Falkner and Boettcher, Appendix A

Average Error: 2.3 → 0.1

Time: 4.9s

Precision: 64

Math

\[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

↓

\[\begin{array}{l} \mathbf{if}\;k \le 8.93171350438392027 \cdot 10^{94}:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}}}{\sqrt{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)\\ \end{array}\]

C

double f(double a, double k, double m) {
        double r323236 = a;
        double r323237 = k;
        double r323238 = m;
        double r323239 = pow(r323237, r323238);
        double r323240 = r323236 * r323239;
        double r323241 = 1.0;
        double r323242 = 10.0;
        double r323243 = r323242 * r323237;
        double r323244 = r323241 + r323243;
        double r323245 = r323237 * r323237;
        double r323246 = r323244 + r323245;
        double r323247 = r323240 / r323246;
        return r323247;
}

↓

double f(double a, double k, double m) {
        double r323248 = k;
        double r323249 = 8.93171350438392e+94;
        bool r323250 = r323248 <= r323249;
        double r323251 = a;
        double r323252 = m;
        double r323253 = pow(r323248, r323252);
        double r323254 = r323251 * r323253;
        double r323255 = 1.0;
        double r323256 = 10.0;
        double r323257 = 1.0;
        double r323258 = fma(r323256, r323248, r323257);
        double r323259 = fma(r323248, r323248, r323258);
        double r323260 = sqrt(r323259);
        double r323261 = r323255 / r323260;
        double r323262 = r323261 / r323260;
        double r323263 = r323254 * r323262;
        double r323264 = -1.0;
        double r323265 = r323255 / r323248;
        double r323266 = log(r323265);
        double r323267 = r323252 * r323266;
        double r323268 = r323264 * r323267;
        double r323269 = exp(r323268);
        double r323270 = r323269 / r323248;
        double r323271 = r323251 / r323248;
        double r323272 = 99.0;
        double r323273 = r323251 * r323269;
        double r323274 = 4.0;
        double r323275 = pow(r323248, r323274);
        double r323276 = r323273 / r323275;
        double r323277 = r323272 * r323276;
        double r323278 = 3.0;
        double r323279 = pow(r323248, r323278);
        double r323280 = r323273 / r323279;
        double r323281 = r323256 * r323280;
        double r323282 = r323277 - r323281;
        double r323283 = fma(r323270, r323271, r323282);
        double r323284 = r323250 ? r323263 : r323283;
        return r323284;
}

Error

Bits error versus a

Bits error versus k

Bits error versus m

Derivation

1. Split input into 2 regimes

2. if k < 8.93171350438392e+94

   1. Initial program 0.1

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
   2. Using strategy rm

   3. Applied div-inv 0.1

      \[\leadsto \color{blue}{\left(a \cdot {k}^{m}\right) \cdot \frac{1}{\left(1 + 10 \cdot k\right) + k \cdot k}}\]

   4. Simplified 0.1

      \[\leadsto \left(a \cdot {k}^{m}\right) \cdot \color{blue}{\frac{1}{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 0.2

      \[\leadsto \left(a \cdot {k}^{m}\right) \cdot \frac{1}{\color{blue}{\sqrt{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)} \cdot \sqrt{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}}}\]

   7. Applied associate-/r* 0.2

      \[\leadsto \left(a \cdot {k}^{m}\right) \cdot \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}}}{\sqrt{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}}}\]

   if 8.93171350438392e+94 < k

   1. Initial program 8.3

      \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]

   2. Taylor expanded around inf 8.3

      \[\leadsto \color{blue}{\left(\frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{2}} + 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}}\right) - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}}\]

   3. Simplified 0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \le 8.93171350438392027 \cdot 10^{94}:\\ \;\;\;\;\left(a \cdot {k}^{m}\right) \cdot \frac{\frac{1}{\sqrt{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}}}{\sqrt{\mathsf{fma}\left(k, k, \mathsf{fma}\left(10, k, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{k}, \frac{a}{k}, 99 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{4}} - 10 \cdot \frac{a \cdot e^{-1 \cdot \left(m \cdot \log \left(\frac{1}{k}\right)\right)}}{{k}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  :precision binary64
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))