Logistic regression 2

Percentage Accurate: 99.3% → 99.3%

Time: 9.6s

Alternatives: 9

Speedup: 1.8×

• could not determine a ground truth

  1. x = 2.289167972776327e+145
  2. y = 5.952258450052088e+67

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: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0 - y, x, \mathsf{log1p}\left(e^{x}\right)\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (- 0.0 y) x (log1p (exp x))))

C

double code(double x, double y) {
	return fma((0.0 - y), x, log1p(exp(x)));
}

Julia

function code(x, y)
	return fma(Float64(0.0 - y), x, log1p(exp(x)))
end

Wolfram

code[x_, y_] := N[(N[(0.0 - y), $MachinePrecision] * x + N[Log[1 + N[Exp[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(y\right)\right), \color{blue}{x}, \log \left(1 + e^{x}\right)\right) \]

   6. neg-sub0 N/A

      \[\leadsto \mathsf{fma.f64}\left(\left(0 - y\right), x, \log \left(1 + e^{x}\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, y\right), x, \log \left(1 + e^{x}\right)\right) \]

   8. accelerator-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, y\right), x, \mathsf{log1p.f64}\left(\left(e^{x}\right)\right)\right) \]

   9. exp-lowering-exp.f64 99.6%

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, y\right), x, \mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(x\right)\right)\right) \]

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0 - y, x, \mathsf{log1p}\left(e^{x}\right)\right)} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- (log (+ (exp x) 1.0)) (* y x)))

C

double code(double x, double y) {
	return log((exp(x) + 1.0)) - (y * x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = log((exp(x) + 1.0d0)) - (y * x)
end function

Java

public static double code(double x, double y) {
	return Math.log((Math.exp(x) + 1.0)) - (y * x);
}

Python

def code(x, y):
	return math.log((math.exp(x) + 1.0)) - (y * x)

Julia

function code(x, y)
	return Float64(log(Float64(exp(x) + 1.0)) - Float64(y * x))
end

MATLAB

function tmp = code(x, y)
	tmp = log((exp(x) + 1.0)) - (y * x);
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 3: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6)
   (- 0.0 (* y x))
   (+
    (log 2.0)
    (* x (+ 0.5 (- (* x (+ 0.125 (* -0.005208333333333333 (* x x)))) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * 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 <= (-2.6d0)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = log(2.0d0) + (x * (0.5d0 + ((x * (0.125d0 + ((-0.005208333333333333d0) * (x * x)))) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.6:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(0.5 + Float64(Float64(x * Float64(0.125 + Float64(-0.005208333333333333 * Float64(x * x)))) - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 + N[(N[(x * N[(0.125 + N[(-0.005208333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000009

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

      2. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{y}\right)\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      2. neg-lowering-neg.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(y\right)\right) \]

   7. Applied egg-rr 100.0%

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

   if -2.60000000000000009 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{x} \cdot \left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right) - y\right)}\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) - y\right)}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right) - y\right)}\right)\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\left(x \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right), \color{blue}{y}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{1}{8} + \frac{-1}{192} \cdot {x}^{2}\right)\right), y\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \left(\frac{-1}{192} \cdot {x}^{2}\right)\right)\right), y\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \left({x}^{2}\right)\right)\right)\right), y\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \left(x \cdot x\right)\right)\right)\right), y\right)\right)\right)\right) \]

      11. *-lowering-*.f64 99.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(\frac{-1}{192}, \mathsf{*.f64}\left(x, x\right)\right)\right)\right), y\right)\right)\right)\right) \]

   5. Simplified 99.4%

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -110:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -110.0)
   (- 0.0 (* y x))
   (+ (log 2.0) (* x (- (+ 0.5 (* x 0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -110.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log(2.0) + (x * ((0.5 + (x * 0.125)) - 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 <= (-110.0d0)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = log(2.0d0) + (x * ((0.5d0 + (x * 0.125d0)) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -110.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) + (x * ((0.5 + (x * 0.125)) - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -110.0:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0) + (x * ((0.5 + (x * 0.125)) - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -110.0)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(Float64(0.5 + Float64(x * 0.125)) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -110.0)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0) + (x * ((0.5 + (x * 0.125)) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -110.0], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(N[(0.5 + N[(x * 0.125), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -110

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

      2. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{y}\right)\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      2. neg-lowering-neg.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(y\right)\right) \]

   7. Applied egg-rr 100.0%

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

   if -110 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{x} \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right)}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right), \color{blue}{y}\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{8} \cdot x\right)\right), y\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{1}{8}\right)\right), y\right)\right)\right) \]

      7. *-lowering-*.f64 99.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{1}{8}\right)\right), y\right)\right)\right) \]

