Average Error: 34.4 → 10.3
Time: 5.3s
Precision: 64
\[\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]
\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 -1.0674124610604968 \cdot 10^{-82}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\mathbf{elif}\;b \le 5.96876625840091586 \cdot 10^{107}:\\
\;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\

\mathbf{else}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\end{array}
double f(double a, double b, double c) {
        double r81577 = b;
        double r81578 = -r81577;
        double r81579 = r81577 * r81577;
        double r81580 = 4.0;
        double r81581 = a;
        double r81582 = c;
        double r81583 = r81581 * r81582;
        double r81584 = r81580 * r81583;
        double r81585 = r81579 - r81584;
        double r81586 = sqrt(r81585);
        double r81587 = r81578 - r81586;
        double r81588 = 2.0;
        double r81589 = r81588 * r81581;
        double r81590 = r81587 / r81589;
        return r81590;
}
↓
double f(double a, double b, double c) {
        double r81591 = b;
        double r81592 = -1.0674124610604968e-82;
        bool r81593 = r81591 <= r81592;
        double r81594 = -1.0;
        double r81595 = c;
        double r81596 = r81595 / r81591;
        double r81597 = r81594 * r81596;
        double r81598 = 5.968766258400916e+107;
        bool r81599 = r81591 <= r81598;
        double r81600 = 1.0;
        double r81601 = -r81591;
        double r81602 = r81591 * r81591;
        double r81603 = 4.0;
        double r81604 = a;
        double r81605 = r81604 * r81595;
        double r81606 = r81603 * r81605;
        double r81607 = r81602 - r81606;
        double r81608 = sqrt(r81607);
        double r81609 = r81601 - r81608;
        double r81610 = 2.0;
        double r81611 = r81610 * r81604;
        double r81612 = r81609 / r81611;
        double r81613 = r81600 * r81612;
        double r81614 = 1.0;
        double r81615 = r81591 / r81604;
        double r81616 = r81596 - r81615;
        double r81617 = r81614 * r81616;
        double r81618 = r81599 ? r81613 : r81617;
        double r81619 = r81593 ? r81597 : r81618;
        return r81619;
}

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

Original34.4
Target21.5
Herbie10.3
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.0674124610604968e-82

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 8.9

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

    if -1.0674124610604968e-82 < b < 5.968766258400916e+107

    1. Initial program 13.7

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

    3. Applied clear-num13.9

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

    5. Applied *-un-lft-identity13.9

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

    6. Applied add-cube-cbrt13.9

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot \frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    7. Applied times-frac13.9

      \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    8. Simplified13.9

      \[\leadsto \color{blue}{1} \cdot \frac{\sqrt[3]{1}}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\]

    9. Simplified13.7

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

    if 5.968766258400916e+107 < b

    1. Initial program 50.0

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

    2. Taylor expanded around inf 3.8

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

    3. Simplified3.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.0674124610604968 \cdot 10^{-82}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \le 5.96876625840091586 \cdot 10^{107}:\\ \;\;\;\;1 \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (a b c)
  :name "quadm (p42, negative)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ 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)))