Average Error: 34.2 → 8.9
Time: 18.0s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.290038544730787402474284646819000838132 \cdot 10^{100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.071138839477091811227998523018378891081 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{\frac{a}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\ \mathbf{elif}\;b_2 \le 1.4947314724287081461477196158248694513 \cdot 10^{91}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -1.290038544730787402474284646819000838132 \cdot 10^{100}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 1.071138839477091811227998523018378891081 \cdot 10^{-291}:\\
\;\;\;\;\frac{\frac{\frac{a}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\

\mathbf{elif}\;b_2 \le 1.4947314724287081461477196158248694513 \cdot 10^{91}:\\
\;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{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 r132541 = b_2;
        double r132542 = -r132541;
        double r132543 = r132541 * r132541;
        double r132544 = a;
        double r132545 = c;
        double r132546 = r132544 * r132545;
        double r132547 = r132543 - r132546;
        double r132548 = sqrt(r132547);
        double r132549 = r132542 - r132548;
        double r132550 = r132549 / r132544;
        return r132550;
}
↓
double f(double a, double b_2, double c) {
        double r132551 = b_2;
        double r132552 = -1.2900385447307874e+100;
        bool r132553 = r132551 <= r132552;
        double r132554 = -0.5;
        double r132555 = c;
        double r132556 = r132555 / r132551;
        double r132557 = r132554 * r132556;
        double r132558 = 1.0711388394770918e-291;
        bool r132559 = r132551 <= r132558;
        double r132560 = a;
        double r132561 = r132551 * r132551;
        double r132562 = r132560 * r132555;
        double r132563 = r132561 - r132562;
        double r132564 = sqrt(r132563);
        double r132565 = r132564 - r132551;
        double r132566 = sqrt(r132565);
        double r132567 = r132566 / r132555;
        double r132568 = r132560 / r132567;
        double r132569 = r132568 / r132566;
        double r132570 = r132569 / r132560;
        double r132571 = 1.4947314724287081e+91;
        bool r132572 = r132551 <= r132571;
        double r132573 = -r132551;
        double r132574 = r132573 - r132564;
        double r132575 = 1.0;
        double r132576 = r132575 / r132560;
        double r132577 = r132574 * r132576;
        double r132578 = 0.5;
        double r132579 = r132578 * r132556;
        double r132580 = 2.0;
        double r132581 = r132551 / r132560;
        double r132582 = r132580 * r132581;
        double r132583 = r132579 - r132582;
        double r132584 = r132572 ? r132577 : r132583;
        double r132585 = r132559 ? r132570 : r132584;
        double r132586 = r132553 ? r132557 : r132585;
        return r132586;
}

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 4 regimes

  2. if b_2 < -1.2900385447307874e+100

    1. Initial program 59.4

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

    2. Taylor expanded around -inf 2.6

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

    if -1.2900385447307874e+100 < b_2 < 1.0711388394770918e-291

    1. Initial program 31.7

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

    3. Applied flip--31.8

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

    4. Simplified17.3

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

    5. Simplified17.3

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

    7. Applied add-sqr-sqrt17.5

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

    8. Applied associate-/r*17.5

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

    9. Simplified16.0

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

    if 1.0711388394770918e-291 < b_2 < 1.4947314724287081e+91

    1. Initial program 8.7

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

    3. Applied div-inv8.8

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

    if 1.4947314724287081e+91 < b_2

    1. Initial program 45.7

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

    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 4 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.290038544730787402474284646819000838132 \cdot 10^{100}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 1.071138839477091811227998523018378891081 \cdot 10^{-291}:\\ \;\;\;\;\frac{\frac{\frac{a}{\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{c}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\\ \mathbf{elif}\;b_2 \le 1.4947314724287081461477196158248694513 \cdot 10^{91}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))