Average Error: 6.0 → 2.6
Time: 8.0s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 8.802051227216260578823091484650999242207 \cdot 10^{100}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) + \left(\frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\\ \mathbf{elif}\;x \le 4.717658198318118893542685725997000036653 \cdot 10^{216}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\mathsf{fma}\left(0.4000000000000064059868520871532382443547 \cdot x, z, 12.00000000000004796163466380676254630089 \cdot x - 0.1009522780952416126654114236771420110017 \cdot \left(x \cdot {z}^{2}\right)\right)} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\\ \end{array}\]
\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le 8.802051227216260578823091484650999242207 \cdot 10^{100}:\\
\;\;\;\;\log x \cdot \left(x - 0.5\right) + \left(\frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\\

\mathbf{elif}\;x \le 4.717658198318118893542685725997000036653 \cdot 10^{216}:\\
\;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\mathsf{fma}\left(0.4000000000000064059868520871532382443547 \cdot x, z, 12.00000000000004796163466380676254630089 \cdot x - 0.1009522780952416126654114236771420110017 \cdot \left(x \cdot {z}^{2}\right)\right)} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\\

\end{array}
double f(double x, double y, double z) {
        double r474268 = x;
        double r474269 = 0.5;
        double r474270 = r474268 - r474269;
        double r474271 = log(r474268);
        double r474272 = r474270 * r474271;
        double r474273 = r474272 - r474268;
        double r474274 = 0.91893853320467;
        double r474275 = r474273 + r474274;
        double r474276 = y;
        double r474277 = 0.0007936500793651;
        double r474278 = r474276 + r474277;
        double r474279 = z;
        double r474280 = r474278 * r474279;
        double r474281 = 0.0027777777777778;
        double r474282 = r474280 - r474281;
        double r474283 = r474282 * r474279;
        double r474284 = 0.083333333333333;
        double r474285 = r474283 + r474284;
        double r474286 = r474285 / r474268;
        double r474287 = r474275 + r474286;
        return r474287;
}
↓
double f(double x, double y, double z) {
        double r474288 = x;
        double r474289 = 8.80205122721626e+100;
        bool r474290 = r474288 <= r474289;
        double r474291 = log(r474288);
        double r474292 = 0.5;
        double r474293 = r474288 - r474292;
        double r474294 = r474291 * r474293;
        double r474295 = y;
        double r474296 = 0.0007936500793651;
        double r474297 = r474295 + r474296;
        double r474298 = z;
        double r474299 = r474297 * r474298;
        double r474300 = 0.0027777777777778;
        double r474301 = r474299 - r474300;
        double r474302 = r474301 * r474298;
        double r474303 = 0.083333333333333;
        double r474304 = r474302 + r474303;
        double r474305 = r474304 / r474288;
        double r474306 = 0.91893853320467;
        double r474307 = r474288 - r474306;
        double r474308 = r474305 - r474307;
        double r474309 = r474294 + r474308;
        double r474310 = 4.717658198318119e+216;
        bool r474311 = r474288 <= r474310;
        double r474312 = 2.0;
        double r474313 = pow(r474298, r474312);
        double r474314 = r474313 / r474288;
        double r474315 = r474296 * r474314;
        double r474316 = 1.0;
        double r474317 = r474316 / r474288;
        double r474318 = log(r474317);
        double r474319 = fma(r474318, r474288, r474288);
        double r474320 = r474315 - r474319;
        double r474321 = fma(r474314, r474295, r474320);
        double r474322 = 0.4000000000000064;
        double r474323 = r474322 * r474288;
        double r474324 = 12.000000000000048;
        double r474325 = r474324 * r474288;
        double r474326 = 0.10095227809524161;
        double r474327 = r474288 * r474313;
        double r474328 = r474326 * r474327;
        double r474329 = r474325 - r474328;
        double r474330 = fma(r474323, r474298, r474329);
        double r474331 = r474316 / r474330;
        double r474332 = r474331 - r474307;
        double r474333 = fma(r474291, r474293, r474332);
        double r474334 = r474311 ? r474321 : r474333;
        double r474335 = r474290 ? r474309 : r474334;
        return r474335;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.0
Target1.2
Herbie2.6
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.9189385332046700050057097541866824030876 - x\right)\right) + \frac{0.08333333333333299564049667651488562114537}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - 0.002777777777777800001512975569539776188321\right)\]

Derivation

  1. Split input into 3 regimes

  2. if x < 8.80205122721626e+100

    1. Initial program 1.2

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

    2. Simplified1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - 0.5, \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)}\]
    3. Using strategy rm

    4. Applied fma-udef1.2

      \[\leadsto \color{blue}{\log x \cdot \left(x - 0.5\right) + \left(\frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)}\]

    if 8.80205122721626e+100 < x < 4.717658198318119e+216

    1. Initial program 10.0

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

    2. Simplified10.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - 0.5, \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)}\]

    3. Taylor expanded around inf 10.1

      \[\leadsto \color{blue}{\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - \left(x + x \cdot \log \left(\frac{1}{x}\right)\right)}\]

    4. Simplified6.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)}\]

    if 4.717658198318119e+216 < x

    1. Initial program 15.7

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

    2. Simplified15.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log x, x - 0.5, \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)}\]
    3. Using strategy rm

    4. Applied clear-num15.6

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \color{blue}{\frac{1}{\frac{x}{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}}} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\]

    5. Taylor expanded around 0 14.1

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\color{blue}{\left(0.4000000000000064059868520871532382443547 \cdot \left(x \cdot z\right) + 12.00000000000004796163466380676254630089 \cdot x\right) - 0.1009522780952416126654114236771420110017 \cdot \left(x \cdot {z}^{2}\right)}} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\]

    6. Simplified2.9

      \[\leadsto \mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\color{blue}{\mathsf{fma}\left(0.4000000000000064059868520871532382443547 \cdot x, z, 12.00000000000004796163466380676254630089 \cdot x - 0.1009522780952416126654114236771420110017 \cdot \left(x \cdot {z}^{2}\right)\right)}} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification2.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 8.802051227216260578823091484650999242207 \cdot 10^{100}:\\ \;\;\;\;\log x \cdot \left(x - 0.5\right) + \left(\frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\\ \mathbf{elif}\;x \le 4.717658198318118893542685725997000036653 \cdot 10^{216}:\\ \;\;\;\;\mathsf{fma}\left(\frac{{z}^{2}}{x}, y, 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} - \mathsf{fma}\left(\log \left(\frac{1}{x}\right), x, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log x, x - 0.5, \frac{1}{\mathsf{fma}\left(0.4000000000000064059868520871532382443547 \cdot x, z, 12.00000000000004796163466380676254630089 \cdot x - 0.1009522780952416126654114236771420110017 \cdot \left(x \cdot {z}^{2}\right)\right)} - \left(x - 0.9189385332046700050057097541866824030876\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))