Logistic regression 2

Average Error: 0.5 → 0.5

Time: 8.2s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((1.0d0 + exp(x))) - (x * y)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((1.0d0 + exp(x))) - (x * y)
end function

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.log((1.0 + 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.log((1.0 + 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(log(Float64(1.0 + exp(x))) - Float64(x * y))
end

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

↓

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (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[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Target

Original	0.5
Target	0.0
Herbie	0.5

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

Derivation

1. Initial program 0.5

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

2. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -80000000000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 - y\right) \cdot x + \log 2\\ \end{array} \]

Alternative 2
Error	1.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -950000000000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 3
Error	12.0
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-33}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-44}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 - y\right) \cdot x\\ \end{array} \]

Alternative 4
Error	34.1
Cost	256

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

Alternative 5
Error	61.8
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023092 
(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)))