Logistic regression 2

Percentage Accurate: 99.3% → 99.4%

Time: 8.3s

Alternatives: 9

Speedup: 1.8×

• could not determine a ground truth

  1. x = 2.497807129154998e+232
  2. y = 1.206709554115137e+179

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 99.2%

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

   1. sub-neg 99.2%

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

   2. +-commutative 99.2%

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

   3. distribute-rgt-neg-in 99.2%

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

   4. fma-def 99.2%

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

   5. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

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 99.2%

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

   1. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 3: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -54.0) (* x (- y)) (+ (log 2.0) (* x (+ (- 0.5 y) (* x 0.125))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -54.0) {
		tmp = x * -y;
	} else {
		tmp = log(2.0) + (x * ((0.5 - y) + (x * 0.125)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-54.0d0)) then
        tmp = x * -y
    else
        tmp = log(2.0d0) + (x * ((0.5d0 - y) + (x * 0.125d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -54.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) + (x * ((0.5 - y) + (x * 0.125)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -54.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) + (x * ((0.5 - y) + (x * 0.125)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -54.0)
		tmp = Float64(x * Float64(-y));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(Float64(0.5 - y) + Float64(x * 0.125))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -54.0)
		tmp = x * -y;
	else
		tmp = log(2.0) + (x * ((0.5 - y) + (x * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -54.0], N[(x * (-y)), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(N[(0.5 - y), $MachinePrecision] + N[(x * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -54

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   6. Simplified 100.0%

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

   if -54 < x

   1. Initial program 98.7%

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

      1. log1p-def 98.8%

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

   3. Simplified 98.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   4. Step-by-step derivation

      1. *-un-lft-identity 98.8%

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

      2. exp-prod 98.8%

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

   5. Applied egg-rr 98.8%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{{\left(e^{1}\right)}^{x}}\right) - x \cdot y \]
   6. Step-by-step derivation

      1. exp-1-e 98.8%

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

   7. Simplified 98.8%

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

   8. Taylor expanded in x around 0 98.5%

      \[\leadsto \color{blue}{\log 2 + \left(0.5 \cdot \left(\left(-0.25 \cdot {\log e}^{2} + 0.5 \cdot {\log e}^{2}\right) \cdot {x}^{2}\right) + \left(0.5 \cdot \log e - y\right) \cdot x\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 98.5%

         \[\leadsto \log 2 + \left(\color{blue}{\left(0.5 \cdot \left(-0.25 \cdot {\log e}^{2} + 0.5 \cdot {\log e}^{2}\right)\right) \cdot {x}^{2}} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      2. distribute-rgt-out 98.5%

         \[\leadsto \log 2 + \left(\left(0.5 \cdot \color{blue}{\left({\log e}^{2} \cdot \left(-0.25 + 0.5\right)\right)}\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      3. log-E 98.5%

         \[\leadsto \log 2 + \left(\left(0.5 \cdot \left({\color{blue}{1}}^{2} \cdot \left(-0.25 + 0.5\right)\right)\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      4. metadata-eval 98.5%

         \[\leadsto \log 2 + \left(\left(0.5 \cdot \left(\color{blue}{1} \cdot \left(-0.25 + 0.5\right)\right)\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      5. metadata-eval 98.5%

         \[\leadsto \log 2 + \left(\left(0.5 \cdot \left(\color{blue}{{1}^{3}} \cdot \left(-0.25 + 0.5\right)\right)\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      6. log-E 98.5%

         \[\leadsto \log 2 + \left(\left(0.5 \cdot \left({\color{blue}{\log e}}^{3} \cdot \left(-0.25 + 0.5\right)\right)\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      7. metadata-eval 98.5%

         \[\leadsto \log 2 + \left(\left(0.5 \cdot \left({\log e}^{3} \cdot \color{blue}{0.25}\right)\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      8. *-commutative 98.5%

         \[\leadsto \log 2 + \left(\left(0.5 \cdot \color{blue}{\left(0.25 \cdot {\log e}^{3}\right)}\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      9. log-E 98.5%

         \[\leadsto \log 2 + \left(\left(0.5 \cdot \left(0.25 \cdot {\color{blue}{1}}^{3}\right)\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      10. metadata-eval 98.5%

          \[\leadsto \log 2 + \left(\left(0.5 \cdot \left(0.25 \cdot \color{blue}{1}\right)\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      11. metadata-eval 98.5%

          \[\leadsto \log 2 + \left(\left(0.5 \cdot \color{blue}{0.25}\right) \cdot {x}^{2} + \left(0.5 \cdot \log e - y\right) \cdot x\right) \]

      12. metadata-eval 98.5%

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

      13. unpow2 98.5%

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

      14. *-commutative 98.5%

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

   10. Simplified 98.5%

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

       1. *-un-lft-identity 98.5%

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

       2. *-commutative 98.5%

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

       3. +-commutative 98.5%

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

       4. *-commutative 98.5%

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

       5. associate-*r* 98.5%

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

       6. distribute-rgt-out 98.5%

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

       7. *-commutative 98.5%

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

   12. Applied egg-rr 98.5%

       \[\leadsto \color{blue}{\left(\log 2 + x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\right) \cdot 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -54:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 - y\right) + x \cdot 0.125\right)\\ \end{array} \]

