Average Error: 33.2 → 10.7
Time: 24.0s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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 -2.2415082771065304 \cdot 10^{-131}:\\
\;\;\;\;-\frac{c}{b}\\

\mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\
\;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\

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

\end{array}
double f(double a, double b, double c) {
        double r2143435 = b;
        double r2143436 = -r2143435;
        double r2143437 = r2143435 * r2143435;
        double r2143438 = 4.0;
        double r2143439 = a;
        double r2143440 = c;
        double r2143441 = r2143439 * r2143440;
        double r2143442 = r2143438 * r2143441;
        double r2143443 = r2143437 - r2143442;
        double r2143444 = sqrt(r2143443);
        double r2143445 = r2143436 - r2143444;
        double r2143446 = 2.0;
        double r2143447 = r2143446 * r2143439;
        double r2143448 = r2143445 / r2143447;
        return r2143448;
}
↓
double f(double a, double b, double c) {
        double r2143449 = b;
        double r2143450 = -2.2415082771065304e-131;
        bool r2143451 = r2143449 <= r2143450;
        double r2143452 = c;
        double r2143453 = r2143452 / r2143449;
        double r2143454 = -r2143453;
        double r2143455 = 2.559678284282607e+69;
        bool r2143456 = r2143449 <= r2143455;
        double r2143457 = -r2143449;
        double r2143458 = r2143449 * r2143449;
        double r2143459 = a;
        double r2143460 = r2143452 * r2143459;
        double r2143461 = 4.0;
        double r2143462 = r2143460 * r2143461;
        double r2143463 = r2143458 - r2143462;
        double r2143464 = sqrt(r2143463);
        double r2143465 = r2143457 - r2143464;
        double r2143466 = 0.5;
        double r2143467 = r2143466 / r2143459;
        double r2143468 = r2143465 * r2143467;
        double r2143469 = r2143449 / r2143459;
        double r2143470 = r2143453 - r2143469;
        double r2143471 = r2143456 ? r2143468 : r2143470;
        double r2143472 = r2143451 ? r2143454 : r2143471;
        return r2143472;
}

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

Original33.2
Target19.9
Herbie10.7
\[\begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -2.2415082771065304e-131

    1. Initial program 49.6

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

    2. Taylor expanded around -inf 12.4

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

    3. Simplified12.4

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

    if -2.2415082771065304e-131 < b < 2.559678284282607e+69

    1. Initial program 11.4

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

    3. Applied div-inv11.6

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

    4. Simplified11.5

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

    if 2.559678284282607e+69 < b

    1. Initial program 38.9

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

    2. Taylor expanded around inf 4.8

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4}\right) \cdot \frac{\frac{1}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (a b c)
  :name "quadm (p42, negative)"

  :herbie-target
  (if (< b 0) (/ c (* a (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))) (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))

  (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))