Average Error: 0.5 → 0.9
Time: 38.7s
Precision: 64
Internal Precision: 640
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00010381768890495736:\\ \;\;\;\;\left(\log \left(1 - e^{x} \cdot e^{x}\right) - \log \left(1 - e^{x}\right)\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\frac{1}{2} \cdot {x}^{2}\right)}^{3} + {\left(2 + x\right)}^{3}\right) - \left(\log \left(\left(x + 2\right) \cdot \left(x + 2\right) - \left(\left(x + 2\right) - \left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right) + y \cdot x\right)\\ \end{array}\]

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.0
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \le 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00010381768890495736

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied log-div0.1

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

    5. Applied simplify0.1

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

    if -0.00010381768890495736 < x

    1. Initial program 0.6

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

    2. Taylor expanded around 0 0.5

      \[\leadsto \log \color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + \left(2 + x\right)\right)} - x \cdot y\]
    3. Using strategy rm

    4. Applied flip3-+0.5

      \[\leadsto \log \color{blue}{\left(\frac{{\left(\frac{1}{2} \cdot {x}^{2}\right)}^{3} + {\left(2 + x\right)}^{3}}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \left(\left(2 + x\right) \cdot \left(2 + x\right) - \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(2 + x\right)\right)}\right)} - x \cdot y\]

    5. Applied log-div1.2

      \[\leadsto \color{blue}{\left(\log \left({\left(\frac{1}{2} \cdot {x}^{2}\right)}^{3} + {\left(2 + x\right)}^{3}\right) - \log \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \left(\left(2 + x\right) \cdot \left(2 + x\right) - \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(2 + x\right)\right)\right)\right)} - x \cdot y\]

    6. Applied associate--l-1.2

      \[\leadsto \color{blue}{\log \left({\left(\frac{1}{2} \cdot {x}^{2}\right)}^{3} + {\left(2 + x\right)}^{3}\right) - \left(\log \left(\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(\frac{1}{2} \cdot {x}^{2}\right) + \left(\left(2 + x\right) \cdot \left(2 + x\right) - \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \left(2 + x\right)\right)\right) + x \cdot y\right)}\]

    7. Applied simplify1.2

      \[\leadsto \log \left({\left(\frac{1}{2} \cdot {x}^{2}\right)}^{3} + {\left(2 + x\right)}^{3}\right) - \color{blue}{\left(\log \left(\left(x + 2\right) \cdot \left(x + 2\right) - \left(\left(x + 2\right) - \left(x \cdot x\right) \cdot \frac{1}{2}\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{2}\right)\right) + y \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 38.7s)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' 
(FPCore (x y)
  :name "Logistic regression 2"

  :herbie-target
  (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y))))

  (- (log (+ 1 (exp x))) (* x y)))