Average Error: 34.5 → 10.2
Time: 4.1s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.3645547041066157 \cdot 10^{-80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -8.3645547041066157 \cdot 10^{-80}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r87686 = b;
        double r87687 = -r87686;
        double r87688 = r87686 * r87686;
        double r87689 = 4.0;
        double r87690 = a;
        double r87691 = c;
        double r87692 = r87690 * r87691;
        double r87693 = r87689 * r87692;
        double r87694 = r87688 - r87693;
        double r87695 = sqrt(r87694);
        double r87696 = r87687 - r87695;
        double r87697 = 2.0;
        double r87698 = r87697 * r87690;
        double r87699 = r87696 / r87698;
        return r87699;
}
↓
double f(double a, double b, double c) {
        double r87700 = b;
        double r87701 = -8.364554704106616e-80;
        bool r87702 = r87700 <= r87701;
        double r87703 = -1.0;
        double r87704 = c;
        double r87705 = r87704 / r87700;
        double r87706 = r87703 * r87705;
        double r87707 = 4.1199128263687574e+46;
        bool r87708 = r87700 <= r87707;
        double r87709 = -r87700;
        double r87710 = r87700 * r87700;
        double r87711 = 4.0;
        double r87712 = a;
        double r87713 = r87712 * r87704;
        double r87714 = r87711 * r87713;
        double r87715 = r87710 - r87714;
        double r87716 = sqrt(r87715);
        double r87717 = r87709 - r87716;
        double r87718 = 1.0;
        double r87719 = 2.0;
        double r87720 = r87719 * r87712;
        double r87721 = r87718 / r87720;
        double r87722 = r87717 * r87721;
        double r87723 = 1.0;
        double r87724 = r87700 / r87712;
        double r87725 = r87705 - r87724;
        double r87726 = r87723 * r87725;
        double r87727 = r87708 ? r87722 : r87726;
        double r87728 = r87702 ? r87706 : r87727;
        return r87728;
}

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

Original34.5
Target21.1
Herbie10.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -8.364554704106616e-80

    1. Initial program 53.8

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

    2. Taylor expanded around -inf 9.1

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

    if -8.364554704106616e-80 < b < 4.1199128263687574e+46

    1. Initial program 13.8

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

    3. Applied div-inv13.9

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

    if 4.1199128263687574e+46 < b

    1. Initial program 36.8

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

    2. Taylor expanded around inf 5.2

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

    3. Simplified5.2

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.3645547041066157 \cdot 10^{-80}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 4.1199128263687574 \cdot 10^{46}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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)))