Average Error: 0.4 → 0.4
Time: 3.4s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\sqrt[3]{{\left({\log \left(1 + e^{x}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}} - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\sqrt[3]{{\left({\log \left(1 + e^{x}\right)}^{\left(\sqrt{3}\right)}\right)}^{\left(\sqrt{3}\right)}} - x \cdot y
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y)
 :precision binary64
 (- (cbrt (pow (pow (log (+ 1.0 (exp x))) (sqrt 3.0)) (sqrt 3.0))) (* x y)))
double code(double x, double y) {
	return log(1.0 + exp(x)) - (x * y);
}
↓
double code(double x, double y) {
	return cbrt(pow(pow(log(1.0 + exp(x)), sqrt(3.0)), sqrt(3.0))) - (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
Target0.0
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.4

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

  3. Applied add-cbrt-cube_binary64_28420.4

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

  4. Simplified0.4

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

  6. Applied add-sqr-sqrt_binary64_28281.0

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

  7. Applied pow-unpow_binary64_28830.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020346 
(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)))