Average Error: 33.9 → 10.0
Time: 4.6s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -8.7237121667270365 \cdot 10^{113}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.51349304478110011 \cdot 10^{-103}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -8.7237121667270365 \cdot 10^{113}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.51349304478110011 \cdot 10^{-103}:\\
\;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\

\end{array}
double f(double a, double b_2, double c) {
        double r79474 = b_2;
        double r79475 = -r79474;
        double r79476 = r79474 * r79474;
        double r79477 = a;
        double r79478 = c;
        double r79479 = r79477 * r79478;
        double r79480 = r79476 - r79479;
        double r79481 = sqrt(r79480);
        double r79482 = r79475 - r79481;
        double r79483 = r79482 / r79477;
        return r79483;
}
↓
double f(double a, double b_2, double c) {
        double r79484 = b_2;
        double r79485 = -8.723712166727036e+113;
        bool r79486 = r79484 <= r79485;
        double r79487 = -0.5;
        double r79488 = c;
        double r79489 = r79488 / r79484;
        double r79490 = r79487 * r79489;
        double r79491 = 1.5134930447811e-103;
        bool r79492 = r79484 <= r79491;
        double r79493 = 1.0;
        double r79494 = r79493 / r79488;
        double r79495 = r79484 * r79484;
        double r79496 = a;
        double r79497 = r79496 * r79488;
        double r79498 = r79495 - r79497;
        double r79499 = sqrt(r79498);
        double r79500 = r79499 - r79484;
        double r79501 = r79494 * r79500;
        double r79502 = r79493 / r79501;
        double r79503 = 0.5;
        double r79504 = r79503 * r79489;
        double r79505 = 2.0;
        double r79506 = r79484 / r79496;
        double r79507 = r79505 * r79506;
        double r79508 = r79504 - r79507;
        double r79509 = r79492 ? r79502 : r79508;
        double r79510 = r79486 ? r79490 : r79509;
        return r79510;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -8.723712166727036e+113

    1. Initial program 60.6

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 2.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]

    if -8.723712166727036e+113 < b_2 < 1.5134930447811e-103

    1. Initial program 26.8

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied flip--28.5

      \[\leadsto \frac{\color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}}{a}\]

    4. Simplified17.0

      \[\leadsto \frac{\frac{\color{blue}{0 + a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\]

    5. Simplified17.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity17.0

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}}{a}\]

    8. Applied *-un-lft-identity17.0

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(0 + a \cdot c\right)}}{1 \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}{a}\]

    9. Applied times-frac17.0

      \[\leadsto \frac{\color{blue}{\frac{1}{1} \cdot \frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]

    10. Applied associate-/l*17.2

      \[\leadsto \color{blue}{\frac{\frac{1}{1}}{\frac{a}{\frac{0 + a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}}\]

    11. Simplified16.4

      \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{a}{a \cdot c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}}\]
    12. Using strategy rm

    13. Applied clear-num16.4

      \[\leadsto \frac{\frac{1}{1}}{\color{blue}{\frac{1}{\frac{a \cdot c}{a}}} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\]

    14. Simplified12.1

      \[\leadsto \frac{\frac{1}{1}}{\frac{1}{\color{blue}{c}} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\]

    if 1.5134930447811e-103 < b_2

    1. Initial program 25.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 12.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -8.7237121667270365 \cdot 10^{113}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.51349304478110011 \cdot 10^{-103}:\\ \;\;\;\;\frac{1}{\frac{1}{c} \cdot \left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))