Logistic regression 2

Average Accuracy: 99.3% → 99.3%

Time: 9.9s

Precision: binary64

Cost: 19456

Math

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

↓

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

FPCore

(FPCore (x y) :precision binary64 (- (log (+ 1.0 (exp x))) (* x y)))

↓

(FPCore (x y) :precision binary64 (fma x (- y) (log1p (exp x))))

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

↓

double code(double x, double y) {
	return fma(x, -y, log1p(exp(x)));
}

Julia

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

↓

function code(x, y)
	return fma(x, Float64(-y), log1p(exp(x)))
end

Wolfram

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

↓

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

Target

Original	99.3%
Target	99.9%
Herbie	99.3%

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

Derivation

1. Initial program 99.3%

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

2. Simplified 99.3%

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

   [Start] 99.3	\[ \log \left(1 + e^{x}\right) - x \cdot y \]
   sub-neg [=>] 99.3	\[ \color{blue}{\log \left(1 + e^{x}\right) + \left(-x \cdot y\right)} \]
   +-commutative [=>] 99.3	\[ \color{blue}{\left(-x \cdot y\right) + \log \left(1 + e^{x}\right)} \]
   distribute-rgt-neg-in [=>] 99.3	\[ \color{blue}{x \cdot \left(-y\right)} + \log \left(1 + e^{x}\right) \]
   fma-def [=>] 99.3	\[ \color{blue}{\mathsf{fma}\left(x, -y, \log \left(1 + e^{x}\right)\right)} \]
   log1p-def [=>] 99.3	\[ \mathsf{fma}\left(x, -y, \color{blue}{\mathsf{log1p}\left(e^{x}\right)}\right) \]

3. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	13120

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

Alternative 2
Accuracy	98.6%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -52000000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

Alternative 3
Accuracy	80.4%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00037:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]

Alternative 4
Accuracy	98.1%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -52000000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 5
Accuracy	79.9%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 6
Accuracy	47.3%
Cost	256

\[y \cdot \left(-x\right) \]

Alternative 7
Accuracy	3.5%
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2023151 
(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)))