Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B

Average Error: 6.0 → 0.6

Time: 26.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \le -3.633422022558080812827674410669942517416 \cdot 10^{-4}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right)\right) - x\right)\right) + \left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right)\\ \mathbf{elif}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \le 3.405492295865614600845076579600426971829 \cdot 10^{297}:\\ \;\;\;\;\left(\sqrt{0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)} \cdot \sqrt{0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right) + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right)\right) - x\right)\right) + \left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r22937222 = x;
        double r22937223 = 0.5;
        double r22937224 = r22937222 - r22937223;
        double r22937225 = log(r22937222);
        double r22937226 = r22937224 * r22937225;
        double r22937227 = r22937226 - r22937222;
        double r22937228 = 0.91893853320467;
        double r22937229 = r22937227 + r22937228;
        double r22937230 = y;
        double r22937231 = 0.0007936500793651;
        double r22937232 = r22937230 + r22937231;
        double r22937233 = z;
        double r22937234 = r22937232 * r22937233;
        double r22937235 = 0.0027777777777778;
        double r22937236 = r22937234 - r22937235;
        double r22937237 = r22937236 * r22937233;
        double r22937238 = 0.083333333333333;
        double r22937239 = r22937237 + r22937238;
        double r22937240 = r22937239 / r22937222;
        double r22937241 = r22937229 + r22937240;
        return r22937241;
}

↓

double f(double x, double y, double z) {
        double r22937242 = z;
        double r22937243 = 0.0007936500793651;
        double r22937244 = y;
        double r22937245 = r22937243 + r22937244;
        double r22937246 = r22937245 * r22937242;
        double r22937247 = 0.0027777777777778;
        double r22937248 = r22937246 - r22937247;
        double r22937249 = r22937242 * r22937248;
        double r22937250 = -0.0003633422022558081;
        bool r22937251 = r22937249 <= r22937250;
        double r22937252 = 0.91893853320467;
        double r22937253 = x;
        double r22937254 = 0.5;
        double r22937255 = r22937253 - r22937254;
        double r22937256 = cbrt(r22937253);
        double r22937257 = log(r22937256);
        double r22937258 = r22937255 * r22937257;
        double r22937259 = r22937257 + r22937257;
        double r22937260 = r22937259 * r22937255;
        double r22937261 = r22937258 + r22937260;
        double r22937262 = r22937261 - r22937253;
        double r22937263 = r22937252 + r22937262;
        double r22937264 = r22937253 / r22937242;
        double r22937265 = r22937242 / r22937264;
        double r22937266 = r22937265 * r22937245;
        double r22937267 = r22937242 * r22937247;
        double r22937268 = r22937267 / r22937253;
        double r22937269 = r22937266 - r22937268;
        double r22937270 = r22937263 + r22937269;
        double r22937271 = 3.4054922958656146e+297;
        bool r22937272 = r22937249 <= r22937271;
        double r22937273 = sqrt(r22937253);
        double r22937274 = log(r22937273);
        double r22937275 = r22937255 * r22937274;
        double r22937276 = r22937275 - r22937253;
        double r22937277 = r22937252 + r22937276;
        double r22937278 = sqrt(r22937277);
        double r22937279 = r22937278 * r22937278;
        double r22937280 = r22937279 + r22937275;
        double r22937281 = 0.083333333333333;
        double r22937282 = r22937281 + r22937249;
        double r22937283 = r22937282 / r22937253;
        double r22937284 = r22937280 + r22937283;
        double r22937285 = r22937272 ? r22937284 : r22937270;
        double r22937286 = r22937251 ? r22937270 : r22937285;
        return r22937286;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.2
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < -0.0003633422022558081 or 3.4054922958656146e+297 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

   1. Initial program 34.8

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

   3. Applied add-cube-cbrt 34.8

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   4. Applied log-prod 34.9

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   5. Applied distribute-rgt-in 34.9

      \[\leadsto \left(\left(\color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   6. Simplified 34.9

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   7. Taylor expanded around inf 35.8

      \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \color{blue}{\left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + \frac{{z}^{2} \cdot y}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)}\]

   8. Simplified 1.5

      \[\leadsto \left(\left(\left(\left(x - 0.5\right) \cdot \left(\log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \color{blue}{\left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right)}\]

   if -0.0003633422022558081 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 3.4054922958656146e+297

   1. Initial program 0.2

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

   3. Applied add-sqr-sqrt 0.2

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

   4. Applied log-prod 0.2

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

   5. Applied distribute-rgt-in 0.2

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

   6. Applied associate--l+ 0.3

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

   7. Applied associate-+l+ 0.3

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

   9. Applied add-sqr-sqrt 0.4

      \[\leadsto \left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) + \color{blue}{\sqrt{\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876} \cdot \sqrt{\left(\log \left(\sqrt{x}\right) \cdot \left(x - 0.5\right) - x\right) + 0.9189385332046700050057097541866824030876}}\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \le -3.633422022558080812827674410669942517416 \cdot 10^{-4}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right)\right) - x\right)\right) + \left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right)\\ \mathbf{elif}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \le 3.405492295865614600845076579600426971829 \cdot 10^{297}:\\ \;\;\;\;\left(\sqrt{0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)} \cdot \sqrt{0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right) - x\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt{x}\right)\right) + \frac{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(0.9189385332046700050057097541866824030876 + \left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right) \cdot \left(x - 0.5\right)\right) - x\right)\right) + \left(\frac{z}{\frac{x}{z}} \cdot \left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"

  :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)))