Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 12.2s

Alternatives: 9

Speedup: 1.8×

• could not determine a ground truth

  1. x = 1.2644041164815007e+148
  2. y = -4.788148475591599e+302

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: 99.3% 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: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. cancel-sign-sub-inv 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-lft-neg-out 99.9%

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

   4. distribute-rgt-neg-out 99.9%

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

   5. fma-def 100.0%

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

   6. log1p-def 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 99.4% 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 99.9%

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

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

5. Final simplification 100.0%

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

Alternative 3: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -255:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(x \cdot 0.125 + \left(0.5 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -255.0) (* x (- y)) (+ (log 2.0) (* x (+ (* x 0.125) (- 0.5 y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -255.0) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) + (x * ((x * 0.125) + (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 <= (-255.0d0)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) + (x * ((x * 0.125d0) + (0.5d0 - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -255.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * ((x * 0.125) + (0.5 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -255.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) + (x * ((x * 0.125) + (0.5 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -255.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(Float64(x * 0.125) + Float64(0.5 - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -255.0)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * ((x * 0.125) + (0.5 - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -255.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(N[(x * 0.125), $MachinePrecision] + N[(0.5 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -255

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

   5. Simplified 100.0%

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

   if -255 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 98.3%

      \[\leadsto \color{blue}{\log 2 + \left(0.125 \cdot {x}^{2} + x \cdot \left(0.5 - y\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \log 2 + \left(\color{blue}{{x}^{2} \cdot 0.125} + x \cdot \left(0.5 - y\right)\right) \]

      2. unpow2 98.3%

         \[\leadsto \log 2 + \left(\color{blue}{\left(x \cdot x\right)} \cdot 0.125 + x \cdot \left(0.5 - y\right)\right) \]

      3. associate-*l* 98.3%

         \[\leadsto \log 2 + \left(\color{blue}{x \cdot \left(x \cdot 0.125\right)} + x \cdot \left(0.5 - y\right)\right) \]

      4. sub-neg 98.3%

         \[\leadsto \log 2 + \left(x \cdot \left(x \cdot 0.125\right) + x \cdot \color{blue}{\left(0.5 + \left(-y\right)\right)}\right) \]

      5. mul-1-neg 98.3%

         \[\leadsto \log 2 + \left(x \cdot \left(x \cdot 0.125\right) + x \cdot \left(0.5 + \color{blue}{-1 \cdot y}\right)\right) \]

      6. distribute-lft-out 98.3%

         \[\leadsto \log 2 + \color{blue}{x \cdot \left(x \cdot 0.125 + \left(0.5 + -1 \cdot y\right)\right)} \]

      7. mul-1-neg 98.3%

         \[\leadsto \log 2 + x \cdot \left(x \cdot 0.125 + \left(0.5 + \color{blue}{\left(-y\right)}\right)\right) \]

      8. sub-neg 98.3%

         \[\leadsto \log 2 + x \cdot \left(x \cdot 0.125 + \color{blue}{\left(0.5 - y\right)}\right) \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(x \cdot 0.125 + \left(0.5 - y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -255:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(x \cdot 0.125 + \left(0.5 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.38:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.38) (* x (- y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.38) {
		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 <= (-1.38d0)) 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 <= -1.38) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.38:
		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 <= -1.38)
		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 <= -1.38)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.38], 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 < -1.3799999999999999

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 98.8%

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

      1. mul-1-neg 98.8%

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

      2. distribute-rgt-neg-in 98.8%

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

   5. Simplified 98.8%

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

   if -1.3799999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 97.9%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.38:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-77}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-54}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.4e-77)
   (* x (- y))
   (if (<= x 2.35e-54) (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-77) {
		tmp = x * -y;
	} else if (x <= 2.35e-54) {
		tmp = log(2.0);
	} else {
		tmp = 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 <= (-3.4d-77)) then
        tmp = x * -y
    else if (x <= 2.35d-54) then
        tmp = log(2.0d0)
    else
        tmp = x * (0.5d0 - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.4e-77) {
		tmp = x * -y;
	} else if (x <= 2.35e-54) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.4e-77:
		tmp = x * -y
	elif x <= 2.35e-54:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.4e-77)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 2.35e-54)
		tmp = log(2.0);
	else
		tmp = Float64(x * Float64(0.5 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.4e-77)
		tmp = x * -y;
	elseif (x <= 2.35e-54)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.4e-77], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 2.35e-54], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.39999999999999983e-77

