Logistic regression 2

Average Error: 0.6 → 0.5

Time: 7.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := 0.5 \cdot \mathsf{log1p}\left(e^{x}\right)\\ t_0 + \left(t_0 - x \cdot y\right) \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.5 (log1p (exp x))))) (+ t_0 (- t_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 t_0 = 0.5 * log1p(exp(x));
	return t_0 + (t_0 - (x * y));
}

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 t_0 = 0.5 * Math.log1p(Math.exp(x));
	return t_0 + (t_0 - (x * y));
}

Python

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

↓

def code(x, y):
	t_0 = 0.5 * math.log1p(math.exp(x))
	return t_0 + (t_0 - (x * y))

Julia

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

↓

function code(x, y)
	t_0 = Float64(0.5 * log1p(exp(x)))
	return Float64(t_0 + Float64(t_0 - Float64(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_] := Block[{t$95$0 = N[(0.5 * N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(t$95$0 + N[(t$95$0 - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Error

Bits error versus x

Bits error versus y

Target

Original	0.6
Target	0.1
Herbie	0.5

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

Derivation

1. Initial program 0.6

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

2. Applied add-sqr-sqrt_binary64 1.4

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

3. Applied log-prod_binary64 1.1

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

4. Applied associate--l+_binary64 1.1

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

5. Simplified 1.1

   \[\leadsto \log \left(\sqrt{1 + e^{x}}\right) + \color{blue}{\left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right) - x \cdot y\right)} \]

6. Applied pow1/2_binary64 1.1

   \[\leadsto \log \color{blue}{\left({\left(1 + e^{x}\right)}^{0.5}\right)} + \left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right) - x \cdot y\right) \]

7. Applied log-pow_binary64 0.6

   \[\leadsto \color{blue}{0.5 \cdot \log \left(1 + e^{x}\right)} + \left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right) - x \cdot y\right) \]

8. Simplified 0.5

   \[\leadsto 0.5 \cdot \color{blue}{\mathsf{log1p}\left(e^{x}\right)} + \left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right) - x \cdot y\right) \]

9. Final simplification 0.5

   \[\leadsto 0.5 \cdot \mathsf{log1p}\left(e^{x}\right) + \left(0.5 \cdot \mathsf{log1p}\left(e^{x}\right) - x \cdot y\right) \]

Reproduce

herbie shell --seed 2022129 
(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)))