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

Average Error: 15.3 → 0.2

Time: 17.0s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(x, \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right), -z\right)\]

C

double f(double x, double y, double z) {
        double r338184 = x;
        double r338185 = y;
        double r338186 = r338184 / r338185;
        double r338187 = log(r338186);
        double r338188 = r338184 * r338187;
        double r338189 = z;
        double r338190 = r338188 - r338189;
        return r338190;
}

↓

double f(double x, double y, double z) {
        double r338191 = x;
        double r338192 = cbrt(r338191);
        double r338193 = y;
        double r338194 = cbrt(r338193);
        double r338195 = r338192 / r338194;
        double r338196 = log(r338195);
        double r338197 = 2.0;
        double r338198 = cbrt(r338194);
        double r338199 = r338198 * r338198;
        double r338200 = r338198 * r338199;
        double r338201 = r338192 / r338200;
        double r338202 = log(r338201);
        double r338203 = r338197 * r338202;
        double r338204 = r338196 + r338203;
        double r338205 = z;
        double r338206 = -r338205;
        double r338207 = fma(r338191, r338204, r338206);
        return r338207;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.3
Target	7.8
Herbie	0.2

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

Derivation

1. Initial program 15.3

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

2. Simplified 15.3

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log \left(\frac{x}{y}\right), -z\right)}\]
3. Using strategy rm

4. Applied add-cube-cbrt 15.3

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

5. Applied add-cube-cbrt 15.3

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

6. Applied times-frac 15.3

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

7. Applied log-prod 3.6

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

8. Simplified 0.2

   \[\leadsto \mathsf{fma}\left(x, \color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right), -z\right)\]
9. Using strategy rm

10. Applied add-cube-cbrt 0.2

    \[\leadsto \mathsf{fma}\left(x, 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right), -z\right)\]

11. Final simplification 0.2

    \[\leadsto \mathsf{fma}\left(x, \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + 2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right), -z\right)\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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))