Average Error: 33.0 → 10.8
Time: 21.2s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}{a}}{2}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -9.348931433494438 \cdot 10^{+39}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\
\;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}{a}}{2}\\

\mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\
\;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}{a}}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\end{array}
double f(double a, double b, double c) {
        double r3629569 = b;
        double r3629570 = -r3629569;
        double r3629571 = r3629569 * r3629569;
        double r3629572 = 4.0;
        double r3629573 = a;
        double r3629574 = c;
        double r3629575 = r3629573 * r3629574;
        double r3629576 = r3629572 * r3629575;
        double r3629577 = r3629571 - r3629576;
        double r3629578 = sqrt(r3629577);
        double r3629579 = r3629570 + r3629578;
        double r3629580 = 2.0;
        double r3629581 = r3629580 * r3629573;
        double r3629582 = r3629579 / r3629581;
        return r3629582;
}
↓
double f(double a, double b, double c) {
        double r3629583 = b;
        double r3629584 = -9.348931433494438e+39;
        bool r3629585 = r3629583 <= r3629584;
        double r3629586 = c;
        double r3629587 = r3629586 / r3629583;
        double r3629588 = a;
        double r3629589 = r3629583 / r3629588;
        double r3629590 = r3629587 - r3629589;
        double r3629591 = 2.0;
        double r3629592 = r3629590 * r3629591;
        double r3629593 = r3629592 / r3629591;
        double r3629594 = 1.3353078790738604e-121;
        bool r3629595 = r3629583 <= r3629594;
        double r3629596 = -4.0;
        double r3629597 = r3629588 * r3629586;
        double r3629598 = r3629596 * r3629597;
        double r3629599 = r3629583 * r3629583;
        double r3629600 = r3629598 + r3629599;
        double r3629601 = sqrt(r3629600);
        double r3629602 = r3629601 - r3629583;
        double r3629603 = r3629602 / r3629588;
        double r3629604 = r3629603 / r3629591;
        double r3629605 = 1.6168702840263923e-79;
        bool r3629606 = r3629583 <= r3629605;
        double r3629607 = -2.0;
        double r3629608 = r3629607 * r3629587;
        double r3629609 = r3629608 / r3629591;
        double r3629610 = 1.546013236023957e-67;
        bool r3629611 = r3629583 <= r3629610;
        double r3629612 = r3629611 ? r3629604 : r3629609;
        double r3629613 = r3629606 ? r3629609 : r3629612;
        double r3629614 = r3629595 ? r3629604 : r3629613;
        double r3629615 = r3629585 ? r3629593 : r3629614;
        return r3629615;
}

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

Original33.0
Target20.1
Herbie10.8
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -9.348931433494438e+39

    1. Initial program 34.0

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

    2. Simplified34.0

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

    3. Taylor expanded around -inf 6.2

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c}{b} - 2 \cdot \frac{b}{a}}}{2}\]

    4. Simplified6.2

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

    if -9.348931433494438e+39 < b < 1.3353078790738604e-121 or 1.6168702840263923e-79 < b < 1.546013236023957e-67

    1. Initial program 12.9

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

    2. Simplified12.9

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

    4. Applied div-inv13.0

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

    6. Applied associate-*r/12.9

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

    7. Simplified12.9

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

    if 1.3353078790738604e-121 < b < 1.6168702840263923e-79 or 1.546013236023957e-67 < b

    1. Initial program 50.8

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

    2. Simplified50.8

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

    3. Taylor expanded around inf 11.2

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -9.348931433494438 \cdot 10^{+39}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.3353078790738604 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}{a}}{2}\\ \mathbf{elif}\;b \le 1.6168702840263923 \cdot 10^{-79}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 1.546013236023957 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\sqrt{-4 \cdot \left(a \cdot c\right) + b \cdot b} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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)))