Average Error: 33.8 → 6.8
Time: 6.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.6559623908913229 \cdot 10^{-256}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.1445535679869069 \cdot 10^{60}:\\ \;\;\;\;\frac{1}{\left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{0.5}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\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 -3.12428337420519208 \cdot 10^{57}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le -2.6559623908913229 \cdot 10^{-256}:\\
\;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\

\mathbf{elif}\;b \le 4.1445535679869069 \cdot 10^{60}:\\
\;\;\;\;\frac{1}{\left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{0.5}{c}}\\

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

\end{array}
double f(double a, double b, double c) {
        double r56673 = b;
        double r56674 = -r56673;
        double r56675 = r56673 * r56673;
        double r56676 = 4.0;
        double r56677 = a;
        double r56678 = r56676 * r56677;
        double r56679 = c;
        double r56680 = r56678 * r56679;
        double r56681 = r56675 - r56680;
        double r56682 = sqrt(r56681);
        double r56683 = r56674 + r56682;
        double r56684 = 2.0;
        double r56685 = r56684 * r56677;
        double r56686 = r56683 / r56685;
        return r56686;
}
↓
double f(double a, double b, double c) {
        double r56687 = b;
        double r56688 = -3.124283374205192e+57;
        bool r56689 = r56687 <= r56688;
        double r56690 = 1.0;
        double r56691 = c;
        double r56692 = r56691 / r56687;
        double r56693 = a;
        double r56694 = r56687 / r56693;
        double r56695 = r56692 - r56694;
        double r56696 = r56690 * r56695;
        double r56697 = -2.655962390891323e-256;
        bool r56698 = r56687 <= r56697;
        double r56699 = -r56687;
        double r56700 = r56687 * r56687;
        double r56701 = 4.0;
        double r56702 = r56701 * r56693;
        double r56703 = r56702 * r56691;
        double r56704 = r56700 - r56703;
        double r56705 = sqrt(r56704);
        double r56706 = r56699 + r56705;
        double r56707 = 2.0;
        double r56708 = r56707 * r56693;
        double r56709 = r56706 / r56708;
        double r56710 = 4.144553567986907e+60;
        bool r56711 = r56687 <= r56710;
        double r56712 = 1.0;
        double r56713 = 2.0;
        double r56714 = pow(r56687, r56713);
        double r56715 = r56693 * r56691;
        double r56716 = r56701 * r56715;
        double r56717 = r56714 - r56716;
        double r56718 = sqrt(r56717);
        double r56719 = r56699 - r56718;
        double r56720 = 0.5;
        double r56721 = r56720 / r56691;
        double r56722 = r56719 * r56721;
        double r56723 = r56712 / r56722;
        double r56724 = -1.0;
        double r56725 = r56724 * r56692;
        double r56726 = r56711 ? r56723 : r56725;
        double r56727 = r56698 ? r56709 : r56726;
        double r56728 = r56689 ? r56696 : r56727;
        return r56728;
}

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

Original33.8
Target20.4
Herbie6.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -3.124283374205192e+57

    1. Initial program 39.5

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

    2. Taylor expanded around -inf 5.4

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

    3. Simplified5.4

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

    if -3.124283374205192e+57 < b < -2.655962390891323e-256

    1. Initial program 8.1

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

    if -2.655962390891323e-256 < b < 4.144553567986907e+60

    1. Initial program 28.5

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

    3. Applied flip-+28.5

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

    4. Simplified16.8

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

    6. Applied clear-num16.9

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

    7. Simplified16.6

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

    8. Taylor expanded around 0 10.1

      \[\leadsto \frac{1}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right) \cdot \color{blue}{\frac{0.5}{c}}}\]

    9. Taylor expanded around 0 10.1

      \[\leadsto \frac{1}{\left(\left(-b\right) - \sqrt{\color{blue}{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}\right) \cdot \frac{0.5}{c}}\]

    if 4.144553567986907e+60 < b

    1. Initial program 58.0

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

    2. Taylor expanded around inf 3.1

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

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.12428337420519208 \cdot 10^{57}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le -2.6559623908913229 \cdot 10^{-256}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{elif}\;b \le 4.1445535679869069 \cdot 10^{60}:\\ \;\;\;\;\frac{1}{\left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{0.5}{c}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))