Average Error: 0.5 → 0.6
Time: 3.0s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \mathsf{fma}\left(x, y, \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right)\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \mathsf{fma}\left(x, y, \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right)
double f(double x, double y) {
        double r162536 = 1.0;
        double r162537 = x;
        double r162538 = exp(r162537);
        double r162539 = r162536 + r162538;
        double r162540 = log(r162539);
        double r162541 = y;
        double r162542 = r162537 * r162541;
        double r162543 = r162540 - r162542;
        return r162543;
}
↓
double f(double x, double y) {
        double r162544 = 1.0;
        double r162545 = 3.0;
        double r162546 = pow(r162544, r162545);
        double r162547 = x;
        double r162548 = exp(r162547);
        double r162549 = pow(r162548, r162545);
        double r162550 = r162546 + r162549;
        double r162551 = log(r162550);
        double r162552 = y;
        double r162553 = r162544 * r162544;
        double r162554 = r162548 * r162548;
        double r162555 = r162544 * r162548;
        double r162556 = r162554 - r162555;
        double r162557 = r162553 + r162556;
        double r162558 = log(r162557);
        double r162559 = fma(r162547, r162552, r162558);
        double r162560 = r162551 - r162559;
        return r162560;
}

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.0
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 0.5

    \[\log \left(1 + e^{x}\right) - x \cdot y\]
  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. Applied log-div0.6

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

  5. Applied associate--l-0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2020024 +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)))