Average Error: 34.0 → 12.4
Time: 6.0s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.64064137297236348 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.5846768506177574 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -1.64064137297236348 \cdot 10^{109}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.5846768506177574 \cdot 10^{-31}:\\
\;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r241647 = b;
        double r241648 = -r241647;
        double r241649 = r241647 * r241647;
        double r241650 = 4.0;
        double r241651 = a;
        double r241652 = r241650 * r241651;
        double r241653 = c;
        double r241654 = r241652 * r241653;
        double r241655 = r241649 - r241654;
        double r241656 = sqrt(r241655);
        double r241657 = r241648 + r241656;
        double r241658 = 2.0;
        double r241659 = r241658 * r241651;
        double r241660 = r241657 / r241659;
        return r241660;
}
↓
double f(double a, double b, double c) {
        double r241661 = b;
        double r241662 = -1.6406413729723635e+109;
        bool r241663 = r241661 <= r241662;
        double r241664 = 1.0;
        double r241665 = c;
        double r241666 = r241665 / r241661;
        double r241667 = a;
        double r241668 = r241661 / r241667;
        double r241669 = r241666 - r241668;
        double r241670 = r241664 * r241669;
        double r241671 = 1.5846768506177574e-31;
        bool r241672 = r241661 <= r241671;
        double r241673 = 4.0;
        double r241674 = 2.0;
        double r241675 = pow(r241661, r241674);
        double r241676 = r241675 - r241675;
        double r241677 = r241667 * r241665;
        double r241678 = r241673 * r241677;
        double r241679 = r241676 + r241678;
        double r241680 = r241679 / r241677;
        double r241681 = r241673 / r241680;
        double r241682 = -r241661;
        double r241683 = r241661 * r241661;
        double r241684 = r241673 * r241667;
        double r241685 = r241684 * r241665;
        double r241686 = r241683 - r241685;
        double r241687 = sqrt(r241686);
        double r241688 = r241682 + r241687;
        double r241689 = r241681 * r241688;
        double r241690 = 2.0;
        double r241691 = r241690 * r241667;
        double r241692 = r241689 / r241691;
        double r241693 = -1.0;
        double r241694 = r241693 * r241666;
        double r241695 = r241672 ? r241692 : r241694;
        double r241696 = r241663 ? r241670 : r241695;
        return r241696;
}

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
Target20.8
Herbie12.4
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.6406413729723635e+109

    1. Initial program 49.5

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

    2. Taylor expanded around -inf 3.2

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

    3. Simplified3.2

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

    if -1.6406413729723635e+109 < b < 1.5846768506177574e-31

    1. Initial program 14.5

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

    3. Applied flip-+31.2

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

    4. Simplified28.6

      \[\leadsto \frac{\frac{\color{blue}{0 + 4 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
    5. Using strategy rm

    6. Applied flip--31.5

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}}{2 \cdot a}\]

    7. Applied associate-/r/31.6

      \[\leadsto \frac{\color{blue}{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

    8. Simplified18.6

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

    if 1.5846768506177574e-31 < b

    1. Initial program 54.3

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

    2. Taylor expanded around inf 7.7

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

  4. Final simplification12.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.64064137297236348 \cdot 10^{109}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.5846768506177574 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{4}{\frac{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)}{a \cdot c}} \cdot \left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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)))