Average Error: 34.1 → 8.2
Time: 6.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\
\;\;\;\;\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 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\
\;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r220667 = b;
        double r220668 = -r220667;
        double r220669 = r220667 * r220667;
        double r220670 = 4.0;
        double r220671 = a;
        double r220672 = r220670 * r220671;
        double r220673 = c;
        double r220674 = r220672 * r220673;
        double r220675 = r220669 - r220674;
        double r220676 = sqrt(r220675);
        double r220677 = r220668 + r220676;
        double r220678 = 2.0;
        double r220679 = r220678 * r220671;
        double r220680 = r220677 / r220679;
        return r220680;
}
↓
double f(double a, double b, double c) {
        double r220681 = b;
        double r220682 = -3.974595954042362e+78;
        bool r220683 = r220681 <= r220682;
        double r220684 = 1.0;
        double r220685 = c;
        double r220686 = r220685 / r220681;
        double r220687 = a;
        double r220688 = r220681 / r220687;
        double r220689 = r220686 - r220688;
        double r220690 = r220684 * r220689;
        double r220691 = -3.106669992168921e-249;
        bool r220692 = r220681 <= r220691;
        double r220693 = -r220681;
        double r220694 = r220681 * r220681;
        double r220695 = 4.0;
        double r220696 = r220695 * r220687;
        double r220697 = r220696 * r220685;
        double r220698 = r220694 - r220697;
        double r220699 = sqrt(r220698);
        double r220700 = r220693 + r220699;
        double r220701 = 1.0;
        double r220702 = 2.0;
        double r220703 = r220702 * r220687;
        double r220704 = r220701 / r220703;
        double r220705 = r220700 * r220704;
        double r220706 = 3.3294485045805704e-14;
        bool r220707 = r220681 <= r220706;
        double r220708 = r220701 / r220687;
        double r220709 = r220693 - r220699;
        double r220710 = r220709 / r220685;
        double r220711 = r220708 * r220710;
        double r220712 = r220695 / r220711;
        double r220713 = r220701 * r220712;
        double r220714 = r220713 / r220703;
        double r220715 = -1.0;
        double r220716 = r220715 * r220686;
        double r220717 = r220707 ? r220714 : r220716;
        double r220718 = r220692 ? r220705 : r220717;
        double r220719 = r220683 ? r220690 : r220718;
        return r220719;
}

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

Original34.1
Target20.7
Herbie8.2
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.974595954042362e+78

    1. Initial program 41.5

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

    2. Taylor expanded around -inf 4.2

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

    3. Simplified4.2

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

    if -3.974595954042362e+78 < b < -3.106669992168921e-249

    1. Initial program 8.2

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

    3. Applied div-inv8.3

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

    if -3.106669992168921e-249 < b < 3.3294485045805704e-14

    1. Initial program 23.6

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

    3. Applied flip-+23.7

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

    4. Simplified17.5

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

    6. Applied *-un-lft-identity17.5

      \[\leadsto \frac{\frac{0 + 4 \cdot \left(a \cdot c\right)}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

    7. Applied *-un-lft-identity17.5

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + 4 \cdot \left(a \cdot c\right)\right)}}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}{2 \cdot a}\]

    8. Applied times-frac17.5

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

    9. Simplified17.5

      \[\leadsto \frac{\color{blue}{1} \cdot \frac{0 + 4 \cdot \left(a \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]

    10. Simplified17.5

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

    12. Applied *-un-lft-identity17.5

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

    13. Applied times-frac14.8

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

    if 3.3294485045805704e-14 < b

    1. Initial program 55.3

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

    2. Taylor expanded around inf 5.9

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

  4. Final simplification8.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.974595954042361881691403492534140168485 \cdot 10^{78}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -3.1066699921689209753658832131181641309 \cdot 10^{-249}:\\ \;\;\;\;\left(\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 3.329448504580570356283886843099725433358 \cdot 10^{-14}:\\ \;\;\;\;\frac{1 \cdot \frac{4}{\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{c}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(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)))