Average Error: 34.2 → 13.9
Time: 19.9s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -5.791348048249166002460683130439961414293 \cdot 10^{138}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 1}{a}\\ \mathbf{elif}\;b \le 1.832703499755311344543445048213516250218 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{b}{a \cdot c}}}{a}\\ \end{array}\]
\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 -5.791348048249166002460683130439961414293 \cdot 10^{138}:\\
\;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 1}{a}\\

\mathbf{elif}\;b \le 1.832703499755311344543445048213516250218 \cdot 10^{-46}:\\
\;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r70359 = b;
        double r70360 = -r70359;
        double r70361 = r70359 * r70359;
        double r70362 = 4.0;
        double r70363 = a;
        double r70364 = c;
        double r70365 = r70363 * r70364;
        double r70366 = r70362 * r70365;
        double r70367 = r70361 - r70366;
        double r70368 = sqrt(r70367);
        double r70369 = r70360 + r70368;
        double r70370 = 2.0;
        double r70371 = r70370 * r70363;
        double r70372 = r70369 / r70371;
        return r70372;
}
↓
double f(double a, double b, double c) {
        double r70373 = b;
        double r70374 = -5.791348048249166e+138;
        bool r70375 = r70373 <= r70374;
        double r70376 = a;
        double r70377 = c;
        double r70378 = r70373 / r70377;
        double r70379 = r70376 / r70378;
        double r70380 = r70379 - r70373;
        double r70381 = 1.0;
        double r70382 = r70380 * r70381;
        double r70383 = r70382 / r70376;
        double r70384 = 1.8327034997553113e-46;
        bool r70385 = r70373 <= r70384;
        double r70386 = r70373 * r70373;
        double r70387 = 4.0;
        double r70388 = r70387 * r70376;
        double r70389 = r70388 * r70377;
        double r70390 = r70386 - r70389;
        double r70391 = sqrt(r70390);
        double r70392 = r70391 - r70373;
        double r70393 = 2.0;
        double r70394 = r70392 / r70393;
        double r70395 = r70394 / r70376;
        double r70396 = -1.0;
        double r70397 = r70376 * r70377;
        double r70398 = r70373 / r70397;
        double r70399 = r70396 / r70398;
        double r70400 = r70399 / r70376;
        double r70401 = r70385 ? r70395 : r70400;
        double r70402 = r70375 ? r70383 : r70401;
        return r70402;
}

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

Original34.2
Target21.0
Herbie13.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -5.791348048249166e+138

    1. Initial program 58.5

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

    2. Simplified58.5

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

    3. Taylor expanded around -inf 9.2

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

    4. Simplified2.0

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

    if -5.791348048249166e+138 < b < 1.8327034997553113e-46

    1. Initial program 13.0

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

    2. Simplified13.0

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

    if 1.8327034997553113e-46 < b

    1. Initial program 54.2

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

    2. Simplified54.2

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

    3. Taylor expanded around inf 18.5

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

    4. Simplified18.9

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

  4. Final simplification13.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -5.791348048249166002460683130439961414293 \cdot 10^{138}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} - b\right) \cdot 1}{a}\\ \mathbf{elif}\;b \le 1.832703499755311344543445048213516250218 \cdot 10^{-46}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{\frac{b}{a \cdot c}}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4.0 (* a c))))) (* 2.0 a)))