Average Error: 0.5 → 1.0
Time: 2.8s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) \cdot 2\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) \cdot 2\right)\right) - x \cdot y
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y)
 :precision binary64
 (-
  (+
   (log (sqrt (+ 1.0 (exp x))))
   (+
    (log (cbrt (sqrt (+ 1.0 (exp x)))))
    (* (log (cbrt (sqrt (+ 1.0 (exp x))))) 2.0)))
  (* 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) (((double) log(((double) sqrt(((double) (1.0 + ((double) exp(x)))))))) + ((double) (((double) log(((double) cbrt(((double) sqrt(((double) (1.0 + ((double) exp(x)))))))))) + ((double) (((double) log(((double) cbrt(((double) sqrt(((double) (1.0 + ((double) exp(x)))))))))) * 2.0)))))) - ((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

Original0.5
Target0.1
Herbie1.0
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

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

  3. Applied add-sqr-sqrt1.3

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

  4. Applied log-prod1.0

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

  6. Applied add-cube-cbrt1.0

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

  7. Applied log-prod1.0

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

  8. Simplified1.0

    \[\leadsto \left(\log \left(\sqrt{1 + e^{x}}\right) + \left(\color{blue}{\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) \cdot 2} + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right)\right)\right) - x \cdot y\]

  9. Final simplification1.0

    \[\leadsto \left(\log \left(\sqrt{1 + e^{x}}\right) + \left(\log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt[3]{\sqrt{1 + e^{x}}}\right) \cdot 2\right)\right) - x \cdot y\]

Reproduce

herbie shell --seed 2020198 
(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)))