Average Error: 0.5 → 0.6
Time: 4.5s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\sqrt[3]{{\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right)\right)}^{3}} - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\sqrt[3]{{\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right)\right)}^{3}} - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right) - x \cdot y
double code(double x, double y) {
	return (log((1.0 + exp(x))) - (x * y));
}
↓
double code(double x, double y) {
	return ((cbrt(pow(log((pow(1.0, 3.0) + pow(exp(x), 3.0))), 3.0)) - log(((1.0 * 1.0) + ((exp(x) * exp(x)) - (1.0 * exp(x)))))) - (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

Original0.5
Target0.0
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  3. Applied flip3-+0.6

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

  4. Applied log-div0.6

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

  6. Applied add-cbrt-cube0.6

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

  7. Simplified0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))