Average Error: 0.5 → 0.5
Time: 2.5s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(1 + {\left(e^{x}\right)}^{3}\right) - \log \left(1 + \left({\left(e^{x}\right)}^{2} - e^{x}\right)\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left(1 + {\left(e^{x}\right)}^{3}\right) - \log \left(1 + \left({\left(e^{x}\right)}^{2} - e^{x}\right)\right)\right) - x \cdot y
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y)
 :precision binary64
 (-
  (-
   (log (+ 1.0 (pow (exp x) 3.0)))
   (log (+ 1.0 (- (pow (exp x) 2.0) (exp x)))))
  (* x y)))
double code(double x, double y) {
	return log(1.0 + exp(x)) - (x * y);
}
↓
double code(double x, double y) {
	return (log(1.0 + pow(exp(x), 3.0)) - log(1.0 + (pow(exp(x), 2.0) - 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
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  3. Applied flip3-+_binary640.5

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

  4. Applied log-div_binary640.5

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

  5. Simplified0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2020232 
(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)))