Average Error: 0.6 → 1.1
Time: 3.5s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(e^{\log \left(\log \left(e^{x} + 1\right)\right)} \cdot \frac{2}{3} + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right)\right) + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) - x \cdot y\right)\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(e^{\log \left(\log \left(e^{x} + 1\right)\right)} \cdot \frac{2}{3} + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right)\right) + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) - x \cdot y\right)
double code(double x, double y) {
	return ((double) (((double) log(((double) (1.0 + ((double) exp(x)))))) - ((double) (x * y))));
}
↓
double code(double x, double y) {
	return ((double) (((double) (((double) (((double) exp(((double) log(((double) log(((double) (((double) exp(x)) + 1.0)))))))) * 0.6666666666666666)) + ((double) log(((double) cbrt(((double) sqrt(((double) (1.0 + ((double) exp(x)))))))))))) + ((double) (((double) log(((double) cbrt(((double) sqrt(((double) (1.0 + ((double) exp(x)))))))))) - ((double) (x * y))))));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original0.6
Target0.1
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.6

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  2. Using strategy rm

  3. Applied add-cube-cbrt1.8

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

  4. Applied log-prod1.9

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

  5. Applied associate--l+1.9

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

  7. Applied add-exp-log1.9

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

  8. Simplified1.4

    \[\leadsto e^{\color{blue}{\log \frac{2}{3} + \log \left(\log \left(1 + e^{x}\right)\right)}} + \left(\log \left(\sqrt[3]{1 + e^{x}}\right) - x \cdot y\right)\]
  9. Using strategy rm

  10. Applied add-sqr-sqrt1.4

    \[\leadsto e^{\log \frac{2}{3} + \log \left(\log \left(1 + e^{x}\right)\right)} + \left(\log \left(\sqrt[3]{\color{blue}{\sqrt{1 + e^{x}} \cdot \sqrt{1 + e^{x}}}}\right) - x \cdot y\right)\]

  11. Applied cbrt-prod0.6

    \[\leadsto e^{\log \frac{2}{3} + \log \left(\log \left(1 + e^{x}\right)\right)} + \left(\log \color{blue}{\left(\sqrt[3]{\sqrt{1 + e^{x}}} \cdot \sqrt[3]{\sqrt{1 + e^{x}}}\right)} - x \cdot y\right)\]

  12. Applied log-prod1.1

    \[\leadsto e^{\log \frac{2}{3} + \log \left(\log \left(1 + e^{x}\right)\right)} + \left(\color{blue}{\left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right)\right)} - x \cdot y\right)\]

  13. Applied associate--l+1.1

    \[\leadsto e^{\log \frac{2}{3} + \log \left(\log \left(1 + e^{x}\right)\right)} + \color{blue}{\left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) - x \cdot y\right)\right)}\]

  14. Applied associate-+r+1.1

    \[\leadsto \color{blue}{\left(e^{\log \frac{2}{3} + \log \left(\log \left(1 + e^{x}\right)\right)} + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right)\right) + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) - x \cdot y\right)}\]

  15. Simplified1.1

    \[\leadsto \color{blue}{\left(e^{\log \left(\log \left(e^{x} + 1\right)\right)} \cdot \frac{2}{3} + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right)\right)} + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) - x \cdot y\right)\]

  16. Final simplification1.1

    \[\leadsto \left(e^{\log \left(\log \left(e^{x} + 1\right)\right)} \cdot \frac{2}{3} + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right)\right) + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) - x \cdot y\right)\]

Reproduce

herbie shell --seed 2020171 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (neg x)))) (* (neg x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))