Average Error: 0.6 → 0.6
Time: 4.9s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x} - 1, 1 \cdot 1\right)}\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left(\frac{{1}^{3} + {\left(e^{x}\right)}^{3}}{\mathsf{fma}\left(e^{x}, e^{x} - 1, 1 \cdot 1\right)}\right) - x \cdot y
double f(double x, double y) {
        double r153273 = 1.0;
        double r153274 = x;
        double r153275 = exp(r153274);
        double r153276 = r153273 + r153275;
        double r153277 = log(r153276);
        double r153278 = y;
        double r153279 = r153274 * r153278;
        double r153280 = r153277 - r153279;
        return r153280;
}
↓
double f(double x, double y) {
        double r153281 = 1.0;
        double r153282 = 3.0;
        double r153283 = pow(r153281, r153282);
        double r153284 = x;
        double r153285 = exp(r153284);
        double r153286 = pow(r153285, r153282);
        double r153287 = r153283 + r153286;
        double r153288 = r153285 - r153281;
        double r153289 = r153281 * r153281;
        double r153290 = fma(r153285, r153288, r153289);
        double r153291 = r153287 / r153290;
        double r153292 = log(r153291);
        double r153293 = y;
        double r153294 = r153284 * r153293;
        double r153295 = r153292 - r153294;
        return r153295;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.6
Target0.1
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \le 0.0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Initial program 0.6

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

  3. Applied flip3-+0.6

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

  4. Simplified0.6

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

  5. Final simplification0.6

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

Reproduce

herbie shell --seed 2019353 +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)))