Logistic regression 2

Average Error: 0.5 → 0.7

Time: 5.7s

Precision: 64

• could not determine a ground truth for program body

  1. x = 3.479121160436394e+34
  2. y = -1.6040526032817678e-230

Math

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

↓

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

C

double f(double x, double y) {
        double r128056 = 1.0;
        double r128057 = x;
        double r128058 = exp(r128057);
        double r128059 = r128056 + r128058;
        double r128060 = log(r128059);
        double r128061 = y;
        double r128062 = r128057 * r128061;
        double r128063 = r128060 - r128062;
        return r128063;
}

↓

double f(double x, double y) {
        double r128064 = 1.0;
        double r128065 = 3.0;
        double r128066 = pow(r128064, r128065);
        double r128067 = x;
        double r128068 = exp(r128067);
        double r128069 = pow(r128068, r128065);
        double r128070 = r128066 + r128069;
        double r128071 = log(r128070);
        double r128072 = y;
        double r128073 = r128064 * r128064;
        double r128074 = r128068 * r128068;
        double r128075 = r128064 * r128068;
        double r128076 = r128074 - r128075;
        double r128077 = r128073 + r128076;
        double r128078 = log(r128077);
        double r128079 = cbrt(r128078);
        double r128080 = r128079 * r128079;
        double r128081 = exp(r128068);
        double r128082 = r128068 - r128064;
        double r128083 = pow(r128081, r128082);
        double r128084 = log(r128083);
        double r128085 = r128073 + r128084;
        double r128086 = log(r128085);
        double r128087 = cbrt(r128086);
        double r128088 = r128080 * r128087;
        double r128089 = fma(r128067, r128072, r128088);
        double r128090 = r128071 - r128089;
        return r128090;
}

Error

Bits error versus x

Bits error versus y

Target

Original	0.5
Target	0.1
Herbie	0.7

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

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-div 0.5

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

5. Applied associate--l- 0.5

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

6. Simplified 0.5

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

8. Applied add-cube-cbrt 0.5

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

10. Applied add-log-exp 0.6

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

11. Applied add-log-exp 0.7

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

12. Applied diff-log 0.7

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

13. Simplified 0.7

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

14. Final simplification 0.7

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

Reproduce

herbie shell --seed 2020047 +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)))