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

Average Error: 15.0 → 0.2

Time: 22.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r389137 = x;
        double r389138 = y;
        double r389139 = r389137 / r389138;
        double r389140 = log(r389139);
        double r389141 = r389137 * r389140;
        double r389142 = z;
        double r389143 = r389141 - r389142;
        return r389143;
}

↓

double f(double x, double y, double z) {
        double r389144 = x;
        double r389145 = cbrt(r389144);
        double r389146 = y;
        double r389147 = cbrt(r389146);
        double r389148 = r389145 / r389147;
        double r389149 = log(r389148);
        double r389150 = 2.0;
        double r389151 = cbrt(r389145);
        double r389152 = r389151 * r389151;
        double r389153 = r389151 * r389152;
        double r389154 = r389153 / r389147;
        double r389155 = log(r389154);
        double r389156 = r389150 * r389155;
        double r389157 = r389149 + r389156;
        double r389158 = z;
        double r389159 = -r389158;
        double r389160 = fma(r389144, r389157, r389159);
        return r389160;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.0
Target	7.9
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 15.0

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

2. Simplified 15.0

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

4. Applied add-cube-cbrt 15.0

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

5. Applied add-cube-cbrt 15.0

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

6. Applied times-frac 15.0

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

7. Applied log-prod 3.3

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

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

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019196 +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))