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

Average Error: 5.9 → 5.5

Time: 16.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 3.428668489872041303924036672141566098601 \cdot 10^{72}:\\ \;\;\;\;\left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{x}}\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.08333333333333299564049667651488562114537 \cdot \frac{1}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r331173 = x;
        double r331174 = 0.5;
        double r331175 = r331173 - r331174;
        double r331176 = log(r331173);
        double r331177 = r331175 * r331176;
        double r331178 = r331177 - r331173;
        double r331179 = 0.91893853320467;
        double r331180 = r331178 + r331179;
        double r331181 = y;
        double r331182 = 0.0007936500793651;
        double r331183 = r331181 + r331182;
        double r331184 = z;
        double r331185 = r331183 * r331184;
        double r331186 = 0.0027777777777778;
        double r331187 = r331185 - r331186;
        double r331188 = r331187 * r331184;
        double r331189 = 0.083333333333333;
        double r331190 = r331188 + r331189;
        double r331191 = r331190 / r331173;
        double r331192 = r331180 + r331191;
        return r331192;
}

↓

double f(double x, double y, double z) {
        double r331193 = x;
        double r331194 = 3.4286684898720413e+72;
        bool r331195 = r331193 <= r331194;
        double r331196 = 0.3333333333333333;
        double r331197 = pow(r331193, r331196);
        double r331198 = log(r331197);
        double r331199 = 3.0;
        double r331200 = r331199 * r331193;
        double r331201 = 1.5;
        double r331202 = r331200 - r331201;
        double r331203 = r331198 * r331202;
        double r331204 = r331203 - r331193;
        double r331205 = 0.91893853320467;
        double r331206 = r331204 + r331205;
        double r331207 = y;
        double r331208 = 0.0007936500793651;
        double r331209 = r331207 + r331208;
        double r331210 = z;
        double r331211 = r331209 * r331210;
        double r331212 = 0.0027777777777778;
        double r331213 = r331211 - r331212;
        double r331214 = r331213 * r331210;
        double r331215 = 0.083333333333333;
        double r331216 = r331214 + r331215;
        double r331217 = r331216 / r331193;
        double r331218 = r331206 + r331217;
        double r331219 = 0.5;
        double r331220 = r331193 - r331219;
        double r331221 = 2.0;
        double r331222 = cbrt(r331193);
        double r331223 = log(r331222);
        double r331224 = r331221 * r331223;
        double r331225 = r331222 * r331222;
        double r331226 = cbrt(r331225);
        double r331227 = log(r331226);
        double r331228 = r331224 + r331227;
        double r331229 = r331220 * r331228;
        double r331230 = cbrt(r331222);
        double r331231 = log(r331230);
        double r331232 = r331220 * r331231;
        double r331233 = r331229 + r331232;
        double r331234 = r331233 - r331193;
        double r331235 = r331234 + r331205;
        double r331236 = pow(r331210, r331221);
        double r331237 = r331236 / r331193;
        double r331238 = r331208 * r331237;
        double r331239 = 1.0;
        double r331240 = r331239 / r331193;
        double r331241 = r331215 * r331240;
        double r331242 = r331238 + r331241;
        double r331243 = r331210 / r331193;
        double r331244 = r331212 * r331243;
        double r331245 = r331242 - r331244;
        double r331246 = r331235 + r331245;
        double r331247 = r331195 ? r331218 : r331246;
        return r331247;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	5.9
Target	1.1
Herbie	5.5

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

Derivation

1. Split input into 2 regimes

2. if x < 3.4286684898720413e+72

   1. Initial program 0.6

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

   3. Applied add-cube-cbrt 0.6

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

   4. Applied log-prod 0.6

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

   5. Applied distribute-lft-in 0.6

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

   6. Simplified 0.6

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

   7. Taylor expanded around 0 0.6

      \[\leadsto \left(\left(\color{blue}{\left(3 \cdot \left(x \cdot \log \left({x}^{\frac{1}{3}}\right)\right) - 1.5 \cdot \log \left({x}^{\frac{1}{3}}\right)\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   8. Simplified 0.6

      \[\leadsto \left(\left(\color{blue}{\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right)} - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]

   if 3.4286684898720413e+72 < x

   1. Initial program 12.2

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

   3. Applied add-cube-cbrt 12.2

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

   4. Applied log-prod 12.3

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

   5. Applied distribute-lft-in 12.3

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

   6. Simplified 12.3

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 12.3

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

   9. Applied cbrt-prod 12.3

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

   10. Applied log-prod 12.3

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

   11. Applied distribute-lft-in 12.3

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

   12. Applied associate-+r+ 12.3

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

   13. Simplified 12.3

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

   14. Taylor expanded around 0 11.3

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 3.428668489872041303924036672141566098601 \cdot 10^{72}:\\ \;\;\;\;\left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right) + \log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{x}}\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right) + \left(\left(7.936500793651000149400709382518925849581 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.08333333333333299564049667651488562114537 \cdot \frac{1}{x}\right) - 0.002777777777777800001512975569539776188321 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 1978988140 
(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)) (- 0.91893853320467001 x)) (/ 0.0833333333333329956 x)) (* (/ z x) (- (* z (+ y 7.93650079365100015e-4)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467001) (/ (+ (* (- (* (+ y 7.93650079365100015e-4) z) 0.0027777777777778) z) 0.0833333333333329956) x)))