Average Error: 33.2 → 10.7
Time: 19.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.2415082771065304 \cdot 10^{-131}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le 2.559678284282607 \cdot 10^{+69}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]
\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}:\\
\;\;\;\;\frac{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}{a \cdot 2}\\

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

\end{array}
double f(double a, double b, double c) {
        double r903526 = b;
        double r903527 = -r903526;
        double r903528 = r903526 * r903526;
        double r903529 = 4.0;
        double r903530 = a;
        double r903531 = c;
        double r903532 = r903530 * r903531;
        double r903533 = r903529 * r903532;
        double r903534 = r903528 - r903533;
        double r903535 = sqrt(r903534);
        double r903536 = r903527 - r903535;
        double r903537 = 2.0;
        double r903538 = r903537 * r903530;
        double r903539 = r903536 / r903538;
        return r903539;
}
↓
double f(double a, double b, double c) {
        double r903540 = b;
        double r903541 = -2.2415082771065304e-131;
        bool r903542 = r903540 <= r903541;
        double r903543 = c;
        double r903544 = r903543 / r903540;
        double r903545 = -r903544;
        double r903546 = 2.559678284282607e+69;
        bool r903547 = r903540 <= r903546;
        double r903548 = -r903540;
        double r903549 = -4.0;
        double r903550 = a;
        double r903551 = r903549 * r903550;
        double r903552 = r903551 * r903543;
        double r903553 = r903540 * r903540;
        double r903554 = r903552 + r903553;
        double r903555 = sqrt(r903554);
        double r903556 = r903548 - r903555;
        double r903557 = 2.0;
        double r903558 = r903550 * r903557;
        double r903559 = r903556 / r903558;
        double r903560 = r903540 / r903550;
        double r903561 = r903544 - r903560;
        double r903562 = r903547 ? r903559 : r903561;
        double r903563 = r903542 ? r903545 : r903562;
        return r903563;
}

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. Using strategy rm

    3. Applied div-inv49.6

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

    4. Simplified49.6

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

    5. Taylor expanded around -inf 12.4

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

    6. 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. Taylor expanded around -inf 11.4

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

    3. Simplified11.4

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

    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 38.9

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

    3. Simplified39.0

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

    4. 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

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

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)))