Logistic regression 2

Percentage Accurate: 79.3% → 99.7%

Time: 10.1s

Precision: binary64

Cost: 26436

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 10^{+248}:\\ \;\;\;\;\mathsf{log1p}\left(e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x - x \cdot y\right) + \log 2\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (- (log (+ 1.0 (exp x))) (* x y)) 1e+248)
   (- (log1p (exp x)) (* x y))
   (+ (- x (* x 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 ((log((1.0 + exp(x))) - (x * y)) <= 1e+248) {
		tmp = log1p(exp(x)) - (x * y);
	} else {
		tmp = (x - (x * y)) + log(2.0);
	}
	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 ((Math.log((1.0 + Math.exp(x))) - (x * y)) <= 1e+248) {
		tmp = Math.log1p(Math.exp(x)) - (x * y);
	} else {
		tmp = (x - (x * 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 (math.log((1.0 + math.exp(x))) - (x * y)) <= 1e+248:
		tmp = math.log1p(math.exp(x)) - (x * y)
	else:
		tmp = (x - (x * 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 (Float64(log(Float64(1.0 + exp(x))) - Float64(x * y)) <= 1e+248)
		tmp = Float64(log1p(exp(x)) - Float64(x * y));
	else
		tmp = Float64(Float64(x - Float64(x * y)) + log(2.0));
	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[N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], 1e+248], N[(N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Target

Original	79.3%
Target	99.9%
Herbie	99.7%

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 1 (exp.f64 x))) (*.f64 x y)) < 1.00000000000000005e248

   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 1.00000000000000005e248 < (-.f64 (log.f64 (+.f64 1 (exp.f64 x))) (*.f64 x y))

   1. Initial program 43.7%

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

   2. Simplified 43.7%

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

      [Start] 43.7%	\[ \log \left(1 + e^{x}\right) - x \cdot y \]
      log1p-def [=>] 43.7%	\[ \color{blue}{\mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

   3. Taylor expanded in x around 0 67.8%

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

   4. Applied egg-rr 10.1%

      \[\leadsto \color{blue}{\left({\left(x \cdot \left(0.5 - y\right)\right)}^{2} - {\log 2}^{2}\right) \cdot \frac{1}{x \cdot \left(0.5 - y\right) - \log 2}} \]
      Step-by-step derivation

      [Start] 67.8%	\[ \left(0.5 - y\right) \cdot x + \log 2 \]
      flip-+ [=>] 10.1%	\[ \color{blue}{\frac{\left(\left(0.5 - y\right) \cdot x\right) \cdot \left(\left(0.5 - y\right) \cdot x\right) - \log 2 \cdot \log 2}{\left(0.5 - y\right) \cdot x - \log 2}} \]
      div-inv [=>] 10.1%	\[ \color{blue}{\left(\left(\left(0.5 - y\right) \cdot x\right) \cdot \left(\left(0.5 - y\right) \cdot x\right) - \log 2 \cdot \log 2\right) \cdot \frac{1}{\left(0.5 - y\right) \cdot x - \log 2}} \]
      pow2 [=>] 10.1%	\[ \left(\color{blue}{{\left(\left(0.5 - y\right) \cdot x\right)}^{2}} - \log 2 \cdot \log 2\right) \cdot \frac{1}{\left(0.5 - y\right) \cdot x - \log 2} \]
      *-commutative [=>] 10.1%	\[ \left({\color{blue}{\left(x \cdot \left(0.5 - y\right)\right)}}^{2} - \log 2 \cdot \log 2\right) \cdot \frac{1}{\left(0.5 - y\right) \cdot x - \log 2} \]
      pow2 [=>] 10.1%	\[ \left({\left(x \cdot \left(0.5 - y\right)\right)}^{2} - \color{blue}{{\log 2}^{2}}\right) \cdot \frac{1}{\left(0.5 - y\right) \cdot x - \log 2} \]
      *-commutative [=>] 10.1%	\[ \left({\left(x \cdot \left(0.5 - y\right)\right)}^{2} - {\log 2}^{2}\right) \cdot \frac{1}{\color{blue}{x \cdot \left(0.5 - y\right)} - \log 2} \]

   5. Taylor expanded in y around inf 67.8%

      \[\leadsto \color{blue}{\left(\log 2 + \left(x + -1 \cdot \left(y \cdot x\right)\right)\right) - 0.5 \cdot x} \]

   6. Simplified 67.8%

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

      [Start] 67.8%	\[ \left(\log 2 + \left(x + -1 \cdot \left(y \cdot x\right)\right)\right) - 0.5 \cdot x \]
      +-commutative [=>] 67.8%	\[ \color{blue}{\left(\left(x + -1 \cdot \left(y \cdot x\right)\right) + \log 2\right)} - 0.5 \cdot x \]
      associate--l+ [=>] 67.8%	\[ \color{blue}{\left(x + -1 \cdot \left(y \cdot x\right)\right) + \left(\log 2 - 0.5 \cdot x\right)} \]
      *-commutative [<=] 67.8%	\[ \left(x + -1 \cdot \color{blue}{\left(x \cdot y\right)}\right) + \left(\log 2 - 0.5 \cdot x\right) \]
      neg-mul-1 [<=] 67.8%	\[ \left(x + \color{blue}{\left(-x \cdot y\right)}\right) + \left(\log 2 - 0.5 \cdot x\right) \]
      unsub-neg [=>] 67.8%	\[ \color{blue}{\left(x - x \cdot y\right)} + \left(\log 2 - 0.5 \cdot x\right) \]
      *-commutative [=>] 67.8%	\[ \left(x - \color{blue}{y \cdot x}\right) + \left(\log 2 - 0.5 \cdot x\right) \]
      sub-neg [=>] 67.8%	\[ \left(x - y \cdot x\right) + \color{blue}{\left(\log 2 + \left(-0.5 \cdot x\right)\right)} \]
      *-commutative [=>] 67.8%	\[ \left(x - y \cdot x\right) + \left(\log 2 + \left(-\color{blue}{x \cdot 0.5}\right)\right) \]
      distribute-rgt-neg-in [=>] 67.8%	\[ \left(x - y \cdot x\right) + \left(\log 2 + \color{blue}{x \cdot \left(-0.5\right)}\right) \]
      metadata-eval [=>] 67.8%	\[ \left(x - y \cdot x\right) + \left(\log 2 + x \cdot \color{blue}{-0.5}\right) \]

   7. Taylor expanded in x around 0 98.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 10^{+248}:\\ \;\;\;\;\mathsf{log1p}\left(e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x - x \cdot y\right) + \log 2\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.7%
Cost	26436

\[\begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 10^{+248}:\\ \;\;\;\;\mathsf{log1p}\left(e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(x - x \cdot y\right) + \log 2\\ \end{array} \]

Alternative 2
Accuracy	88.9%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -58:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.4:\\ \;\;\;\;\log 2 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 3
Accuracy	89.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -2.05:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 4
Accuracy	99.1%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -2.05:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - x \cdot y\right) + \log 2\\ \end{array} \]

Alternative 5
Accuracy	76.4%
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-39}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-47}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 6
Accuracy	52.4%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 7
Accuracy	50.2%
Cost	256

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

Alternative 8
Accuracy	5.4%
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2023272 
(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)))