Average Error: 0.4 → 0.4

Time: 4.8s

Precision: binary64

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

↓

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

\log \left(1 + e^{x}\right) - x \cdot y

↓

\log \left(1 + e^{x}\right) - x \cdot y

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

↓

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (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 + exp(x)) - (x * y);
}

Error

The X axis uses an exponential scale — Bits error versus `x`

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	0.4
Target	0.1
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

Initial program 0.4
\[\log \left(1 + e^{x}\right) - x \cdot y\]
Final simplification0.4
\[\leadsto \log \left(1 + e^{x}\right) - x \cdot y\]

Reproduce

herbie shell --seed 2020231 
(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)))