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

Average Error: 5.8 → 2.1

Time: 28.8s

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) = -\infty:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot 7.936500793651000149400709382518925849581 \cdot 10^{-4}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \mathbf{elif}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \le 2.413037256038914791422685382082884363974 \cdot 10^{296}:\\ \;\;\;\;\frac{1}{\frac{x}{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot 7.936500793651000149400709382518925849581 \cdot 10^{-4}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r16327243 = x;
        double r16327244 = 0.5;
        double r16327245 = r16327243 - r16327244;
        double r16327246 = log(r16327243);
        double r16327247 = r16327245 * r16327246;
        double r16327248 = r16327247 - r16327243;
        double r16327249 = 0.91893853320467;
        double r16327250 = r16327248 + r16327249;
        double r16327251 = y;
        double r16327252 = 0.0007936500793651;
        double r16327253 = r16327251 + r16327252;
        double r16327254 = z;
        double r16327255 = r16327253 * r16327254;
        double r16327256 = 0.0027777777777778;
        double r16327257 = r16327255 - r16327256;
        double r16327258 = r16327257 * r16327254;
        double r16327259 = 0.083333333333333;
        double r16327260 = r16327258 + r16327259;
        double r16327261 = r16327260 / r16327243;
        double r16327262 = r16327250 + r16327261;
        return r16327262;
}

↓

double f(double x, double y, double z) {
        double r16327263 = z;
        double r16327264 = 0.0007936500793651;
        double r16327265 = y;
        double r16327266 = r16327264 + r16327265;
        double r16327267 = r16327266 * r16327263;
        double r16327268 = 0.0027777777777778;
        double r16327269 = r16327267 - r16327268;
        double r16327270 = r16327263 * r16327269;
        double r16327271 = -inf.0;
        bool r16327272 = r16327270 <= r16327271;
        double r16327273 = 1.0;
        double r16327274 = x;
        double r16327275 = r16327274 / r16327263;
        double r16327276 = r16327275 / r16327263;
        double r16327277 = r16327276 / r16327265;
        double r16327278 = r16327263 * r16327265;
        double r16327279 = r16327278 * r16327278;
        double r16327280 = r16327274 / r16327279;
        double r16327281 = 6.298804484762296e-07;
        double r16327282 = r16327281 / r16327265;
        double r16327283 = r16327280 * r16327282;
        double r16327284 = r16327277 + r16327283;
        double r16327285 = r16327280 * r16327264;
        double r16327286 = r16327284 - r16327285;
        double r16327287 = r16327273 / r16327286;
        double r16327288 = cbrt(r16327274);
        double r16327289 = r16327288 * r16327288;
        double r16327290 = log(r16327289);
        double r16327291 = 0.5;
        double r16327292 = r16327274 - r16327291;
        double r16327293 = r16327290 * r16327292;
        double r16327294 = 0.91893853320467;
        double r16327295 = 0.3333333333333333;
        double r16327296 = pow(r16327274, r16327295);
        double r16327297 = log(r16327296);
        double r16327298 = r16327292 * r16327297;
        double r16327299 = r16327298 - r16327274;
        double r16327300 = r16327294 + r16327299;
        double r16327301 = r16327293 + r16327300;
        double r16327302 = r16327287 + r16327301;
        double r16327303 = 2.4130372560389148e+296;
        bool r16327304 = r16327270 <= r16327303;
        double r16327305 = 0.083333333333333;
        double r16327306 = r16327305 + r16327270;
        double r16327307 = r16327274 / r16327306;
        double r16327308 = r16327273 / r16327307;
        double r16327309 = r16327308 + r16327301;
        double r16327310 = r16327304 ? r16327309 : r16327302;
        double r16327311 = r16327272 ? r16327302 : r16327310;
        return r16327311;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.8
Target	1.2
Herbie	2.1

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

Derivation

1. Split input into 2 regimes

2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < -inf.0 or 2.4130372560389148e+296 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

   1. Initial program 60.5

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

   3. Applied add-cube-cbrt 60.5

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

   4. Applied log-prod 60.5

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

   5. Applied distribute-lft-in 60.5

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

   6. Applied associate--l+ 60.5

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

   7. Applied associate-+l+ 60.5

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

   9. Applied pow1/3 60.5

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left({x}^{\frac{1}{3}}\right)} - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
   10. Using strategy rm

   11. Applied clear-num 60.5

       \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \color{blue}{\frac{1}{\frac{x}{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}}}\]

   12. Taylor expanded around inf 62.1

       \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{1}{\color{blue}{\left(6.29880448476229567390436391016717010416 \cdot 10^{-7} \cdot \frac{x}{{z}^{2} \cdot {y}^{3}} + \frac{x}{{z}^{2} \cdot y}\right) - 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{x}{{z}^{2} \cdot {y}^{2}}}}\]

   13. Simplified 19.7

       \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{1}{\color{blue}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(y \cdot z\right) \cdot \left(y \cdot z\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - 7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{x}{\left(y \cdot z\right) \cdot \left(y \cdot z\right)}}}\]

   if -inf.0 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 2.4130372560389148e+296

   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-cube-cbrt 0.2

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

   4. Applied log-prod 0.3

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

   5. Applied distribute-lft-in 0.3

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

   6. Applied associate--l+ 0.3

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

   7. Applied associate-+l+ 0.3

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

   9. Applied pow1/3 0.2

      \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left({x}^{\frac{1}{3}}\right)} - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
   10. Using strategy rm

   11. Applied clear-num 0.3

       \[\leadsto \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\right) + \color{blue}{\frac{1}{\frac{x}{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) = -\infty:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot 7.936500793651000149400709382518925849581 \cdot 10^{-4}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \mathbf{elif}\;z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \le 2.413037256038914791422685382082884363974 \cdot 10^{296}:\\ \;\;\;\;\frac{1}{\frac{x}{0.08333333333333299564049667651488562114537 + z \cdot \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} + y\right) \cdot z - 0.002777777777777800001512975569539776188321\right)}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{\frac{\frac{x}{z}}{z}}{y} + \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot \frac{6.29880448476229567390436391016717010416 \cdot 10^{-7}}{y}\right) - \frac{x}{\left(z \cdot y\right) \cdot \left(z \cdot y\right)} \cdot 7.936500793651000149400709382518925849581 \cdot 10^{-4}} + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(0.9189385332046700050057097541866824030876 + \left(\left(x - 0.5\right) \cdot \log \left({x}^{\frac{1}{3}}\right) - x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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)))