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

Average Error: 6.2 → 4.3

Time: 33.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 4755205523904898028011520:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, -{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}\right) + \left(\left(0.9189385332046700050057097541866824030876 - x\right) + x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, -e^{\log x}\right) + \left(\left(0.9189385332046700050057097541866824030876 - x\right) + x\right)\right) + \mathsf{fma}\left(0.002777777777777800001512975569539776188321, -\frac{z}{x}, \frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r313938 = x;
        double r313939 = 0.5;
        double r313940 = r313938 - r313939;
        double r313941 = log(r313938);
        double r313942 = r313940 * r313941;
        double r313943 = r313942 - r313938;
        double r313944 = 0.91893853320467;
        double r313945 = r313943 + r313944;
        double r313946 = y;
        double r313947 = 0.0007936500793651;
        double r313948 = r313946 + r313947;
        double r313949 = z;
        double r313950 = r313948 * r313949;
        double r313951 = 0.0027777777777778;
        double r313952 = r313950 - r313951;
        double r313953 = r313952 * r313949;
        double r313954 = 0.083333333333333;
        double r313955 = r313953 + r313954;
        double r313956 = r313955 / r313938;
        double r313957 = r313945 + r313956;
        return r313957;
}

↓

double f(double x, double y, double z) {
        double r313958 = x;
        double r313959 = 4.755205523904898e+24;
        bool r313960 = r313958 <= r313959;
        double r313961 = 0.5;
        double r313962 = r313958 - r313961;
        double r313963 = log(r313958);
        double r313964 = cbrt(r313958);
        double r313965 = 2.0;
        double r313966 = pow(r313964, r313965);
        double r313967 = r313966 * r313964;
        double r313968 = -r313967;
        double r313969 = fma(r313962, r313963, r313968);
        double r313970 = 0.91893853320467;
        double r313971 = r313970 - r313958;
        double r313972 = r313971 + r313958;
        double r313973 = r313969 + r313972;
        double r313974 = y;
        double r313975 = 0.0007936500793651;
        double r313976 = r313974 + r313975;
        double r313977 = z;
        double r313978 = r313976 * r313977;
        double r313979 = 0.0027777777777778;
        double r313980 = r313978 - r313979;
        double r313981 = r313980 * r313977;
        double r313982 = 0.083333333333333;
        double r313983 = r313981 + r313982;
        double r313984 = r313983 / r313958;
        double r313985 = r313973 + r313984;
        double r313986 = exp(r313963);
        double r313987 = -r313986;
        double r313988 = fma(r313962, r313963, r313987);
        double r313989 = r313988 + r313972;
        double r313990 = r313977 / r313958;
        double r313991 = -r313990;
        double r313992 = pow(r313977, r313965);
        double r313993 = r313992 / r313958;
        double r313994 = r313993 * r313976;
        double r313995 = fma(r313979, r313991, r313994);
        double r313996 = r313989 + r313995;
        double r313997 = r313960 ? r313985 : r313996;
        return r313997;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	6.2
Target	1.2
Herbie	4.3

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

Derivation

1. Split input into 2 regimes

2. if x < 4.755205523904898e+24

   1. Initial program 0.3

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

   3. Applied add-sqr-sqrt 0.3

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

   4. Applied prod-diff 0.3

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

   5. Applied associate-+l+ 0.3

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

   6. Simplified 0.3

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

   8. Applied add-exp-log 0.3

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

   9. Applied add-exp-log 0.3

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

   10. Applied prod-exp 0.3

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

   11. Simplified 0.3

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

   13. Applied add-cube-cbrt 0.3

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

   14. Applied log-prod 0.3

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

   15. Applied exp-sum 0.3

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

   16. Simplified 0.3

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

   17. Simplified 0.3

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

   if 4.755205523904898e+24 < x

   1. Initial program 11.2

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

   3. Applied add-sqr-sqrt 11.2

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

   4. Applied prod-diff 11.1

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

   5. Applied associate-+l+ 11.1

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

   6. Simplified 11.1

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

   8. Applied add-exp-log 11.0

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

   9. Applied add-exp-log 10.9

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

   10. Applied prod-exp 10.9

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

   11. Simplified 10.9

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

   12. Taylor expanded around inf 11.1

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

   13. Simplified 7.6

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

4. Final simplification 4.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4755205523904898028011520:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, -{\left(\sqrt[3]{x}\right)}^{2} \cdot \sqrt[3]{x}\right) + \left(\left(0.9189385332046700050057097541866824030876 - x\right) + x\right)\right) + \frac{\left(\left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right) \cdot z - 0.002777777777777800001512975569539776188321\right) \cdot z + 0.08333333333333299564049667651488562114537}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{fma}\left(x - 0.5, \log x, -e^{\log x}\right) + \left(\left(0.9189385332046700050057097541866824030876 - x\right) + x\right)\right) + \mathsf{fma}\left(0.002777777777777800001512975569539776188321, -\frac{z}{x}, \frac{{z}^{2}}{x} \cdot \left(y + 7.936500793651000149400709382518925849581 \cdot 10^{-4}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +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)) (- 0.91893853320467001 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)))