4406773;
        return r4406774;
}
↓
double f(double a, double b, double c) {
        double r4406775 = b;
        double r4406776 = -9.332433396832084e-58;
        bool r4406777 = r4406775 <= r4406776;
        double r4406778 = -1.0;
        double r4406779 = c;
        double r4406780 = r4406779 / r4406775;
        double r4406781 = r4406778 * r4406780;
        double r4406782 = 4.8254787200880607e+107;
        bool r4406783 = r4406775 <= r4406782;
        double r4406784 = 2.0;
        double r4406785 = a;
        double r4406786 = r4406784 * r4406785;
        double r4406787 = r4406775 / r4406786;
        double r4406788 = -r4406787;
        double r4406789 = r4406775 * r4406775;
        double r4406790 = 4.0;
        double r4406791 = r4406785 * r4406779;
        double r4406792 = r4406790 * r4406791;
        double r4406793 = r4406789 - r4406792;
        double r4406794 = sqrt(r4406793);
        double r4406795 = r4406794 / r4406786;
        double r4406796 = r4406788 - r4406795;
        double r4406797 = r4406775 / r4406785;
        double r4406798 = 0.5;
        double r4406799 = r4406797 * r4406798;
        double r4406800 = r4406788 - r4406799;
        double r4406801 = r4406783 ? r4406796 : r4406800;
        double r4406802 = r4406777 ? r4406781 : r4406801;
        return r4406802;
}

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
Target21.4
Herbie10.5
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.332433396832084e-58

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 8.7

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

    if -9.332433396832084e-58 < b < 4.8254787200880607e+107

    1. Initial program 14.1

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

    3. Applied div-sub14.1

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

    if 4.8254787200880607e+107 < b

    1. Initial program 49.2

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

    3. Applied div-sub49.2

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

    5. Applied clear-num49.2

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

    6. Taylor expanded around 0 3.7

      \[\leadsto \frac{-b}{2 \cdot a} - \color{blue}{0.5 \cdot \frac{b}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.332433396832084322962138528577137922234 \cdot 10^{-58}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.825478720088060668779950456669858064189 \cdot 10^{107}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{b}{2 \cdot a}\right) - \frac{b}{a} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

  :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)))