Logistic regression 2

Percentage Accurate: 99.2% → 99.3%

Time: 8.5s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.3760261185539682e+85
  2. y = -1.4787425111215293e-102

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, 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 99.7%

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

   1. sub-neg 99.7%

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

   2. +-commutative 99.7%

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

   3. distribute-rgt-neg-in 99.7%

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

   4. fma-def 99.8%

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

   5. 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. Final simplification 100.0%

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

Alternative 2: 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 99.7%

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

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

4. Final simplification 100.0%

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

Alternative 3: 82.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 0.045:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e-45)
   (* x (- y))
   (if (<= x 0.045) (+ (log 2.0) (* x 0.5)) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e-45) {
		tmp = x * -y;
	} else if (x <= 0.045) {
		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 <= (-1.32d-45)) then
        tmp = x * -y
    else if (x <= 0.045d0) 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 <= -1.32e-45) {
		tmp = x * -y;
	} else if (x <= 0.045) {
		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 <= -1.32e-45:
		tmp = x * -y
	elif x <= 0.045:
		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 <= -1.32e-45)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 0.045)
		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 <= -1.32e-45)
		tmp = x * -y;
	elseif (x <= 0.045)
		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, -1.32e-45], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 0.045], 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 < -1.32000000000000005e-45

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 94.9%

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

      1. mul-1-neg 94.9%

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

      2. distribute-rgt-neg-in 94.9%

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

   6. Simplified 94.9%

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

   if -1.32000000000000005e-45 < x < 0.044999999999999998

   1. Initial program 99.9%

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

      1. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 99.5%

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

   5. Taylor expanded in y around 0 77.4%

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

   if 0.044999999999999998 < x

   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 89.9%

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

   5. Taylor expanded in x around inf 89.8%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 0.045:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 4: 99.0% 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 99.3%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 97.7%

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

      1. mul-1-neg 97.7%

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

      2. distribute-rgt-neg-in 97.7%

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

   6. Simplified 97.7%

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

   if -1.3999999999999999 < x

   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 98.4%

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

4. Final simplification 98.2%

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

Alternative 5: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -160:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -160.0) (* x (- y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -160.0) {
		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 <= (-160.0d0)) 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 <= -160.0) {
		tmp = x * -y;
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -160.0:
		tmp = x * -y
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -160.0)
		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 <= -160.0)
		tmp = x * -y;
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -160.0], 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 < -160

   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-in 100.0%

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

   6. Simplified 100.0%

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

   if -160 < x

   1. Initial program 99.6%

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

      1. log1p-udef 99.6%

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

      2. add-sqr-sqrt 98.0%

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

      3. log-prod 98.6%

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

      4. +-commutative 98.6%

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

      5. +-commutative 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. count-2 98.6%

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

      2. +-commutative 98.6%

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

   7. Simplified 98.6%

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

   8. Taylor expanded in x around 0 96.0%

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

      1. prod-diff 89.2%

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

      2. *-commutative 89.2%

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

      3. fma-neg 89.2%

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

      4. add-log-exp 67.4%

         \[\leadsto \color{blue}{\log \left(e^{2 \cdot \log \left(\sqrt{2}\right) - x \cdot y}\right)} + \mathsf{fma}\left(-y, x, y \cdot x\right) \]

      5. exp-diff 67.4%

         \[\leadsto \log \color{blue}{\left(\frac{e^{2 \cdot \log \left(\sqrt{2}\right)}}{e^{x \cdot y}}\right)} + \mathsf{fma}\left(-y, x, y \cdot x\right) \]

      6. *-commutative 67.4%

         \[\leadsto \log \left(\frac{e^{\color{blue}{\log \left(\sqrt{2}\right) \cdot 2}}}{e^{x \cdot y}}\right) + \mathsf{fma}\left(-y, x, y \cdot x\right) \]

      7. exp-lft-sqr 65.8%

         \[\leadsto \log \left(\frac{\color{blue}{e^{\log \left(\sqrt{2}\right)} \cdot e^{\log \left(\sqrt{2}\right)}}}{e^{x \cdot y}}\right) + \mathsf{fma}\left(-y, x, y \cdot x\right) \]

