Logistic regression 2

Average Error: 0.6 → 0.9

Time: 9.0s

Precision: binary64

Cost: 13252

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -9294685459.320797:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= x -9294685459.320797) (* y (- x)) (fma x (- 0.5 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 (x <= -9294685459.320797) {
		tmp = y * -x;
	} else {
		tmp = fma(x, (0.5 - y), 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 (x <= -9294685459.320797)
		tmp = Float64(y * Float64(-x));
	else
		tmp = fma(x, Float64(0.5 - 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[x, -9294685459.320797], N[(y * (-x)), $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Target

Original	0.6
Target	0.1
Herbie	0.9

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

Derivation

1. Split input into 2 regimes

2. if x < -9294685459.320797

   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 inf 0

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

   4. Simplified 0

      \[\leadsto \color{blue}{x \cdot \left(-y\right)} \]
      Proof
      (*.f64 x (neg.f64 y)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x y))): 0 points increase in error, 0 points decrease in error
      (neg.f64 (Rewrite<= *-commutative_binary64 (*.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 -9294685459.320797 < x

   1. Initial program 0.8

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
      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 1.2

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

   4. Simplified 1.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 - y, \log 2\right)} \]
      Proof
      (fma.f64 x (-.f64 1/2 y) (log.f64 2)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x (-.f64 1/2 y)) (log.f64 2))): 1 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1/2 y) x)) (log.f64 2)): 0 points increase in error, 0 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9294685459.320797:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	13120

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

Alternative 2
Error	12.0
Cost	6984

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3604006377644673 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.2332186922492038 \cdot 10^{-54}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 3
Error	0.9
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq -9294685459.320797:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 4
Error	1.1
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -9294685459.320797:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 5
Error	12.1
Cost	6728

\[\begin{array}{l} \mathbf{if}\;x \leq -3.3604006377644673 \cdot 10^{-6}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;x \leq 1.2332186922492038 \cdot 10^{-54}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 6
Error	34.3
Cost	256

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

Alternative 7
Error	61.7
Cost	192

\[x \cdot 0.5 \]

Reproduce

herbie shell --seed 2022291 
(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)))