Average Error: 0.4 → 0.6
Time: 3.4s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \log \left(e \cdot e^{e^{x}}\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \log \left(e \cdot e^{e^{x}}\right) - x \cdot y
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y) :precision binary64 (- (log (log (* E (exp (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(log(((double) M_E) * exp(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

Original0.4
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 0.4

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

  3. Applied add-log-exp_binary640.6

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

  4. Applied add-log-exp_binary640.6

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

  5. Applied sum-log_binary640.6

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

  6. Final simplification0.6

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

Alternatives

Reproduce

herbie shell --seed 2021110 
(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)))