Alternative 4: 84.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 0.16:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.5e-8)
   (* x (- y))
   (if (<= x 0.16) (+ (log 2.0) (* x 0.5)) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.5e-8) {
		tmp = x * -y;
	} else if (x <= 0.16) {
		tmp = log(2.0) + (x * 0.5);
	} 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 <= (-6.5d-8)) then
        tmp = x * -y
    else if (x <= 0.16d0) then
        tmp = log(2.0d0) + (x * 0.5d0)
    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 <= -6.5e-8) {
		tmp = x * -y;
	} else if (x <= 0.16) {
		tmp = Math.log(2.0) + (x * 0.5);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.5e-8:
		tmp = x * -y
	elif x <= 0.16:
		tmp = math.log(2.0) + (x * 0.5)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.5e-8)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 0.16)
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	else
		tmp = Float64(x * Float64(0.5 - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.5e-8)
		tmp = x * -y;
	elseif (x <= 0.16)
		tmp = log(2.0) + (x * 0.5);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.5e-8], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 0.16], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.49999999999999997e-8

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 97.9%

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

      1. mul-1-neg 97.9%

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

      2. distribute-rgt-neg-out 97.9%

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

   6. Simplified 97.9%

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

   if -6.49999999999999997e-8 < x < 0.160000000000000003

   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. Taylor expanded in x around 0 98.8%

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

   5. Taylor expanded in y around 0 76.1%

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

   if 0.160000000000000003 < x

   1. Initial program 86.9%

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

      1. log1p-def 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in x around 0 94.6%

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

   5. Taylor expanded in x around inf 94.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 0.16:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.9× speedup

Math

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

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 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   6. Simplified 100.0%

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

   if -1.3999999999999999 < x

   1. Initial program 98.7%

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

      1. log1p-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 97.9%

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

4. Final simplification 98.7%

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

Alternative 6: 99.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -42:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -42.0) (* x (- y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -42.0) {
		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 <= -42.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -42.0:
		tmp = x * -y
	else:
		tmp = math.log1p(1.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -42.0)
		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, -42.0], 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 < -42

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-out 100.0%

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

   6. Simplified 100.0%

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

   if -42 < x

   1. Initial program 98.7%

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

      1. log1p-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in x around 0 96.7%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -42:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \]

Alternative 7: 84.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.12 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-47}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.12e-29)
   (* x (- y))
   (if (<= x 7.6e-47) (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.12e-29) {
		tmp = x * -y;
	} else if (x <= 7.6e-47) {
		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 <= (-2.12d-29)) then
        tmp = x * -y
    else if (x <= 7.6d-47) 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 <= -2.12e-29) {
		tmp = x * -y;
	} else if (x <= 7.6e-47) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.12e-29:
		tmp = x * -y
	elif x <= 7.6e-47:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.12e-29)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 7.6e-47)
		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 <= -2.12e-29)
		tmp = x * -y;
	elseif (x <= 7.6e-47)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.12e-29], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 7.6e-47], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.1199999999999999e-29

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around inf 95.2%

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

      1. mul-1-neg 95.2%

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

      2. distribute-rgt-neg-out 95.2%

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

   6. Simplified 95.2%

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

   if -2.1199999999999999e-29 < x < 7.60000000000000029e-47

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. Taylor expanded in x around 0 100.0%

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

      1. add-cube-cbrt 99.6%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(0.5 - y\right) \cdot x} \cdot \sqrt[3]{\left(0.5 - y\right) \cdot x}\right) \cdot \sqrt[3]{\left(0.5 - y\right) \cdot x}} + \log 2 \]

      2. pow3 99.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(0.5 - y\right) \cdot x}\right)}^{3}} + \log 2 \]

      3. *-commutative 99.6%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \left(0.5 - y\right)}}\right)}^{3} + \log 2 \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \left(0.5 - y\right)}\right)}^{3}} + \log 2 \]

   7. Taylor expanded in x around 0 79.4%

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

   if 7.60000000000000029e-47 < x

   1. Initial program 92.3%

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

      1. log1p-def 92.4%

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

   3. Simplified 92.4%

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

   4. Taylor expanded in x around 0 90.7%

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

   5. Taylor expanded in x around inf 73.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.12 \cdot 10^{-29}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-47}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 8: 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 99.2%

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

   1. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around inf 56.1%

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

   1. mul-1-neg 56.1%

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

   2. distribute-rgt-neg-out 56.1%

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

6. Simplified 56.1%

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

7. Final simplification 56.1%

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

Alternative 9: 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 99.2%

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

   1. log1p-def 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in x around 0 80.4%

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

5. Taylor expanded in y around 0 45.7%

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

6. Taylor expanded in x around inf 3.7%

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

7. Final simplification 3.7%

   \[\leadsto x \cdot 0.5 \]

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 2023221 
(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)))