Logistic regression 2

Percentage Accurate: 99.4% → 99.5%

Time: 6.6s

Alternatives: 8

Speedup: 1.7×

• could not determine a ground truth

  1. x = 3.413344329022236e+247
  2. y = -1.7112591790580352e-213

Specification

Math

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

FPCore

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

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

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

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

Wolfram

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return log((1.0 + exp(x))) - (x * y);
}

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

Java

public static double code(double x, double y) {
	return Math.log((1.0 + Math.exp(x))) - (x * y);
}

Python

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

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = log((1.0 + exp(x))) - (x * y);
end

Wolfram

code[x_, y_] := N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (- y) x (log1p (exp x))))

C

double code(double x, double y) {
	return fma(-y, x, log1p(exp(x)));
}

Julia

function code(x, y)
	return fma(Float64(-y), x, log1p(exp(x)))
end

Wolfram

code[x_, y_] := N[((-y) * x + N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.6%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(y\right), x, \log \left(1 + e^{x}\right)\right)} \]

   8. lower-neg.f64 98.6

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

   9. lift-log.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), x, \color{blue}{\log \left(1 + e^{x}\right)}\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), x, \log \color{blue}{\left(1 + e^{x}\right)}\right) \]

   11. lower-log1p.f64 98.9

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

4. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-y, x, \mathsf{log1p}\left(e^{x}\right)\right)} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(e^{x} + 1\right) - y \cdot x\\ t_1 := \left(-y\right) \cdot x\\ \mathbf{if}\;t\_0 \leq 0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ (exp x) 1.0)) (* y x))) (t_1 (* (- y) x)))
   (if (<= t_0 0.2) t_1 (if (<= t_0 20.0) (fma x 0.5 (log 2.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((exp(x) + 1.0)) - (y * x);
	double t_1 = -y * x;
	double tmp;
	if (t_0 <= 0.2) {
		tmp = t_1;
	} else if (t_0 <= 20.0) {
		tmp = fma(x, 0.5, log(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(log(Float64(exp(x) + 1.0)) - Float64(y * x))
	t_1 = Float64(Float64(-y) * x)
	tmp = 0.0
	if (t_0 <= 0.2)
		tmp = t_1;
	elseif (t_0 <= 20.0)
		tmp = fma(x, 0.5, log(2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(N[Exp[x], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-y) * x), $MachinePrecision]}, If[LessEqual[t$95$0, 0.2], t$95$1, If[LessEqual[t$95$0, 20.0], N[(x * 0.5 + N[Log[2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 0.20000000000000001 or 20 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 98.9%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x \cdot y\right)} \]

      2. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \left(\mathsf{neg}\left(y\right)\right)} \]

      4. lower-neg.f64 97.7

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

   5. Applied rewrites 97.7%

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

   if 0.20000000000000001 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 20

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log \left(1 + e^{x}\right)} \]
   4. Step-by-step derivation

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log 2 + \color{blue}{\frac{1}{2} \cdot x} \]
   7. Step-by-step derivation

   8. Applied rewrites 97.9%

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{0.5}, \log 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{x} + 1\right) - y \cdot x \leq 0.2:\\ \;\;\;\;\left(-y\right) \cdot x\\ \mathbf{elif}\;\log \left(e^{x} + 1\right) - y \cdot x \leq 20:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0)
   (- (log (+ 1.0 (exp x))) (* x y))
   (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = log((1.0 + exp(x))) - (x * y);
	} else {
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0d0) then
        tmp = log((1.0d0 + exp(x))) - (x * y)
    else
        tmp = log((1.0d0 + exp(-x))) - (-x * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = Math.log((1.0 + Math.exp(x))) - (x * y);
	} else {
		tmp = Math.log((1.0 + Math.exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0:
		tmp = math.log((1.0 + math.exp(x))) - (x * y)
	else:
		tmp = math.log((1.0 + math.exp(-x))) - (-x * (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0)
		tmp = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y));
	else
		tmp = Float64(log(Float64(1.0 + exp(Float64(-x)))) - Float64(Float64(-x) * Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0)
		tmp = log((1.0 + exp(x))) - (x * y);
	else
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0], N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[((-x) * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024229 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y)))))

  (- (log (+ 1.0 (exp x))) (* x y)))