Average Error: 33.0 → 10.8
Time: 16.3s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.9862044966069494 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.990519652731023 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}{2}\\ \mathbf{elif}\;b \le 1.0350377446088803 \cdot 10^{-69}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 3.325219738594455 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]
\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 -2.9862044966069494 \cdot 10^{+41}:\\
\;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\

\mathbf{elif}\;b \le 1.990519652731023 \cdot 10^{-106}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}{2}\\

\mathbf{elif}\;b \le 1.0350377446088803 \cdot 10^{-69}:\\
\;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\

\mathbf{elif}\;b \le 3.325219738594455 \cdot 10^{-21}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}{2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2559528 = b;
        double r2559529 = -r2559528;
        double r2559530 = r2559528 * r2559528;
        double r2559531 = 4.0;
        double r2559532 = a;
        double r2559533 = c;
        double r2559534 = r2559532 * r2559533;
        double r2559535 = r2559531 * r2559534;
        double r2559536 = r2559530 - r2559535;
        double r2559537 = sqrt(r2559536);
        double r2559538 = r2559529 + r2559537;
        double r2559539 = 2.0;
        double r2559540 = r2559539 * r2559532;
        double r2559541 = r2559538 / r2559540;
        return r2559541;
}
↓
double f(double a, double b, double c) {
        double r2559542 = b;
        double r2559543 = -2.9862044966069494e+41;
        bool r2559544 = r2559542 <= r2559543;
        double r2559545 = c;
        double r2559546 = r2559545 / r2559542;
        double r2559547 = a;
        double r2559548 = r2559542 / r2559547;
        double r2559549 = r2559546 - r2559548;
        double r2559550 = 2.0;
        double r2559551 = r2559549 * r2559550;
        double r2559552 = r2559551 / r2559550;
        double r2559553 = 1.990519652731023e-106;
        bool r2559554 = r2559542 <= r2559553;
        double r2559555 = r2559542 * r2559542;
        double r2559556 = 4.0;
        double r2559557 = r2559556 * r2559547;
        double r2559558 = r2559557 * r2559545;
        double r2559559 = r2559555 - r2559558;
        double r2559560 = sqrt(r2559559);
        double r2559561 = r2559560 - r2559542;
        double r2559562 = r2559561 / r2559547;
        double r2559563 = r2559562 / r2559550;
        double r2559564 = 1.0350377446088803e-69;
        bool r2559565 = r2559542 <= r2559564;
        double r2559566 = -2.0;
        double r2559567 = r2559566 * r2559546;
        double r2559568 = r2559567 / r2559550;
        double r2559569 = 3.325219738594455e-21;
        bool r2559570 = r2559542 <= r2559569;
        double r2559571 = r2559570 ? r2559563 : r2559568;
        double r2559572 = r2559565 ? r2559568 : r2559571;
        double r2559573 = r2559554 ? r2559563 : r2559572;
        double r2559574 = r2559544 ? r2559552 : r2559573;
        return r2559574;
}

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 < -2.9862044966069494e+41

    1. Initial program 34.1

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

    2. Simplified34.1

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

    3. Taylor expanded around -inf 6.3

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

    4. Simplified6.3

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

    if -2.9862044966069494e+41 < b < 1.990519652731023e-106 or 1.0350377446088803e-69 < b < 3.325219738594455e-21

    1. Initial program 14.8

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

    2. Simplified14.8

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

    4. Applied *-un-lft-identity14.8

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

    5. Applied *-un-lft-identity14.8

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

    6. Applied times-frac14.8

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

    7. Simplified14.8

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

    if 1.990519652731023e-106 < b < 1.0350377446088803e-69 or 3.325219738594455e-21 < b

    1. Initial program 53.0

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

    2. Simplified53.1

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

    3. Taylor expanded around inf 8.7

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.9862044966069494 \cdot 10^{+41}:\\ \;\;\;\;\frac{\left(\frac{c}{b} - \frac{b}{a}\right) \cdot 2}{2}\\ \mathbf{elif}\;b \le 1.990519652731023 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}{2}\\ \mathbf{elif}\;b \le 1.0350377446088803 \cdot 10^{-69}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \mathbf{elif}\;b \le 3.325219738594455 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{a}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \frac{c}{b}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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)))