Average Error: 34.1 → 10.0
Time: 14.9s
Precision: 64
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b_2 \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b_2 \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]
\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\

\mathbf{elif}\;b_2 \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\
\;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\

\mathbf{elif}\;b_2 \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\
\;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\

\mathbf{elif}\;b_2 \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\
\;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\

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

\end{array}
double f(double a, double b_2, double c) {
        double r58092 = b_2;
        double r58093 = -r58092;
        double r58094 = r58092 * r58092;
        double r58095 = a;
        double r58096 = c;
        double r58097 = r58095 * r58096;
        double r58098 = r58094 - r58097;
        double r58099 = sqrt(r58098);
        double r58100 = r58093 - r58099;
        double r58101 = r58100 / r58095;
        return r58101;
}
↓
double f(double a, double b_2, double c) {
        double r58102 = b_2;
        double r58103 = -5.84038254482515e+46;
        bool r58104 = r58102 <= r58103;
        double r58105 = c;
        double r58106 = r58105 / r58102;
        double r58107 = -0.5;
        double r58108 = r58106 * r58107;
        double r58109 = -7.877985662156599e-94;
        bool r58110 = r58102 <= r58109;
        double r58111 = a;
        double r58112 = r58111 * r58105;
        double r58113 = r58102 * r58102;
        double r58114 = r58113 - r58112;
        double r58115 = sqrt(r58114);
        double r58116 = r58115 - r58102;
        double r58117 = r58112 / r58116;
        double r58118 = r58117 / r58111;
        double r58119 = -6.596302400897662e-136;
        bool r58120 = r58102 <= r58119;
        double r58121 = 7.501979458872916e+77;
        bool r58122 = r58102 <= r58121;
        double r58123 = 1.0;
        double r58124 = -r58102;
        double r58125 = r58124 - r58115;
        double r58126 = r58111 / r58125;
        double r58127 = r58123 / r58126;
        double r58128 = -2.0;
        double r58129 = r58102 / r58111;
        double r58130 = r58128 * r58129;
        double r58131 = r58122 ? r58127 : r58130;
        double r58132 = r58120 ? r58108 : r58131;
        double r58133 = r58110 ? r58118 : r58132;
        double r58134 = r58104 ? r58108 : r58133;
        return r58134;
}

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 < -5.84038254482515e+46 or -7.877985662156599e-94 < b_2 < -6.596302400897662e-136

    1. Initial program 54.3

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

    2. Taylor expanded around -inf 8.3

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

    3. Simplified8.3

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

    if -5.84038254482515e+46 < b_2 < -7.877985662156599e-94

    1. Initial program 40.2

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

    3. Applied flip--40.2

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

    4. Simplified15.2

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

    5. Simplified15.2

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

    if -6.596302400897662e-136 < b_2 < 7.501979458872916e+77

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    if 7.501979458872916e+77 < b_2

    1. Initial program 42.5

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

    3. Applied clear-num42.6

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

    4. Taylor expanded around 0 5.1

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

    5. Simplified5.1

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.840382544825149510322162525528307154775 \cdot 10^{46}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b_2 \le -7.877985662156598668725484528840176897607 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{a \cdot c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}\\ \mathbf{elif}\;b_2 \le -6.596302400897661869317839215315745353488 \cdot 10^{-136}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;b_2 \le 7.501979458872916117674264090696641915837 \cdot 10^{77}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))