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

Average Error: 6.0 → 0.4

Time: 27.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 406117785587.8438720703125:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) + 0.08333333333333299564049667651488562114537}{x} + \left(0.9189385332046700050057097541866824030876 + \left(\sqrt[3]{\left(\left(\left(x - 0.5\right) \cdot \log x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x\right)\right) \cdot \left(\left(x - 0.5\right) \cdot \log x\right)} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right) + \left(\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r28060938 = x;
        double r28060939 = 0.5;
        double r28060940 = r28060938 - r28060939;
        double r28060941 = log(r28060938);
        double r28060942 = r28060940 * r28060941;
        double r28060943 = r28060942 - r28060938;
        double r28060944 = 0.91893853320467;
        double r28060945 = r28060943 + r28060944;
        double r28060946 = y;
        double r28060947 = 0.0007936500793651;
        double r28060948 = r28060946 + r28060947;
        double r28060949 = z;
        double r28060950 = r28060948 * r28060949;
        double r28060951 = 0.0027777777777778;
        double r28060952 = r28060950 - r28060951;
        double r28060953 = r28060952 * r28060949;
        double r28060954 = 0.083333333333333;
        double r28060955 = r28060953 + r28060954;
        double r28060956 = r28060955 / r28060938;
        double r28060957 = r28060945 + r28060956;
        return r28060957;
}

↓

double f(double x, double y, double z) {
        double r28060958 = x;
        double r28060959 = 406117785587.8439;
        bool r28060960 = r28060958 <= r28060959;
        double r28060961 = z;
        double r28060962 = y;
        double r28060963 = 0.0007936500793651;
        double r28060964 = r28060962 + r28060963;
        double r28060965 = r28060964 * r28060961;
        double r28060966 = 0.0027777777777778;
        double r28060967 = r28060965 - r28060966;
        double r28060968 = r28060961 * r28060967;
        double r28060969 = 0.083333333333333;
        double r28060970 = r28060968 + r28060969;
        double r28060971 = r28060970 / r28060958;
        double r28060972 = 0.91893853320467;
        double r28060973 = 0.5;
        double r28060974 = r28060958 - r28060973;
        double r28060975 = log(r28060958);
        double r28060976 = r28060974 * r28060975;
        double r28060977 = r28060976 * r28060976;
        double r28060978 = r28060977 * r28060976;
        double r28060979 = cbrt(r28060978);
        double r28060980 = r28060979 - r28060958;
        double r28060981 = r28060972 + r28060980;
        double r28060982 = r28060971 + r28060981;
        double r28060983 = r28060958 / r28060961;
        double r28060984 = r28060961 / r28060983;
        double r28060985 = r28060984 * r28060964;
        double r28060986 = r28060961 * r28060966;
        double r28060987 = r28060986 / r28060958;
        double r28060988 = r28060985 - r28060987;
        double r28060989 = cbrt(r28060958);
        double r28060990 = log(r28060989);
        double r28060991 = r28060974 * r28060990;
        double r28060992 = r28060991 + r28060991;
        double r28060993 = r28060991 + r28060992;
        double r28060994 = r28060993 - r28060958;
        double r28060995 = r28060994 + r28060972;
        double r28060996 = r28060988 + r28060995;
        double r28060997 = r28060960 ? r28060982 : r28060996;
        return r28060997;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.0
Target	1.2
Herbie	0.4

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

Derivation

1. Split input into 2 regimes

2. if x < 406117785587.8439

   1. Initial program 0.1

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

   3. Applied add-cbrt-cube 0.1

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

   if 406117785587.8439 < x

   1. Initial program 10.2

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

   3. Applied add-cube-cbrt 10.2

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

   4. Applied log-prod 10.3

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

   5. Applied distribute-rgt-in 10.3

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

   6. Simplified 10.3

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

   7. Taylor expanded around inf 10.3

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

   8. Simplified 0.5

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 406117785587.8438720703125:\\ \;\;\;\;\frac{z \cdot \left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) + 0.08333333333333299564049667651488562114537}{x} + \left(0.9189385332046700050057097541866824030876 + \left(\sqrt[3]{\left(\left(\left(x - 0.5\right) \cdot \log x\right) \cdot \left(\left(x - 0.5\right) \cdot \log x\right)\right) \cdot \left(\left(x - 0.5\right) \cdot \log x\right)} - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{\frac{x}{z}} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) - \frac{z \cdot 0.002777777777777800001512975569539776188321}{x}\right) + \left(\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)\right) - x\right) + 0.9189385332046700050057097541866824030876\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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)))