Logistic regression 2

Average Error: 0.6 → 0.6

Time: 14.5s

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	0.6
Target	0.1
Herbie	0.6

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

Derivation

1. Initial program 0.6

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

2. Final simplification 0.6

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

Alternatives

Alternative 1
Error	13.3
Cost	7116

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-9}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;\log \left(2 + x\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-113}:\\ \;\;\;\;\left(0.5 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x + \log 2\\ \end{array} \]

Alternative 2
Error	13.2
Cost	6988

\[\begin{array}{l} t_0 := \log \left(2 + x\right)\\ \mathbf{if}\;x \leq -1.02 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-113}:\\ \;\;\;\;\left(0.5 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.6
Cost	6980

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

Alternative 4
Error	13.4
Cost	6860

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-10}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-134}:\\ \;\;\;\;\log 2\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-113}:\\ \;\;\;\;\left(0.5 - y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 5
Error	1.2
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -7800000000000:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 6
Error	33.5
Cost	256

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

Alternative 7
Error	61.7
Cost	192

\[0.5 \cdot x \]

Reproduce

herbie shell --seed 2023077 
(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)))