Logistic regression 2

Average Error: 0.5 → 0.4

Time: 8.2s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

public static double code(double x, double y) {
	return Math.log1p(Math.exp(x)) - (x * y);
}

Python

def code(x, y):
	return math.log((1.0 + math.exp(x))) - (x * y)

↓

def code(x, y):
	return math.log1p(math.exp(x)) - (x * y)

Julia

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

↓

function code(x, y)
	return Float64(log1p(exp(x)) - Float64(x * y))
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[(N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Target

Original	0.5
Target	0.0
Herbie	0.4

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

Derivation

1. Initial program 0.5

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

2. Simplified 0.4

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   Proof
   (-.f64 (log1p.f64 (exp.f64 x)) (*.f64 x y)): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= log1p-def_binary64 (log.f64 (+.f64 1 (exp.f64 x)))) (*.f64 x y)): 2 points increase in error, 1 points decrease in error

3. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.6
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

Alternative 2
Error	12.5
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]

Alternative 3
Error	1.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 4
Error	12.6
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 5
Error	34.4
Cost	256

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

Alternative 6
Error	61.7
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2022329 
(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)))