Average Error: 33.8 → 9.7
Time: 14.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.6257289292067596 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}}}{2}\\ \mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -1.6257289292067596 \cdot 10^{+144}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}}}{2}\\

\mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\
\;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3097668 = b;
        double r3097669 = -r3097668;
        double r3097670 = r3097668 * r3097668;
        double r3097671 = 4.0;
        double r3097672 = a;
        double r3097673 = r3097671 * r3097672;
        double r3097674 = c;
        double r3097675 = r3097673 * r3097674;
        double r3097676 = r3097670 - r3097675;
        double r3097677 = sqrt(r3097676);
        double r3097678 = r3097669 + r3097677;
        double r3097679 = 2.0;
        double r3097680 = r3097679 * r3097672;
        double r3097681 = r3097678 / r3097680;
        return r3097681;
}
↓
double f(double a, double b, double c) {
        double r3097682 = b;
        double r3097683 = -1.6257289292067596e+144;
        bool r3097684 = r3097682 <= r3097683;
        double r3097685 = c;
        double r3097686 = r3097685 / r3097682;
        double r3097687 = a;
        double r3097688 = r3097682 / r3097687;
        double r3097689 = r3097686 - r3097688;
        double r3097690 = 2.0;
        double r3097691 = r3097689 * r3097690;
        double r3097692 = r3097691 / r3097690;
        double r3097693 = 1.739098950628615e-79;
        bool r3097694 = r3097682 <= r3097693;
        double r3097695 = 1.0;
        double r3097696 = r3097682 * r3097682;
        double r3097697 = r3097687 * r3097685;
        double r3097698 = -4.0;
        double r3097699 = r3097697 * r3097698;
        double r3097700 = r3097696 + r3097699;
        double r3097701 = sqrt(r3097700);
        double r3097702 = r3097701 - r3097682;
        double r3097703 = r3097687 / r3097702;
        double r3097704 = r3097695 / r3097703;
        double r3097705 = r3097704 / r3097690;
        double r3097706 = 1.8656332031849816e-25;
        bool r3097707 = r3097682 <= r3097706;
        double r3097708 = -2.0;
        double r3097709 = r3097708 * r3097686;
        double r3097710 = r3097709 / r3097690;
        double r3097711 = 5.297236684235463e-16;
        bool r3097712 = r3097682 <= r3097711;
        double r3097713 = r3097712 ? r3097705 : r3097710;
        double r3097714 = r3097707 ? r3097710 : r3097713;
        double r3097715 = r3097694 ? r3097705 : r3097714;
        double r3097716 = r3097684 ? r3097692 : r3097715;
        return r3097716;
}

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

Original33.8
Target20.3
Herbie9.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.6257289292067596e+144

    1. Initial program 58.0

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

    2. Simplified58.0

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

    3. Taylor expanded around -inf 2.5

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified2.5

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

    if -1.6257289292067596e+144 < b < 1.739098950628615e-79 or 1.8656332031849816e-25 < b < 5.297236684235463e-16

    1. Initial program 12.3

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

    2. Simplified12.3

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

    4. Applied clear-num12.4

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

    6. Applied clear-num12.4

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

    7. Simplified12.4

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

    if 1.739098950628615e-79 < b < 1.8656332031849816e-25 or 5.297236684235463e-16 < b

    1. Initial program 53.2

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

    2. Simplified53.2

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

    3. Taylor expanded around inf 8.2

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6257289292067596 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.739098950628615 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}}}{2}\\ \mathbf{elif}\;b \le 1.8656332031849816 \cdot 10^{-25}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 5.297236684235463 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{1}{\frac{a}{\sqrt{b \cdot b + \left(a \cdot c\right) \cdot -4} - b}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(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)))