Average Error: 34.3 → 10.2
Time: 6.4s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.4278904486834676 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.4278904486834676 \cdot 10^{-42}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\
\;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

\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 r100326 = b_2;
        double r100327 = -r100326;
        double r100328 = r100326 * r100326;
        double r100329 = a;
        double r100330 = c;
        double r100331 = r100329 * r100330;
        double r100332 = r100328 - r100331;
        double r100333 = sqrt(r100332);
        double r100334 = r100327 - r100333;
        double r100335 = r100334 / r100329;
        return r100335;
}
↓
double f(double a, double b_2, double c) {
        double r100336 = b_2;
        double r100337 = -5.4278904486834676e-42;
        bool r100338 = r100336 <= r100337;
        double r100339 = -0.5;
        double r100340 = c;
        double r100341 = r100340 / r100336;
        double r100342 = r100339 * r100341;
        double r100343 = 2.8046284917653458e+91;
        bool r100344 = r100336 <= r100343;
        double r100345 = -r100336;
        double r100346 = r100336 * r100336;
        double r100347 = a;
        double r100348 = r100347 * r100340;
        double r100349 = r100346 - r100348;
        double r100350 = sqrt(r100349);
        double r100351 = r100345 - r100350;
        double r100352 = r100351 / r100347;
        double r100353 = 0.5;
        double r100354 = r100353 * r100341;
        double r100355 = 2.0;
        double r100356 = r100336 / r100347;
        double r100357 = r100355 * r100356;
        double r100358 = r100354 - r100357;
        double r100359 = r100344 ? r100352 : r100358;
        double r100360 = r100338 ? r100342 : r100359;
        return r100360;
}

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 < -5.4278904486834676e-42

    1. Initial program 54.7

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

    2. Taylor expanded around -inf 7.1

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

    if -5.4278904486834676e-42 < b_2 < 2.8046284917653458e+91

    1. Initial program 14.7

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

    if 2.8046284917653458e+91 < b_2

    1. Initial program 45.3

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

    2. Taylor expanded around inf 4.1

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

  4. Final simplification10.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.4278904486834676 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.8046284917653458 \cdot 10^{91}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))