Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 15.3 → 0.5

Time: 20.9s

Precision: 64

Math

\[x \cdot \log \left(\frac{x}{y}\right) - z\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -4.974644044754423077113473181446697397685 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right) - z\\ \mathbf{elif}\;y \le 3.026653167445680484753712447534489416041 \cdot 10^{-81}:\\ \;\;\;\;\left(\log x \cdot x + \log y \cdot \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) - z\right) + x \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r23070155 = x;
        double r23070156 = y;
        double r23070157 = r23070155 / r23070156;
        double r23070158 = log(r23070157);
        double r23070159 = r23070155 * r23070158;
        double r23070160 = z;
        double r23070161 = r23070159 - r23070160;
        return r23070161;
}

↓

double f(double x, double y, double z) {
        double r23070162 = y;
        double r23070163 = -4.974644044754423e-309;
        bool r23070164 = r23070162 <= r23070163;
        double r23070165 = x;
        double r23070166 = -1.0;
        double r23070167 = r23070166 / r23070162;
        double r23070168 = log(r23070167);
        double r23070169 = r23070166 / r23070165;
        double r23070170 = log(r23070169);
        double r23070171 = r23070168 - r23070170;
        double r23070172 = r23070165 * r23070171;
        double r23070173 = z;
        double r23070174 = r23070172 - r23070173;
        double r23070175 = 3.0266531674456805e-81;
        bool r23070176 = r23070162 <= r23070175;
        double r23070177 = log(r23070165);
        double r23070178 = r23070177 * r23070165;
        double r23070179 = log(r23070162);
        double r23070180 = -r23070165;
        double r23070181 = r23070179 * r23070180;
        double r23070182 = r23070178 + r23070181;
        double r23070183 = r23070182 - r23070173;
        double r23070184 = cbrt(r23070165);
        double r23070185 = cbrt(r23070184);
        double r23070186 = r23070185 / r23070162;
        double r23070187 = log(r23070186);
        double r23070188 = r23070184 * r23070184;
        double r23070189 = cbrt(r23070188);
        double r23070190 = log(r23070189);
        double r23070191 = r23070187 + r23070190;
        double r23070192 = r23070165 * r23070191;
        double r23070193 = r23070192 - r23070173;
        double r23070194 = log(r23070188);
        double r23070195 = r23070165 * r23070194;
        double r23070196 = r23070193 + r23070195;
        double r23070197 = r23070176 ? r23070183 : r23070196;
        double r23070198 = r23070164 ? r23070174 : r23070197;
        return r23070198;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.3
Target	7.8
Herbie	0.5

\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -4.974644044754423e-309

   1. Initial program 15.0

      \[x \cdot \log \left(\frac{x}{y}\right) - z\]

   2. Taylor expanded around -inf 0.3

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right)} - z\]

   if -4.974644044754423e-309 < y < 3.0266531674456805e-81

   1. Initial program 20.1

      \[x \cdot \log \left(\frac{x}{y}\right) - z\]
   2. Using strategy rm

   3. Applied div-inv 20.1

      \[\leadsto x \cdot \log \color{blue}{\left(x \cdot \frac{1}{y}\right)} - z\]

   4. Applied log-prod 0.3

      \[\leadsto x \cdot \color{blue}{\left(\log x + \log \left(\frac{1}{y}\right)\right)} - z\]

   5. Applied distribute-rgt-in 0.3

      \[\leadsto \color{blue}{\left(\log x \cdot x + \log \left(\frac{1}{y}\right) \cdot x\right)} - z\]

   6. Simplified 0.3

      \[\leadsto \left(\log x \cdot x + \color{blue}{x \cdot \left(-\log y\right)}\right) - z\]

   if 3.0266531674456805e-81 < y

   1. Initial program 12.9

      \[x \cdot \log \left(\frac{x}{y}\right) - z\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 12.9

      \[\leadsto x \cdot \log \left(\frac{x}{\color{blue}{1 \cdot y}}\right) - z\]

   4. Applied add-cube-cbrt 12.9

      \[\leadsto x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{1 \cdot y}\right) - z\]

   5. Applied times-frac 12.9

      \[\leadsto x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1} \cdot \frac{\sqrt[3]{x}}{y}\right)} - z\]

   6. Applied log-prod 3.1

      \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \log \left(\frac{\sqrt[3]{x}}{y}\right)\right)} - z\]

   7. Applied distribute-lft-in 3.1

      \[\leadsto \color{blue}{\left(x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{y}\right)\right)} - z\]

   8. Applied associate--l+ 3.1

      \[\leadsto \color{blue}{x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \left(x \cdot \log \left(\frac{\sqrt[3]{x}}{y}\right) - z\right)}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity 3.1

       \[\leadsto x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \left(x \cdot \log \left(\frac{\sqrt[3]{x}}{\color{blue}{1 \cdot y}}\right) - z\right)\]

   11. Applied add-cube-cbrt 3.1

       \[\leadsto x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \left(x \cdot \log \left(\frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{1 \cdot y}\right) - z\right)\]

   12. Applied cbrt-prod 3.1

       \[\leadsto x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \left(x \cdot \log \left(\frac{\color{blue}{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}}}{1 \cdot y}\right) - z\right)\]

   13. Applied times-frac 3.1

       \[\leadsto x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \left(x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{1} \cdot \frac{\sqrt[3]{\sqrt[3]{x}}}{y}\right)} - z\right)\]

   14. Applied log-prod 1.1

       \[\leadsto x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \left(x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{1}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{y}\right)\right)} - z\right)\]

   15. Simplified 1.1

       \[\leadsto x \cdot \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) + \left(x \cdot \left(\color{blue}{\log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)} + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{y}\right)\right) - z\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.974644044754423077113473181446697397685 \cdot 10^{-309}:\\ \;\;\;\;x \cdot \left(\log \left(\frac{-1}{y}\right) - \log \left(\frac{-1}{x}\right)\right) - z\\ \mathbf{elif}\;y \le 3.026653167445680484753712447534489416041 \cdot 10^{-81}:\\ \;\;\;\;\left(\log x \cdot x + \log y \cdot \left(-x\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{y}\right) + \log \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right)\right) - z\right) + x \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))