Average Error: 0.5 → 0.5
Time: 3.5s
Precision: 64
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\left(\log \left({1}^{3} + {\left(e^{x}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(e^{x} \cdot e^{x} - 1 \cdot e^{x}\right)\right)\right) - x \cdot y\]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\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
double f(double x, double y) {
        double r143136 = 1.0;
        double r143137 = x;
        double r143138 = exp(r143137);
        double r143139 = r143136 + r143138;
        double r143140 = log(r143139);
        double r143141 = y;
        double r143142 = r143137 * r143141;
        double r143143 = r143140 - r143142;
        return r143143;
}
↓
double f(double x, double y) {
        double r143144 = 1.0;
        double r143145 = 3.0;
        double r143146 = pow(r143144, r143145);
        double r143147 = x;
        double r143148 = exp(r143147);
        double r143149 = pow(r143148, r143145);
        double r143150 = r143146 + r143149;
        double r143151 = log(r143150);
        double r143152 = r143144 * r143144;
        double r143153 = r143148 * r143148;
        double r143154 = r143144 * r143148;
        double r143155 = r143153 - r143154;
        double r143156 = r143152 + r143155;
        double r143157 = log(r143156);
        double r143158 = r143151 - r143157;
        double r143159 = y;
        double r143160 = r143147 * r143159;
        double r143161 = r143158 - r143160;
        return r143161;
}

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
Target0.1
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 flip3-+0.5

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

  4. Applied log-div0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020021 +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)))