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

   5. Simplified 92.9%

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

   if -3.39999999999999983e-77 < x < 2.35e-54

   1. Initial program 100.0%

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

      1. log1p-udef 100.0%

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

      2. *-un-lft-identity 100.0%

         \[\leadsto \color{blue}{1 \cdot \mathsf{log1p}\left(e^{x}\right)} - x \cdot y \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) \cdot 1} - x \cdot y \]

      4. add-sqr-sqrt 51.3%

         \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - x \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)} \]

      5. sqrt-unprod 72.9%

         \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - x \cdot \color{blue}{\sqrt{y \cdot y}} \]

      6. sqr-neg 72.9%

         \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - x \cdot \sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}} \]

      7. sqrt-unprod 37.9%

         \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - x \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)} \]

      8. add-sqr-sqrt 76.6%

         \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - x \cdot \color{blue}{\left(-y\right)} \]

      9. distribute-rgt-neg-in 76.6%

         \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - \color{blue}{\left(-x \cdot y\right)} \]

      10. neg-mul-1 76.6%

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - \color{blue}{-1 \cdot \left(x \cdot y\right)} \]

      11. add-sqr-sqrt 38.7%

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - -1 \cdot \left(x \cdot \color{blue}{\left(\sqrt{y} \cdot \sqrt{y}\right)}\right) \]

      12. sqrt-unprod 71.5%

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - -1 \cdot \left(x \cdot \color{blue}{\sqrt{y \cdot y}}\right) \]

      13. sqr-neg 71.5%

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - -1 \cdot \left(x \cdot \sqrt{\color{blue}{\left(-y\right) \cdot \left(-y\right)}}\right) \]

      14. sqrt-unprod 48.5%

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - -1 \cdot \left(x \cdot \color{blue}{\left(\sqrt{-y} \cdot \sqrt{-y}\right)}\right) \]

      15. add-sqr-sqrt 100.0%

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - -1 \cdot \left(x \cdot \color{blue}{\left(-y\right)}\right) \]

      16. distribute-rgt-neg-in 100.0%

          \[\leadsto \mathsf{log1p}\left(e^{x}\right) \cdot 1 - -1 \cdot \color{blue}{\left(-x \cdot y\right)} \]

      17. prod-diff 100.0%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(e^{x}\right), 1, -\left(-x \cdot y\right) \cdot -1\right) + \mathsf{fma}\left(-\left(-x \cdot y\right), -1, \left(-x \cdot y\right) \cdot -1\right)} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(e^{x}\right), 1, -\left(x \cdot y\right) \cdot -1\right) + \mathsf{fma}\left(x \cdot y, -1, \left(x \cdot y\right) \cdot -1\right)} \]

   5. Taylor expanded in x around 0 77.2%

      \[\leadsto \color{blue}{\log 2} \]

   if 2.35e-54 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 82.1%

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

   4. Taylor expanded in x around inf 54.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-77}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 2.35 \cdot 10^{-54}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-77}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e-77) (* x (- y)) (+ (log 2.0) (* x 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6e-77) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) + (x * 0.5);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.6d-77)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) + (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.6e-77) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.6e-77:
		tmp = x * -y
	else:
		tmp = math.log(2.0) + (x * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.6e-77)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.6e-77)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6e-77], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999999e-77

   1. Initial program 99.9%

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

   3. Taylor expanded in x around inf 92.9%

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

      1. mul-1-neg 92.9%

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

      2. distribute-rgt-neg-in 92.9%

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

   5. Simplified 92.9%

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

   if -5.5999999999999999e-77 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 74.4%

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

      1. log1p-def 74.4%

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

   5. Simplified 74.4%

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

   6. Taylor expanded in x around 0 72.3%

      \[\leadsto \color{blue}{\log 2 + 0.5 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-77}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -255:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -255.0) (* x (- y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -255.0) {
		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 <= (-255.0d0)) 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 <= -255.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -255.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -255.0)
		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 <= -255.0)
		tmp = x * -y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -255.0], 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 < -255

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

   5. Simplified 100.0%

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

   if -255 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 96.7%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -255:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.2% 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 99.9%

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

3. Taylor expanded in x around inf 51.8%

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

   1. mul-1-neg 51.8%

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

   2. distribute-rgt-neg-in 51.8%

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

5. Simplified 51.8%

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

6. Final simplification 51.8%

   \[\leadsto x \cdot \left(-y\right) \]
7. Add Preprocessing

Alternative 9: 3.5% 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 99.9%

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

3. Taylor expanded in x around 0 80.4%

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

4. Taylor expanded in x around inf 34.5%

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

5. Taylor expanded in y around 0 3.7%

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

6. Final simplification 3.7%

   \[\leadsto x \cdot 0.5 \]
7. Add Preprocessing

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 2024017 
(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)))