Average Error: 32.7 → 10.0
Time: 28.1s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -7.3975762435547 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]
\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 -7.3975762435547 \cdot 10^{+118}:\\
\;\;\;\;\frac{c}{b} - \frac{b}{a}\\

\mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r3274607 = b;
        double r3274608 = -r3274607;
        double r3274609 = r3274607 * r3274607;
        double r3274610 = 4.0;
        double r3274611 = a;
        double r3274612 = c;
        double r3274613 = r3274611 * r3274612;
        double r3274614 = r3274610 * r3274613;
        double r3274615 = r3274609 - r3274614;
        double r3274616 = sqrt(r3274615);
        double r3274617 = r3274608 + r3274616;
        double r3274618 = 2.0;
        double r3274619 = r3274618 * r3274611;
        double r3274620 = r3274617 / r3274619;
        return r3274620;
}
↓
double f(double a, double b, double c) {
        double r3274621 = b;
        double r3274622 = -7.3975762435547e+118;
        bool r3274623 = r3274621 <= r3274622;
        double r3274624 = c;
        double r3274625 = r3274624 / r3274621;
        double r3274626 = a;
        double r3274627 = r3274621 / r3274626;
        double r3274628 = r3274625 - r3274627;
        double r3274629 = 1.3115303715225787e-131;
        bool r3274630 = r3274621 <= r3274629;
        double r3274631 = r3274621 * r3274621;
        double r3274632 = 4.0;
        double r3274633 = r3274624 * r3274626;
        double r3274634 = r3274632 * r3274633;
        double r3274635 = r3274631 - r3274634;
        double r3274636 = sqrt(r3274635);
        double r3274637 = r3274636 - r3274621;
        double r3274638 = 2.0;
        double r3274639 = r3274637 / r3274638;
        double r3274640 = r3274639 / r3274626;
        double r3274641 = -r3274625;
        double r3274642 = r3274630 ? r3274640 : r3274641;
        double r3274643 = r3274623 ? r3274628 : r3274642;
        return r3274643;
}

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

Original32.7
Target20.0
Herbie10.0
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -7.3975762435547e+118

    1. Initial program 49.0

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

    2. Simplified49.0

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

    3. Taylor expanded around -inf 3.0

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

    if -7.3975762435547e+118 < b < 1.3115303715225787e-131

    1. Initial program 10.7

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

    2. Simplified10.7

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

    3. Taylor expanded around inf 10.7

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

    4. Simplified10.7

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

    if 1.3115303715225787e-131 < b

    1. Initial program 50.3

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

    2. Simplified50.3

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

    3. Taylor expanded around inf 50.3

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

    4. Simplified50.3

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

    5. Taylor expanded around inf 11.7

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

    6. Simplified11.7

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -7.3975762435547 \cdot 10^{+118}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.3115303715225787 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(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)))