Average Error: 0.5 → 0.5
Time: 5.0s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left(\sqrt{\sqrt{1 + e^{x}}} \cdot \sqrt{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\left(\log \left(\sqrt{\sqrt{1 + e^{x}}} \cdot \sqrt{\sqrt{1 + e^{x}}}\right) + \log \left(\sqrt{1 + e^{x}}\right)\right) - x \cdot y
double f(double x, double y) {
        double r158222 = 1.0;
        double r158223 = x;
        double r158224 = exp(r158223);
        double r158225 = r158222 + r158224;
        double r158226 = log(r158225);
        double r158227 = y;
        double r158228 = r158223 * r158227;
        double r158229 = r158226 - r158228;
        return r158229;
}
↓
double f(double x, double y) {
        double r158230 = 1.0;
        double r158231 = x;
        double r158232 = exp(r158231);
        double r158233 = r158230 + r158232;
        double r158234 = sqrt(r158233);
        double r158235 = sqrt(r158234);
        double r158236 = r158235 * r158235;
        double r158237 = log(r158236);
        double r158238 = log(r158234);
        double r158239 = r158237 + r158238;
        double r158240 = y;
        double r158241 = r158231 * r158240;
        double r158242 = r158239 - r158241;
        return r158242;
}

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
Target0.0
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.5

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  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-sqr-sqrt1.0

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

  7. Applied sqrt-prod0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020034 
(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)))