Average Error: 33.9 → 6.8
Time: 18.8s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.359953003549156817553996908233908949771 \cdot 10^{103}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.094358742794727790656239317142702500789 \cdot 10^{-239}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 5.099089738165329086098741767888130630655 \cdot 10^{67}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -3.359953003549156817553996908233908949771 \cdot 10^{103}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le 2.094358742794727790656239317142702500789 \cdot 10^{-239}:\\
\;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\

\mathbf{elif}\;b_2 \le 5.099089738165329086098741767888130630655 \cdot 10^{67}:\\
\;\;\;\;\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 r95657 = b_2;
        double r95658 = -r95657;
        double r95659 = r95657 * r95657;
        double r95660 = a;
        double r95661 = c;
        double r95662 = r95660 * r95661;
        double r95663 = r95659 - r95662;
        double r95664 = sqrt(r95663);
        double r95665 = r95658 - r95664;
        double r95666 = r95665 / r95660;
        return r95666;
}
↓
double f(double a, double b_2, double c) {
        double r95667 = b_2;
        double r95668 = -3.359953003549157e+103;
        bool r95669 = r95667 <= r95668;
        double r95670 = -0.5;
        double r95671 = c;
        double r95672 = r95671 / r95667;
        double r95673 = r95670 * r95672;
        double r95674 = 2.094358742794728e-239;
        bool r95675 = r95667 <= r95674;
        double r95676 = -r95667;
        double r95677 = r95667 * r95667;
        double r95678 = a;
        double r95679 = r95678 * r95671;
        double r95680 = r95677 - r95679;
        double r95681 = sqrt(r95680);
        double r95682 = r95676 + r95681;
        double r95683 = r95671 / r95682;
        double r95684 = 5.099089738165329e+67;
        bool r95685 = r95667 <= r95684;
        double r95686 = r95676 - r95681;
        double r95687 = r95686 / r95678;
        double r95688 = 0.5;
        double r95689 = r95688 * r95672;
        double r95690 = 2.0;
        double r95691 = r95667 / r95678;
        double r95692 = r95690 * r95691;
        double r95693 = r95689 - r95692;
        double r95694 = r95685 ? r95687 : r95693;
        double r95695 = r95675 ? r95683 : r95694;
        double r95696 = r95669 ? r95673 : r95695;
        return r95696;
}

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 < -3.359953003549157e+103

    1. Initial program 59.7

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

    2. Taylor expanded around -inf 2.5

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

    if -3.359953003549157e+103 < b_2 < 2.094358742794728e-239

    1. Initial program 30.6

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

    3. Applied clear-num30.6

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

    5. Applied flip--30.7

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

    6. Applied associate-/r/30.8

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

    7. Applied associate-/r*30.8

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

    8. Simplified15.4

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

    9. Taylor expanded around 0 9.5

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

    if 2.094358742794728e-239 < b_2 < 5.099089738165329e+67

    1. Initial program 8.0

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

    if 5.099089738165329e+67 < b_2

    1. Initial program 40.4

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

    2. Taylor expanded around inf 5.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.359953003549156817553996908233908949771 \cdot 10^{103}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le 2.094358742794727790656239317142702500789 \cdot 10^{-239}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\\ \mathbf{elif}\;b_2 \le 5.099089738165329086098741767888130630655 \cdot 10^{67}:\\ \;\;\;\;\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  :precision binary64
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))