Average Error: 0.5 → 0.4
Time: 6.2s
Precision: binary64
\[\log \left(1 + e^{x}\right) - x \cdot y \]
↓
\[\mathsf{fma}\left(1, \mathsf{log1p}\left(e^{x}\right), y \cdot \left(-x\right)\right) \]
(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))
↓
(FPCore (x y) :precision binary64 (fma 1.0 (log1p (exp x)) (* y (- x))))
double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}
↓
double code(double x, double y) {
	return fma(1.0, log1p(exp(x)), (y * -x));
}

function code(x, y)
	return Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
end
↓
function code(x, y)
	return fma(1.0, log1p(exp(x)), Float64(y * Float64(-x)))
end

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(1.0 * N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision] + N[(y * (-x)), $MachinePrecision]), $MachinePrecision]

\log \left(1 + e^{x}\right) - x \cdot y
↓
\mathsf{fma}\left(1, \mathsf{log1p}\left(e^{x}\right), y \cdot \left(-x\right)\right)

Error

Bits error versus x

Bits error versus y

Target

Original0.5
Target0.0
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \]

Derivation

  1. Initial program 0.5

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

  2. Simplified0.4

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

  3. Applied egg-rr15.9

    \[\leadsto \color{blue}{\left({\left(\mathsf{log1p}\left(e^{x}\right)\right)}^{3} - {\left(x \cdot y\right)}^{3}\right) \cdot \frac{1}{{\left(\mathsf{log1p}\left(e^{x}\right)\right)}^{2} + \left(x \cdot y\right) \cdot \left(\mathsf{log1p}\left(e^{x}\right) + x \cdot y\right)}} \]

  4. Applied egg-rr0.4

    \[\leadsto \color{blue}{\mathsf{fma}\left(1, \mathsf{log1p}\left(e^{x}\right), -x \cdot y\right)} \]

  5. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(1, \mathsf{log1p}\left(e^{x}\right), y \cdot \left(-x\right)\right) \]

Reproduce

herbie shell --seed 2022155 
(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)))