Logistic regression 2

Percentage Accurate: 99.2% → 99.3%

Time: 6.2s

Alternatives: 6

Speedup: 1.0×

• could not determine a ground truth

  1. x = 6.085955108699846e+56
  2. y = -3.302347829327934e-209

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: 99.2% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	return Math.log1p(Math.exp(x)) - (x * y);
}

Python

def code(x, y):
	return math.log1p(math.exp(x)) - (x * y)

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. log1p-define 99.6%

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

3. Simplified 99.6%

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

Alternative 2: 99.1% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4) (* x (- y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) + (x * (0.5 - 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 <= (-1.4d0)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) + (x * (0.5d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4:
		tmp = x * -y
	else:
		tmp = math.log(2.0) + (x * (0.5 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(0.5 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.4)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999

   1. Initial program 99.7%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 98.2%

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

      1. mul-1-neg 98.2%

         \[\leadsto \color{blue}{-x \cdot y} \]

      2. distribute-rgt-neg-out 98.2%

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

   7. Simplified 98.2%

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

   if -1.3999999999999999 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.8%

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

Alternative 3: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -18:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -18.0) (* x (- y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -18.0) {
		tmp = x * -y;
	} else {
		tmp = log1p(1.0) - (x * y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -18.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -18.0:
		tmp = x * -y
	else:
		tmp = math.log1p(1.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -18.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log1p(1.0) - Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -18.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[1 + 1.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -18

   1. Initial program 99.7%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

         \[\leadsto \color{blue}{-x \cdot y} \]

      2. distribute-rgt-neg-out 99.2%

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

   7. Simplified 99.2%

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

   if -18 < x

   1. Initial program 99.4%

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

      1. log1p-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1}\right) - x \cdot y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 81.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.4 \cdot 10^{-82}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -8.4e-82) (* x (- y)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.4e-82) {
		tmp = x * -y;
	} else {
		tmp = log(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.4d-82)) then
        tmp = x * -y
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.4e-82) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.4e-82:
		tmp = x * -y
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.4e-82)
		tmp = Float64(x * Float64(-y));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.4e-82)
		tmp = x * -y;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.4e-82], N[(x * (-y)), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.4000000000000001e-82

   1. Initial program 99.7%

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

      1. log1p-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 91.8%

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

      1. mul-1-neg 91.8%

         \[\leadsto \color{blue}{-x \cdot y} \]

      2. distribute-rgt-neg-out 91.8%

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

   7. Simplified 91.8%

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

   if -8.4000000000000001e-82 < x

   1. Initial program 99.3%

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

      1. log1p-define 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 77.6%

      \[\leadsto \color{blue}{\log 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 51.0% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(-y\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x (- y)))

C

double code(double x, double y) {
	return x * -y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * -y
end function

Java

public static double code(double x, double y) {
	return x * -y;
}

Python

def code(x, y):
	return x * -y

Julia

function code(x, y)
	return Float64(x * Float64(-y))
end

MATLAB

function tmp = code(x, y)
	tmp = x * -y;
end

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. log1p-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around inf 54.7%

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

   1. mul-1-neg 54.7%

      \[\leadsto \color{blue}{-x \cdot y} \]

   2. distribute-rgt-neg-out 54.7%

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

7. Simplified 54.7%

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

Alternative 6: 2.3% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x y))

C

double code(double x, double y) {
	return x * y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * y
end function

Java

public static double code(double x, double y) {
	return x * y;
}

Python

def code(x, y):
	return x * y

Julia

function code(x, y)
	return Float64(x * y)
end

MATLAB

function tmp = code(x, y)
	tmp = x * y;
end

Wolfram

code[x_, y_] := N[(x * y), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. log1p-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around inf 54.7%

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

   1. mul-1-neg 54.7%

      \[\leadsto \color{blue}{-x \cdot y} \]

   2. distribute-rgt-neg-out 54.7%

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

7. Simplified 54.7%

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

   1. add-sqr-sqrt 28.3%

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

   2. sqrt-unprod 17.9%

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

   3. sqr-neg 17.9%

      \[\leadsto x \cdot \sqrt{\color{blue}{y \cdot y}} \]

   4. sqrt-unprod 1.2%

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

   5. add-sqr-sqrt 2.3%

      \[\leadsto x \cdot \color{blue}{y} \]

   6. pow1 2.3%

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

9. Applied egg-rr 2.3%

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

    1. unpow1 2.3%

       \[\leadsto \color{blue}{x \cdot y} \]

11. Simplified 2.3%

    \[\leadsto \color{blue}{x \cdot y} \]
12. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× 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 2024191 
(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)))