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

Average Error: 15.3 → 0.5

Time: 21.2s

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 r25267038 = x;
        double r25267039 = y;
        double r25267040 = r25267038 / r25267039;
        double r25267041 = log(r25267040);
        double r25267042 = r25267038 * r25267041;
        double r25267043 = z;
        double r25267044 = r25267042 - r25267043;
        return r25267044;
}

↓

double f(double x, double y, double z) {
        double r25267045 = y;
        double r25267046 = -4.974644044754423e-309;
        bool r25267047 = r25267045 <= r25267046;
        double r25267048 = x;
        double r25267049 = -1.0;
        double r25267050 = r25267049 / r25267045;
        double r25267051 = log(r25267050);
        double r25267052 = r25267049 / r25267048;
        double r25267053 = log(r25267052);
        double r25267054 = r25267051 - r25267053;
        double r25267055 = r25267048 * r25267054;
        double r25267056 = z;
        double r25267057 = r25267055 - r25267056;
        double r25267058 = 3.0266531674456805e-81;
        bool r25267059 = r25267045 <= r25267058;
        double r25267060 = log(r25267048);
        double r25267061 = r25267060 * r25267048;
        double r25267062 = log(r25267045);
        double r25267063 = -r25267048;
        double r25267064 = r25267062 * r25267063;
        double r25267065 = r25267061 + r25267064;
        double r25267066 = r25267065 - r25267056;
        double r25267067 = cbrt(r25267048);
        double r25267068 = cbrt(r25267067);
        double r25267069 = r25267068 / r25267045;
        double r25267070 = log(r25267069);
        double r25267071 = r25267067 * r25267067;
        double r25267072 = cbrt(r25267071);
        double r25267073 = log(r25267072);
        double r25267074 = r25267070 + r25267073;
        double r25267075 = r25267048 * r25267074;
        double r25267076 = r25267075 - r25267056;
        double r25267077 = log(r25267071);
        double r25267078 = r25267048 * r25267077;
        double r25267079 = r25267076 + r25267078;
        double r25267080 = r25267059 ? r25267066 : r25267079;
        double r25267081 = r25267047 ? r25267057 : r25267080;
        return r25267081;
}

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-rgt-in 3.1

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

   8. Applied associate--l+ 3.1

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

   10. Applied *-un-lft-identity 3.1

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

   11. Applied add-cube-cbrt 3.1

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

   12. Applied cbrt-prod 3.1

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

   13. Applied times-frac 3.1

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

   14. Applied log-prod 1.1

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

   15. Simplified 1.1

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