Average Error: 0.5 → 0.5
Time: 4.6s
Precision: 64
Internal Precision: 832
\[\log \left(1 + e^{x}\right) - x \cdot y\]
↓
\[\log_* (1 + e^{x}) - y \cdot x\]

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

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Target

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

Derivation

  1. Initial program 0.5

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

  2. Initial simplification0.5

    \[\leadsto \log_* (1 + e^{x}) - y \cdot x\]

  3. Final simplification0.5

    \[\leadsto \log_* (1 + e^{x}) - y \cdot x\]

Runtime

Time bar (total: 4.6s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.50.50.50.0100%
herbie shell --seed 2018340 +o rules:numerics
(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)))