Average Error: 0.5 → 0.5
Time: 6.9s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\sqrt[3]{\log \left(1 + e^{x}\right) \cdot \left(\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right)\right)} - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\sqrt[3]{\log \left(1 + e^{x}\right) \cdot \left(\log \left(1 + e^{x}\right) \cdot \log \left(1 + e^{x}\right)\right)} - x \cdot y
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y)
 :precision binary64
 (-
  (cbrt
   (* (log (+ 1.0 (exp x))) (* (log (+ 1.0 (exp x))) (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 cbrt(log(1.0 + exp(x)) * (log(1.0 + exp(x)) * log(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
Target0.1
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 add-cbrt-cube_binary64_38650.5

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

  4. Final simplification0.5

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

Alternatives

Reproduce

herbie shell --seed 2021098 
(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)))