      8. add-exp-log 65.8%

         \[\leadsto \log \left(\frac{\color{blue}{\sqrt{2}} \cdot e^{\log \left(\sqrt{2}\right)}}{e^{x \cdot y}}\right) + \mathsf{fma}\left(-y, x, y \cdot x\right) \]

      9. add-exp-log 65.8%

         \[\leadsto \log \left(\frac{\sqrt{2} \cdot \color{blue}{\sqrt{2}}}{e^{x \cdot y}}\right) + \mathsf{fma}\left(-y, x, y \cdot x\right) \]

      10. add-sqr-sqrt 67.4%

          \[\leadsto \log \left(\frac{\color{blue}{2}}{e^{x \cdot y}}\right) + \mathsf{fma}\left(-y, x, y \cdot x\right) \]

      11. diff-log 67.4%

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

      12. add-log-exp 90.2%

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

      13. *-commutative 90.2%

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

      14. fma-udef 90.2%

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

   10. Applied egg-rr 90.2%

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

       1. fma-udef 90.2%

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

       2. distribute-rgt-neg-out 90.2%

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

       3. neg-mul-1 90.2%

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

       4. distribute-lft1-in 90.2%

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

       5. metadata-eval 90.2%

          \[\leadsto \left(\log 2 - x \cdot y\right) + \color{blue}{0} \cdot \left(x \cdot y\right) \]

       6. mul0-lft 97.0%

          \[\leadsto \left(\log 2 - x \cdot y\right) + \color{blue}{0} \]

       7. +-rgt-identity 97.0%

          \[\leadsto \color{blue}{\log 2 - x \cdot y} \]

       8. *-commutative 97.0%

          \[\leadsto \log 2 - \color{blue}{y \cdot x} \]

   12. Simplified 97.0%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -160:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]

Alternative 6: 82.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-45)
   (* x (- y))
   (if (<= x 1.7e-16) (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-45) {
		tmp = x * -y;
	} else if (x <= 1.7e-16) {
		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 <= (-1.35d-45)) then
        tmp = x * -y
    else if (x <= 1.7d-16) 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 <= -1.35e-45) {
		tmp = x * -y;
	} else if (x <= 1.7e-16) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e-45:
		tmp = x * -y
	elif x <= 1.7e-16:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-45)
		tmp = Float64(x * Float64(-y));
	elseif (x <= 1.7e-16)
		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 <= -1.35e-45)
		tmp = x * -y;
	elseif (x <= 1.7e-16)
		tmp = log(2.0);
	else
		tmp = x * (0.5 - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-45], N[(x * (-y)), $MachinePrecision], If[LessEqual[x, 1.7e-16], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.34999999999999992e-45

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 94.9%

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

      1. mul-1-neg 94.9%

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

      2. distribute-rgt-neg-in 94.9%

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

   6. Simplified 94.9%

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

   if -1.34999999999999992e-45 < x < 1.7e-16

   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 78.1%

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

   if 1.7e-16 < x

   1. Initial program 99.8%

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

      1. log1p-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 88.1%

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

   5. Taylor expanded in x around inf 82.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-45}:\\ \;\;\;\;x \cdot \left(-y\right)\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-16}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 - y\right)\\ \end{array} \]

Alternative 7: 53.0% 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 99.7%

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

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

4. Taylor expanded in x around inf 53.4%

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

   1. mul-1-neg 53.4%

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

   2. distribute-rgt-neg-in 53.4%

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

6. Simplified 53.4%

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

7. Final simplification 53.4%

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

Alternative 8: 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 99.7%

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

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

4. Taylor expanded in x around 0 83.0%

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

5. Taylor expanded in y around 0 46.5%

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

6. Taylor expanded in x around inf 3.8%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
7. Step-by-step derivation

   1. *-commutative 3.8%

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

8. Simplified 3.8%

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

9. 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 2023257 
(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)))