Logistic regression 2

Average Error: 0.9% → 1.04%

Time: 10.2s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1.36:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -1.36) (* x (- y)) (+ (* x (- 0.5 y)) (log 2.0))))

C

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

↓

double code(double x, double y) {
	double tmp;
	if (x <= -1.36) {
		tmp = x * -y;
	} else {
		tmp = (x * (0.5 - y)) + log(2.0);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (x <= (-1.36d0)) then
        tmp = x * -y
    else
        tmp = (x * (0.5d0 - y)) + log(2.0d0)
    end if
    code = tmp
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) {
	double tmp;
	if (x <= -1.36) {
		tmp = x * -y;
	} else {
		tmp = (x * (0.5 - y)) + Math.log(2.0);
	}
	return tmp;
}

Python

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

↓

def code(x, y):
	tmp = 0
	if x <= -1.36:
		tmp = x * -y
	else:
		tmp = (x * (0.5 - y)) + math.log(2.0)
	return tmp

Julia

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

↓

function code(x, y)
	tmp = 0.0
	if (x <= -1.36)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(Float64(x * Float64(0.5 - y)) + log(2.0));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.36)
		tmp = x * -y;
	else
		tmp = (x * (0.5 - y)) + log(2.0);
	end
	tmp_2 = tmp;
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_] := If[LessEqual[x, -1.36], N[(x * (-y)), $MachinePrecision], N[(N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Target

Original	0.9%
Target	0.12%
Herbie	1.04%

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3600000000000001

   1. Initial program 0.36

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

   2. Simplified 0

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

      [Start] 0.36	\[ \log \left(1 + e^{x}\right) - x \cdot y \]
      log1p-def [=>] 0	\[ \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Taylor expanded in x around inf 0.44

      \[\leadsto \color{blue}{-1 \cdot \left(y \cdot x\right)} \]

   4. Simplified 0.44

      \[\leadsto \color{blue}{x \cdot \left(-y\right)} \]
      Proof

      [Start] 0.44	\[ -1 \cdot \left(y \cdot x\right) \]
      *-commutative [=>] 0.44	\[ -1 \cdot \color{blue}{\left(x \cdot y\right)} \]
      mul-1-neg [=>] 0.44	\[ \color{blue}{-x \cdot y} \]
      distribute-rgt-neg-out [<=] 0.44	\[ \color{blue}{x \cdot \left(-y\right)} \]

   if -1.3600000000000001 < x

   1. Initial program 1.1

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

   2. Simplified 1.09

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

      [Start] 1.1	\[ \log \left(1 + e^{x}\right) - x \cdot y \]
      log1p-def [=>] 1.09	\[ \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Taylor expanded in x around 0 1.27

      \[\leadsto \color{blue}{\left(0.5 - y\right) \cdot x + \log 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.04

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.36:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

Alternatives

Alternative 1
Error	0.79%
Cost	13120

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

Alternative 2
Error	19.67%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-30}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]

Alternative 3
Error	1.54%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -76:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 4
Error	19.91%
Cost	6596

\[\begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-31}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 5
Error	53.22%
Cost	256

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

Alternative 6
Error	96.45%
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2023115 
(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)))