Logistic regression 2

Percentage Accurate: 99.4% → 99.5%

Time: 7.6s

Alternatives: 6

Speedup: 1.0×

• could not determine a ground truth

  1. x = 2.4681162903496214e+22
  2. y = -2.3844274572458948e+272

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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{log1p}\left(e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e-8) (- (log1p (exp x)) (* x y)) (+ (* x (- 0.5 y)) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e-8) {
		tmp = log1p(exp(x)) - (x * y);
	} else {
		tmp = (x * (0.5 - y)) + log(2.0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e-8) {
		tmp = Math.log1p(Math.exp(x)) - (x * y);
	} else {
		tmp = (x * (0.5 - y)) + Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e-8:
		tmp = math.log1p(math.exp(x)) - (x * y)
	else:
		tmp = (x * (0.5 - y)) + math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e-8)
		tmp = Float64(log1p(exp(x)) - Float64(x * y));
	else
		tmp = Float64(Float64(x * Float64(0.5 - y)) + log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e-8], N[(N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9999999999999998e-8

   1. Initial program 99.7%

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

      1. log1p-def 100.0%

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

   3. Simplified 100.0%

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

   if -4.9999999999999998e-8 < x

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{log1p}\left(e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right) + \log 2\\ \end{array} \]

Alternative 2: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[(x * (-y)), $MachinePrecision], N[(N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $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. Taylor expanded in x around inf 98.7%

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

      1. mul-1-neg 98.7%

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

      2. *-commutative 98.7%

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

      3. distribute-rgt-neg-in 98.7%

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

   4. Simplified 98.7%

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

   if -1.3999999999999999 < x

   1. Initial program 97.3%

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

   2. Taylor expanded in x around 0 100.0%

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

4. Final simplification 99.6%

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

Alternative 3: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1000.0) (* x (- y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1000.0) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) - (x * 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 <= (-1000.0d0)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1000.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1000.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1000.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1000.0)
		tmp = x * -y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1000.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e3

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. *-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

   4. Simplified 100.0%

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

   if -1e3 < x

   1. Initial program 97.2%

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

   2. Taylor expanded in x around 0 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1000:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 4: 82.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-61} \lor \neg \left(x \leq 7.5 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.5e-61) (not (<= x 7.5e-11))) (* x (- y)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -3.5e-61) || !(x <= 7.5e-11)) {
		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 <= (-3.5d-61)) .or. (.not. (x <= 7.5d-11))) 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 <= -3.5e-61) || !(x <= 7.5e-11)) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -3.5e-61) or not (x <= 7.5e-11):
		tmp = x * -y
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -3.5e-61) || !(x <= 7.5e-11))
		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 <= -3.5e-61) || ~((x <= 7.5e-11)))
		tmp = x * -y;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -3.5e-61], N[Not[LessEqual[x, 7.5e-11]], $MachinePrecision]], N[(x * (-y)), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5000000000000003e-61 or 7.5e-11 < x

   1. Initial program 95.3%

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

   2. Taylor expanded in x around inf 93.9%

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

      1. mul-1-neg 93.9%

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

      2. *-commutative 93.9%

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

      3. distribute-rgt-neg-in 93.9%

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

   4. Simplified 93.9%

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

   if -3.5000000000000003e-61 < x < 7.5e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 83.1%

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

      1. log1p-def 83.1%

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

   4. Simplified 83.1%

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

   5. Taylor expanded in x around 0 83.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-61} \lor \neg \left(x \leq 7.5 \cdot 10^{-11}\right):\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 5: 52.8% 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 98.0%

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

2. Taylor expanded in x around inf 50.8%

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

   1. mul-1-neg 50.8%

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

   2. *-commutative 50.8%

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

   3. distribute-rgt-neg-in 50.8%

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

4. Simplified 50.8%

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

5. Final simplification 50.8%

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

Alternative 6: 3.6% accurate, 69.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* x 0.5))

C

double code(double x, double y) {
	return x * 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * 0.5d0
end function

Java

public static double code(double x, double y) {
	return x * 0.5;
}

Python

def code(x, y):
	return x * 0.5

Julia

function code(x, y)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = x * 0.5;
end

Wolfram

code[x_, y_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 98.0%

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

2. Taylor expanded in y around 0 55.3%

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

   1. log1p-def 55.4%

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

4. Simplified 55.4%

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

5. Taylor expanded in x around 0 51.4%

   \[\leadsto \color{blue}{0.5 \cdot x + \log 2} \]

6. Taylor expanded in x around inf 3.8%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

7. Final simplification 3.8%

   \[\leadsto x \cdot 0.5 \]

Developer target: 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 2023252 
(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)))