Average Error: 34.1 → 7.1
Time: 43.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.089942740476039 \cdot 10^{+37}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.1606217053284985 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.9653089193303188 \cdot 10^{+135}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -5.089942740476039 \cdot 10^{+37}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.1606217053284985 \cdot 10^{-301}:\\
\;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\

\mathbf{elif}\;b \le 1.9653089193303188 \cdot 10^{+135}:\\
\;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3809555 = b;
        double r3809556 = -r3809555;
        double r3809557 = r3809555 * r3809555;
        double r3809558 = 4.0;
        double r3809559 = a;
        double r3809560 = c;
        double r3809561 = r3809559 * r3809560;
        double r3809562 = r3809558 * r3809561;
        double r3809563 = r3809557 - r3809562;
        double r3809564 = sqrt(r3809563);
        double r3809565 = r3809556 + r3809564;
        double r3809566 = 2.0;
        double r3809567 = r3809566 * r3809559;
        double r3809568 = r3809565 / r3809567;
        return r3809568;
}
↓
double f(double a, double b, double c) {
        double r3809569 = b;
        double r3809570 = -5.089942740476039e+37;
        bool r3809571 = r3809569 <= r3809570;
        double r3809572 = c;
        double r3809573 = r3809572 / r3809569;
        double r3809574 = a;
        double r3809575 = r3809569 / r3809574;
        double r3809576 = r3809573 - r3809575;
        double r3809577 = 1.1606217053284985e-301;
        bool r3809578 = r3809569 <= r3809577;
        double r3809579 = r3809569 * r3809569;
        double r3809580 = 4.0;
        double r3809581 = r3809574 * r3809580;
        double r3809582 = r3809581 * r3809572;
        double r3809583 = r3809579 - r3809582;
        double r3809584 = sqrt(r3809583);
        double r3809585 = r3809584 - r3809569;
        double r3809586 = 2.0;
        double r3809587 = r3809574 * r3809586;
        double r3809588 = r3809585 / r3809587;
        double r3809589 = 1.9653089193303188e+135;
        bool r3809590 = r3809569 <= r3809589;
        double r3809591 = -2.0;
        double r3809592 = r3809572 * r3809591;
        double r3809593 = r3809584 + r3809569;
        double r3809594 = r3809592 / r3809593;
        double r3809595 = -r3809573;
        double r3809596 = r3809590 ? r3809594 : r3809595;
        double r3809597 = r3809578 ? r3809588 : r3809596;
        double r3809598 = r3809571 ? r3809576 : r3809597;
        return r3809598;
}

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
Herbie7.1
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 4 regimes

  2. if b < -5.089942740476039e+37

    1. Initial program 34.9

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

    2. Simplified34.9

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

    4. Applied div-inv35.0

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

    5. Taylor expanded around -inf 6.3

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

    if -5.089942740476039e+37 < b < 1.1606217053284985e-301

    1. Initial program 10.3

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

    2. Simplified10.3

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

    if 1.1606217053284985e-301 < b < 1.9653089193303188e+135

    1. Initial program 34.6

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

    2. Simplified34.6

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

    4. Applied div-inv34.6

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

    6. Applied flip--34.7

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

    7. Applied associate-*l/34.7

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

    8. Simplified14.6

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

    9. Taylor expanded around 0 8.3

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

    if 1.9653089193303188e+135 < b

    1. Initial program 61.2

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

    2. Simplified61.2

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

    3. Taylor expanded around inf 2.1

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

    4. Simplified2.1

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.089942740476039 \cdot 10^{+37}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.1606217053284985 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.9653089193303188 \cdot 10^{+135}:\\ \;\;\;\;\frac{c \cdot -2}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 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)))