Average Error: 0.4 → 0.4
Time: 2.5s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\sqrt[3]{{\left(\log \left(1 + e^{x}\right)\right)}^{3}} - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\sqrt[3]{{\left(\log \left(1 + e^{x}\right)\right)}^{3}} - x \cdot y
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y)
 :precision binary64
 (- (cbrt (pow (log (+ 1.0 (exp x))) 3.0)) (* x y)))
double code(double x, double y) {
	return ((double) (((double) log(((double) (1.0 + ((double) exp(x)))))) - ((double) (x * y))));
}
↓
double code(double x, double y) {
	return ((double) (((double) cbrt(((double) pow(((double) log(((double) (1.0 + ((double) exp(x)))))), 3.0)))) - ((double) (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
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.4

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

  3. Applied add-cbrt-cube_binary640.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2020219 
(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)))