Average Error: 0.4 → 0.4
Time: 4.9s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(\sqrt{1 + e^{x}}\right) + \left(\left(\log \left(\sqrt{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt{\sqrt{1 + e^{x}}}\right)\right) - x \cdot y\right)\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left(\sqrt{1 + e^{x}}\right) + \left(\left(\log \left(\sqrt{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt{\sqrt{1 + e^{x}}}\right)\right) - x \cdot y\right)
double code(double x, double y) {
	return (log((1.0 + exp(x))) - (x * y));
}
↓
double code(double x, double y) {
	return (log(sqrt((1.0 + exp(x)))) + ((log(sqrt(sqrt((1.0 + exp(x))))) + log(sqrt(sqrt((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

Original0.4
Target0.0
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.4

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

  3. Applied add-sqr-sqrt1.2

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

  4. Applied log-prod0.9

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

  5. Applied associate--l+0.9

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

  7. Applied add-sqr-sqrt0.9

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

  8. Applied sqrt-prod0.4

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

  9. Applied log-prod0.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2020106 +o rules:numerics
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))