Logistic regression 2

Average Error: 0.5 → 0.8

Time: 14.2s

Precision: binary64

Cost: 7364

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -900000000:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \left(0.125 \cdot x + 0.5\right) + \log 2\right) - x \cdot y\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -900000000.0)
   (- (* x y))
   (- (+ (* x (+ (* 0.125 x) 0.5)) (log 2.0)) (* x y))))

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 <= -900000000.0) {
		tmp = -(x * y);
	} else {
		tmp = ((x * ((0.125 * x) + 0.5)) + log(2.0)) - (x * y);
	}
	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 <= (-900000000.0d0)) then
        tmp = -(x * y)
    else
        tmp = ((x * ((0.125d0 * x) + 0.5d0)) + log(2.0d0)) - (x * y)
    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 <= -900000000.0) {
		tmp = -(x * y);
	} else {
		tmp = ((x * ((0.125 * x) + 0.5)) + Math.log(2.0)) - (x * y);
	}
	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 <= -900000000.0:
		tmp = -(x * y)
	else:
		tmp = ((x * ((0.125 * x) + 0.5)) + math.log(2.0)) - (x * y)
	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 <= -900000000.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = Float64(Float64(Float64(x * Float64(Float64(0.125 * x) + 0.5)) + log(2.0)) - Float64(x * y));
	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 <= -900000000.0)
		tmp = -(x * y);
	else
		tmp = ((x * ((0.125 * x) + 0.5)) + log(2.0)) - (x * y);
	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, -900000000.0], (-N[(x * y), $MachinePrecision]), N[(N[(N[(x * N[(N[(0.125 * x), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.5
Target	0.1
Herbie	0.8

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

Derivation

1. Split input into 2 regimes

2. if x < -9e8

   1. Initial program 0

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

   2. Taylor expanded in x around 0 21.9

      \[\leadsto \log \color{blue}{2} - x \cdot y \]

   3. Taylor expanded in x around inf 0

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

   4. Simplified 0

      \[\leadsto \color{blue}{-x \cdot y} \]
      Proof

   if -9e8 < x

   1. Initial program 0.7

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

   2. Taylor expanded in x around 0 1.1

      \[\leadsto \log \color{blue}{\left(2 + \left(0.5 \cdot {x}^{2} + x\right)\right)} - x \cdot y \]

   3. Simplified 1.1

      \[\leadsto \log \color{blue}{\left(2 + \mathsf{fma}\left(0.5, x \cdot x, x\right)\right)} - x \cdot y \]
      Proof

   4. Taylor expanded in x around 0 1.0

      \[\leadsto \color{blue}{\left(0.125 \cdot {x}^{2} + \left(0.5 \cdot x + \log 2\right)\right)} - x \cdot y \]

   5. Simplified 1.0

      \[\leadsto \color{blue}{\left(x \cdot \left(0.125 \cdot x + 0.5\right) + \log 2\right)} - x \cdot y \]
      Proof
3. Recombined 2 regimes into one program.

Alternatives

Alternative 1
Error	0.5
Cost	13120

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

Alternative 2
Error	0.8
Cost	7364

\[\begin{array}{l} \mathbf{if}\;x \leq -900000000:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + 2\right) + 0.5 \cdot \left(x \cdot x\right)\right) - x \cdot y\\ \end{array} \]

Alternative 3
Error	0.8
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -900000000:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot \left(y - 0.5\right)\\ \end{array} \]

Alternative 4
Error	1.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -900000000:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 5
Error	34.2
Cost	256

\[-x \cdot y \]

Reproduce

herbie shell --seed 2023010 
(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)))