Average Error: 34.4 → 7.1
Time: 11.1s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.30260065699914430088466347088364783298 \cdot 10^{85}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.616213321049155371026664508407530431682 \cdot 10^{-298}:\\ \;\;\;\;c \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}\\ \mathbf{elif}\;b_2 \le 1.140745036075281127251828675303402372066 \cdot 10^{117}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -2.30260065699914430088466347088364783298 \cdot 10^{85}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\mathbf{elif}\;b_2 \le -1.616213321049155371026664508407530431682 \cdot 10^{-298}:\\
\;\;\;\;c \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}\\

\mathbf{elif}\;b_2 \le 1.140745036075281127251828675303402372066 \cdot 10^{117}:\\
\;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r71652 = b_2;
        double r71653 = -r71652;
        double r71654 = r71652 * r71652;
        double r71655 = a;
        double r71656 = c;
        double r71657 = r71655 * r71656;
        double r71658 = r71654 - r71657;
        double r71659 = sqrt(r71658);
        double r71660 = r71653 - r71659;
        double r71661 = r71660 / r71655;
        return r71661;
}
↓
double f(double a, double b_2, double c) {
        double r71662 = b_2;
        double r71663 = -2.3026006569991443e+85;
        bool r71664 = r71662 <= r71663;
        double r71665 = -0.5;
        double r71666 = c;
        double r71667 = r71666 / r71662;
        double r71668 = r71665 * r71667;
        double r71669 = -1.6162133210491554e-298;
        bool r71670 = r71662 <= r71669;
        double r71671 = 1.0;
        double r71672 = a;
        double r71673 = r71666 * r71672;
        double r71674 = -r71673;
        double r71675 = fma(r71662, r71662, r71674);
        double r71676 = sqrt(r71675);
        double r71677 = r71676 - r71662;
        double r71678 = r71671 / r71677;
        double r71679 = r71666 * r71678;
        double r71680 = 1.1407450360752811e+117;
        bool r71681 = r71662 <= r71680;
        double r71682 = -r71662;
        double r71683 = r71682 / r71672;
        double r71684 = r71662 * r71662;
        double r71685 = r71672 * r71666;
        double r71686 = r71684 - r71685;
        double r71687 = sqrt(r71686);
        double r71688 = r71687 / r71672;
        double r71689 = r71683 - r71688;
        double r71690 = r71662 / r71672;
        double r71691 = -2.0;
        double r71692 = r71690 * r71691;
        double r71693 = r71681 ? r71689 : r71692;
        double r71694 = r71670 ? r71679 : r71693;
        double r71695 = r71664 ? r71668 : r71694;
        return r71695;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -2.3026006569991443e+85

    1. Initial program 59.1

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

    2. Taylor expanded around -inf 2.7

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

    if -2.3026006569991443e+85 < b_2 < -1.6162133210491554e-298

    1. Initial program 32.4

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

    3. Applied flip--32.4

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

    4. Simplified17.8

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

    5. Simplified17.8

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}}}{a}\]
    6. Using strategy rm

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

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

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

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

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

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

    10. Applied times-frac17.8

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

    11. Applied times-frac17.8

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

    12. Simplified17.8

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

    13. Simplified9.6

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

    15. Applied div-inv9.7

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

    if -1.6162133210491554e-298 < b_2 < 1.1407450360752811e+117

    1. Initial program 9.6

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

    3. Applied div-sub9.5

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

    if 1.1407450360752811e+117 < b_2

    1. Initial program 50.7

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

    3. Applied flip--63.7

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

    4. Simplified62.7

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

    5. Simplified62.7

      \[\leadsto \frac{\frac{0 + a \cdot c}{\color{blue}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied *-un-lft-identity62.7

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

    8. Applied *-un-lft-identity62.7

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

    9. Applied *-un-lft-identity62.7

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

    10. Applied times-frac62.7

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

    11. Applied times-frac62.7

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

    12. Simplified62.7

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

    13. Simplified62.5

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

    14. Taylor expanded around 0 4.0

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

  4. Final simplification7.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.30260065699914430088466347088364783298 \cdot 10^{85}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -1.616213321049155371026664508407530431682 \cdot 10^{-298}:\\ \;\;\;\;c \cdot \frac{1}{\sqrt{\mathsf{fma}\left(b_2, b_2, -c \cdot a\right)} - b_2}\\ \mathbf{elif}\;b_2 \le 1.140745036075281127251828675303402372066 \cdot 10^{117}:\\ \;\;\;\;\frac{-b_2}{a} - \frac{\sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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))