Average Error: 34.0 → 9.6
Time: 5.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.5940112039867074 \cdot 10^{100}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r89664 = b;
        double r89665 = -r89664;
        double r89666 = r89664 * r89664;
        double r89667 = 4.0;
        double r89668 = a;
        double r89669 = r89667 * r89668;
        double r89670 = c;
        double r89671 = r89669 * r89670;
        double r89672 = r89666 - r89671;
        double r89673 = sqrt(r89672);
        double r89674 = r89665 + r89673;
        double r89675 = 2.0;
        double r89676 = r89675 * r89668;
        double r89677 = r89674 / r89676;
        return r89677;
}
↓
double f(double a, double b, double c) {
        double r89678 = b;
        double r89679 = -3.5940112039867074e+100;
        bool r89680 = r89678 <= r89679;
        double r89681 = 1.0;
        double r89682 = c;
        double r89683 = r89682 / r89678;
        double r89684 = a;
        double r89685 = r89678 / r89684;
        double r89686 = r89683 - r89685;
        double r89687 = r89681 * r89686;
        double r89688 = 2.267195199467958e-82;
        bool r89689 = r89678 <= r89688;
        double r89690 = 1.0;
        double r89691 = 2.0;
        double r89692 = r89691 * r89684;
        double r89693 = -r89678;
        double r89694 = r89678 * r89678;
        double r89695 = 4.0;
        double r89696 = r89695 * r89684;
        double r89697 = r89696 * r89682;
        double r89698 = r89694 - r89697;
        double r89699 = sqrt(r89698);
        double r89700 = r89693 + r89699;
        double r89701 = r89692 / r89700;
        double r89702 = r89690 / r89701;
        double r89703 = -1.0;
        double r89704 = r89703 * r89683;
        double r89705 = r89689 ? r89702 : r89704;
        double r89706 = r89680 ? r89687 : r89705;
        return r89706;
}

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

Original34.0
Target20.7
Herbie9.6
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -3.5940112039867074e+100

    1. Initial program 47.3

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

    2. Taylor expanded around -inf 3.8

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

    3. Simplified3.8

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

    if -3.5940112039867074e+100 < b < 2.267195199467958e-82

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 2.267195199467958e-82 < b

    1. Initial program 52.9

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

    2. Taylor expanded around inf 9.0

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

  4. Final simplification9.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.5940112039867074 \cdot 10^{100}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.267195199467958 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (a b c)
  :name "The quadratic formula (r1)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- 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)))