Average Error: 34.3 → 8.7
Time: 6.2s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.8746290509448952 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.1885507100476909 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]
\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 -1.8746290509448952 \cdot 10^{74}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le -1.1885507100476909 \cdot 10^{-281}:\\
\;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\

\mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r77594 = b;
        double r77595 = -r77594;
        double r77596 = r77594 * r77594;
        double r77597 = 4.0;
        double r77598 = a;
        double r77599 = c;
        double r77600 = r77598 * r77599;
        double r77601 = r77597 * r77600;
        double r77602 = r77596 - r77601;
        double r77603 = sqrt(r77602);
        double r77604 = r77595 - r77603;
        double r77605 = 2.0;
        double r77606 = r77605 * r77598;
        double r77607 = r77604 / r77606;
        return r77607;
}
↓
double f(double a, double b, double c) {
        double r77608 = b;
        double r77609 = -1.8746290509448952e+74;
        bool r77610 = r77608 <= r77609;
        double r77611 = -1.0;
        double r77612 = c;
        double r77613 = r77612 / r77608;
        double r77614 = r77611 * r77613;
        double r77615 = -1.1885507100476909e-281;
        bool r77616 = r77608 <= r77615;
        double r77617 = 1.0;
        double r77618 = 2.0;
        double r77619 = r77617 / r77618;
        double r77620 = 2.0;
        double r77621 = pow(r77608, r77620);
        double r77622 = r77621 - r77621;
        double r77623 = 4.0;
        double r77624 = a;
        double r77625 = r77624 * r77612;
        double r77626 = r77623 * r77625;
        double r77627 = r77622 + r77626;
        double r77628 = r77619 * r77627;
        double r77629 = r77628 / r77624;
        double r77630 = -r77608;
        double r77631 = r77608 * r77608;
        double r77632 = r77631 - r77626;
        double r77633 = sqrt(r77632);
        double r77634 = r77630 + r77633;
        double r77635 = r77629 / r77634;
        double r77636 = 1.1800329617120703e+123;
        bool r77637 = r77608 <= r77636;
        double r77638 = r77618 * r77624;
        double r77639 = r77630 - r77633;
        double r77640 = r77638 / r77639;
        double r77641 = r77617 / r77640;
        double r77642 = r77608 / r77624;
        double r77643 = r77611 * r77642;
        double r77644 = r77637 ? r77641 : r77643;
        double r77645 = r77616 ? r77635 : r77644;
        double r77646 = r77610 ? r77614 : r77645;
        return r77646;
}

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

Original34.3
Target21.3
Herbie8.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -1.8746290509448952e+74

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 3.3

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

    if -1.8746290509448952e+74 < b < -1.1885507100476909e-281

    1. Initial program 31.9

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

    3. Applied clear-num32.0

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

    5. Applied flip--32.0

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

    6. Applied associate-/r/32.0

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

    7. Applied associate-/r*32.1

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

    8. Simplified15.9

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

    if -1.1885507100476909e-281 < b < 1.1800329617120703e+123

    1. Initial program 9.3

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

    3. Applied clear-num9.4

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

    if 1.1800329617120703e+123 < b

    1. Initial program 53.2

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

    3. Applied clear-num53.3

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

    4. Taylor expanded around 0 3.2

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.8746290509448952 \cdot 10^{74}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le -1.1885507100476909 \cdot 10^{-281}:\\ \;\;\;\;\frac{\frac{\frac{1}{2} \cdot \left(\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(a \cdot c\right)\right)}{a}}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}\\ \mathbf{elif}\;b \le 1.1800329617120703 \cdot 10^{123}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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)))