Logistic regression 2

Average Error: 0.4 → 0.7

Time: 7.2s

Precision: binary64

Cost: 7108

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -8500:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 + \log 2\right) - x \cdot y\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -8500.0) (* x (- y)) (- (+ (* 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 <= -8500.0) {
		tmp = x * -y;
	} else {
		tmp = ((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 <= (-8500.0d0)) then
        tmp = x * -y
    else
        tmp = ((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 <= -8500.0) {
		tmp = x * -y;
	} else {
		tmp = ((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 <= -8500.0:
		tmp = x * -y
	else:
		tmp = ((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 <= -8500.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(Float64(Float64(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 <= -8500.0)
		tmp = x * -y;
	else
		tmp = ((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, -8500.0], N[(x * (-y)), $MachinePrecision], N[(N[(N[(x * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Target

Original	0.4
Target	0.0
Herbie	0.7

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

Derivation

1. Split input into 2 regimes

2. if x < -8500

   1. Initial program 0

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

   2. Simplified 0

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
      Proof
      (-.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, 0 points decrease in error

   3. Taylor expanded in x around 0 21.9

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

   4. Taylor expanded in x around inf 0

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

   5. Simplified 0

      \[\leadsto \color{blue}{y \cdot \left(-x\right)} \]
      Proof
      (*.f64 y (neg.f64 x)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y x))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 y x))): 0 points increase in error, 0 points decrease in error

   if -8500 < x

   1. Initial program 0.6

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

   2. Simplified 0.5

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
      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, 0 points decrease in error

   3. Taylor expanded in x around 0 1.0

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8500:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 + \log 2\right) - x \cdot y\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	13120

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

Alternative 2
Error	1.0
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -8500:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 3
Error	12.3
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-13}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-32}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 4
Error	34.3
Cost	256

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

Alternative 5
Error	61.7
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2022328 
(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)))