Average Error: 0.6 → 0.6
Time: 4.6s
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 r183015 = 1.0;
        double r183016 = x;
        double r183017 = exp(r183016);
        double r183018 = r183015 + r183017;
        double r183019 = log(r183018);
        double r183020 = y;
        double r183021 = r183016 * r183020;
        double r183022 = r183019 - r183021;
        return r183022;
}
↓
double f(double x, double y) {
        double r183023 = 1.0;
        double r183024 = 3.0;
        double r183025 = pow(r183023, r183024);
        double r183026 = x;
        double r183027 = exp(r183026);
        double r183028 = pow(r183027, r183024);
        double r183029 = r183025 + r183028;
        double r183030 = r183027 - r183023;
        double r183031 = r183023 * r183023;
        double r183032 = fma(r183027, r183030, r183031);
        double r183033 = r183029 / r183032;
        double r183034 = log(r183033);
        double r183035 = y;
        double r183036 = r183026 * r183035;
        double r183037 = r183034 - r183036;
        return r183037;
}

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)))