Logistic regression 2

Percentage Accurate: 79.4% → 89.7%

Time: 16.1s

Precision: binary64

Cost: 13252

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\mathsf{log1p}\left(e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 + \log 2\right) - x \cdot y\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x 700.0)
   (- (log1p (exp x)) (* 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 <= 700.0) {
		tmp = log1p(exp(x)) - (x * y);
	} else {
		tmp = ((x * 0.5) + log(2.0)) - (x * y);
	}
	return tmp;
}

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 <= 700.0) {
		tmp = Math.log1p(Math.exp(x)) - (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 <= 700.0:
		tmp = math.log1p(math.exp(x)) - (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 <= 700.0)
		tmp = Float64(log1p(exp(x)) - Float64(x * y));
	else
		tmp = Float64(Float64(Float64(x * 0.5) + log(2.0)) - Float64(x * y));
	end
	return 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, 700.0], N[(N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Target

Original	79.4%
Target	99.9%
Herbie	89.7%

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

Derivation

1. Split input into 2 regimes

2. if x < 700

   1. Initial program 100.0%

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

   2. Simplified 100.0%

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

      [Start] 100.0%	\[ \log \left(1 + e^{x}\right) - x \cdot y \]
      log1p-def [=>] 100.0%	\[ \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   if 700 < x

   1. Initial program 13.7%

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

   2. Simplified 13.7%

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

      [Start] 13.7%	\[ \log \left(1 + e^{x}\right) - x \cdot y \]
      log1p-def [=>] 13.7%	\[ \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Taylor expanded in x around 0 57.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\mathsf{log1p}\left(e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 + \log 2\right) - x \cdot y\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.7%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;x \leq 700:\\ \;\;\;\;\mathsf{log1p}\left(e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 + \log 2\right) - x \cdot y\\ \end{array} \]

Alternative 2
Accuracy	89.2%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5 + \log 2\right) - x \cdot y\\ \end{array} \]

Alternative 3
Accuracy	88.9%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -55:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\log 2 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 4
Accuracy	89.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 5
Accuracy	77.6%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-39}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 6
Accuracy	52.1%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 7
Accuracy	49.9%
Cost	256

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

Reproduce

herbie shell --seed 2023277 
(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)))