Average Error: 34.0 → 9.3
Time: 6.8s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.5069461462218695 \cdot 10^{125}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.16799708835065593 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -8.5069461462218695 \cdot 10^{125}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\
\;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\

\mathbf{elif}\;b \le 9.16799708835065593 \cdot 10^{-7}:\\
\;\;\;\;\frac{\frac{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}\\

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

\end{array}
double f(double a, double b, double c) {
        double r91782 = b;
        double r91783 = -r91782;
        double r91784 = r91782 * r91782;
        double r91785 = 4.0;
        double r91786 = a;
        double r91787 = r91785 * r91786;
        double r91788 = c;
        double r91789 = r91787 * r91788;
        double r91790 = r91784 - r91789;
        double r91791 = sqrt(r91790);
        double r91792 = r91783 + r91791;
        double r91793 = 2.0;
        double r91794 = r91793 * r91786;
        double r91795 = r91792 / r91794;
        return r91795;
}
↓
double f(double a, double b, double c) {
        double r91796 = b;
        double r91797 = -8.50694614622187e+125;
        bool r91798 = r91796 <= r91797;
        double r91799 = 1.0;
        double r91800 = c;
        double r91801 = r91800 / r91796;
        double r91802 = a;
        double r91803 = r91796 / r91802;
        double r91804 = r91801 - r91803;
        double r91805 = r91799 * r91804;
        double r91806 = 2.2742398392973687e-86;
        bool r91807 = r91796 <= r91806;
        double r91808 = -r91796;
        double r91809 = r91796 * r91796;
        double r91810 = 4.0;
        double r91811 = r91810 * r91802;
        double r91812 = r91811 * r91800;
        double r91813 = r91809 - r91812;
        double r91814 = sqrt(r91813);
        double r91815 = r91808 + r91814;
        double r91816 = 1.0;
        double r91817 = 2.0;
        double r91818 = r91817 * r91802;
        double r91819 = r91816 / r91818;
        double r91820 = r91815 * r91819;
        double r91821 = 9.167997088350656e-07;
        bool r91822 = r91796 <= r91821;
        double r91823 = 0.0;
        double r91824 = r91802 * r91800;
        double r91825 = r91810 * r91824;
        double r91826 = r91823 + r91825;
        double r91827 = r91808 - r91814;
        double r91828 = r91826 / r91827;
        double r91829 = r91828 / r91818;
        double r91830 = -1.0;
        double r91831 = r91830 * r91801;
        double r91832 = r91822 ? r91829 : r91831;
        double r91833 = r91807 ? r91820 : r91832;
        double r91834 = r91798 ? r91805 : r91833;
        return r91834;
}

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
Target20.6
Herbie9.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -8.50694614622187e+125

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 2.7

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

    3. Simplified2.7

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

    if -8.50694614622187e+125 < b < 2.2742398392973687e-86

    1. Initial program 12.3

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

    3. Applied div-inv12.5

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

    if 2.2742398392973687e-86 < b < 9.167997088350656e-07

    1. Initial program 38.3

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

    3. Applied flip-+38.3

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

    4. Simplified18.8

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

    if 9.167997088350656e-07 < b

    1. Initial program 55.8

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

    2. Taylor expanded around inf 5.7

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.5069461462218695 \cdot 10^{125}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 2.2742398392973687 \cdot 10^{-86}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 9.16799708835065593 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))