Average Error: 0.6 → 0.6
Time: 2.9s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \sqrt[3]{{\left(\sqrt[3]{{\left(\log \left(1 \cdot 1 + e^{x} \cdot \left(e^{x} - 1\right)\right)\right)}^{3}}\right)}^{3}}\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \sqrt[3]{{\left(\sqrt[3]{{\left(\log \left(1 \cdot 1 + e^{x} \cdot \left(e^{x} - 1\right)\right)\right)}^{3}}\right)}^{3}}\right) - x \cdot y
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) log(((double) (((double) pow(1.0, 3.0)) + ((double) pow(((double) exp(x)), 3.0)))))) - ((double) cbrt(((double) pow(((double) cbrt(((double) pow(((double) log(((double) (((double) (1.0 * 1.0)) + ((double) (((double) exp(x)) * ((double) (((double) exp(x)) - 1.0)))))))), 3.0)))), 3.0)))))) - ((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
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program Error: 0.6 bits

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

  3. Applied flip3-+Error: 0.6 bits

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

  4. Applied log-divError: 0.6 bits

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

  5. SimplifiedError: 0.6 bits

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

  7. Applied add-cbrt-cubeError: 0.6 bits

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

  8. SimplifiedError: 0.6 bits

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

  10. Applied add-cbrt-cubeError: 0.6 bits

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

  11. SimplifiedError: 0.6 bits

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

  12. Final simplificationError: 0.6 bits

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

Reproduce

herbie shell --seed 2020200 
(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 (- x)))) (* (- x) (- 1.0 y))))

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