Average Error: 0.6 → 0.5
Time: 23.0s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y \]
↓
\[\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right) \]
\log \left(1 + e^{x}\right) - x \cdot y
↓
\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right)

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y) :precision binary64 (fma (- y) x (log1p (exp x))))
double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}
↓
double code(double x, double y) {
	return fma(-y, x, log1p(exp(x)));
}

Error

Bits error versus x

Bits error versus y

Target

Original0.6
Target0.1
Herbie0.5
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \]

Derivation

  1. Initial program 0.6

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

  2. Applied egg-rr0.5

    \[\leadsto \color{blue}{\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right)} \]

  3. Final simplification0.5

    \[\leadsto \mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right) \]

Reproduce

herbie shell --seed 2022129 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :herbie-target
  (if (<= x 0.0) (- (log (+ 1.0 (exp x))) (* x y)) (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y))))

  (- (log (+ 1.0 (exp x))) (* x y)))