   5. Simplified 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -110:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - y \cdot x\\ \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (* y x))))
   (if (<= x -0.0052) t_0 (if (<= x 1.1e-43) (log (+ x 2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (y * x);
	double tmp;
	if (x <= -0.0052) {
		tmp = t_0;
	} else if (x <= 1.1e-43) {
		tmp = log((x + 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (y * x)
    if (x <= (-0.0052d0)) then
        tmp = t_0
    else if (x <= 1.1d-43) then
        tmp = log((x + 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.0 - (y * x);
	double tmp;
	if (x <= -0.0052) {
		tmp = t_0;
	} else if (x <= 1.1e-43) {
		tmp = Math.log((x + 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (y * x)
	tmp = 0
	if x <= -0.0052:
		tmp = t_0
	elif x <= 1.1e-43:
		tmp = math.log((x + 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.0 - Float64(y * x))
	tmp = 0.0
	if (x <= -0.0052)
		tmp = t_0;
	elseif (x <= 1.1e-43)
		tmp = log(Float64(x + 2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.0 - (y * x);
	tmp = 0.0;
	if (x <= -0.0052)
		tmp = t_0;
	elseif (x <= 1.1e-43)
		tmp = log((x + 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0052], t$95$0, If[LessEqual[x, 1.1e-43], N[Log[N[(x + 2.0), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0051999999999999998 or 1.09999999999999999e-43 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

      2. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{y}\right)\right) \]

      5. --lowering--.f64 95.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

   5. Simplified 95.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      2. neg-lowering-neg.f64 95.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(y\right)\right) \]

   7. Applied egg-rr 95.0%

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

   if -0.0051999999999999998 < x < 1.09999999999999999e-43

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \color{blue}{\left(1 + x\right)}\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(x + 1\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

      2. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \mathsf{+.f64}\left(x, 1\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   5. Simplified 99.4%

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

   6. Taylor expanded in y around 0

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

      1. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{log.f64}\left(\left(2 + x\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\left(x + 2\right)\right) \]

      3. +-lowering-+.f64 74.5%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(x, 2\right)\right) \]

   8. Simplified 74.5%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0052:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-43}:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.4) (- 0.0 (* y x)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.4) {
		tmp = 0.0 - (y * x);
	} 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 = 0.0d0 - (y * x)
    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 = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.4:
		tmp = 0.0 - (y * x)
	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(0.0 - Float64(y * x));
	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 = 0.0 - (y * x);
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.4], N[(0.0 - N[(y * x), $MachinePrecision]), $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. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

      2. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{y}\right)\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      2. neg-lowering-neg.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(y\right)\right) \]

   7. Applied egg-rr 100.0%

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

   if -1.3999999999999999 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\frac{1}{2} - y\right)} \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\log 2, \color{blue}{\left(x \cdot \left(\frac{1}{2} - y\right)\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(\color{blue}{x} \cdot \left(\frac{1}{2} - y\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{1}{2} - y\right)}\right)\right) \]

      4. --lowering--.f64 98.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\frac{1}{2}, \color{blue}{y}\right)\right)\right) \]

   5. Simplified 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - y \cdot x\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (* y x))))
   (if (<= x -5.6e-9) t_0 (if (<= x 2.8e-43) (log 2.0) t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (y * x);
	double tmp;
	if (x <= -5.6e-9) {
		tmp = t_0;
	} else if (x <= 2.8e-43) {
		tmp = log(2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (y * x)
    if (x <= (-5.6d-9)) then
        tmp = t_0
    else if (x <= 2.8d-43) then
        tmp = log(2.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.0 - (y * x);
	double tmp;
	if (x <= -5.6e-9) {
		tmp = t_0;
	} else if (x <= 2.8e-43) {
		tmp = Math.log(2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (y * x)
	tmp = 0
	if x <= -5.6e-9:
		tmp = t_0
	elif x <= 2.8e-43:
		tmp = math.log(2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.0 - Float64(y * x))
	tmp = 0.0
	if (x <= -5.6e-9)
		tmp = t_0;
	elseif (x <= 2.8e-43)
		tmp = log(2.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.0 - (y * x);
	tmp = 0.0;
	if (x <= -5.6e-9)
		tmp = t_0;
	elseif (x <= 2.8e-43)
		tmp = log(2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e-9], t$95$0, If[LessEqual[x, 2.8e-43], N[Log[2.0], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.59999999999999969e-9 or 2.7999999999999998e-43 < x

   1. Initial program 99.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

      2. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{y}\right)\right) \]

      5. --lowering--.f64 93.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

   5. Simplified 93.2%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      2. neg-lowering-neg.f64 93.2%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(y\right)\right) \]

   7. Applied egg-rr 93.2%

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

   if -5.59999999999999969e-9 < x < 2.7999999999999998e-43

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2} \]
   4. Step-by-step derivation

      1. log-lowering-log.f64 74.8%

         \[\leadsto \mathsf{log.f64}\left(2\right) \]

   5. Simplified 74.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-9}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{elif}\;x \leq 2.8 \cdot 10^{-43}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -31:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -31.0) (- 0.0 (* y x)) (- (log 2.0) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -31.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log(2.0) - (y * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-31.0d0)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = log(2.0d0) - (y * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -31.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -31.0:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0) - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -31.0)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(log(2.0) - Float64(y * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -31.0)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0) - (y * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -31.0], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -31

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

      2. distribute-rgt-neg-in N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{y}\right)\right) \]

      5. --lowering--.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   6. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(y\right)\right)\right) \]

      2. neg-lowering-neg.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(y\right)\right) \]

   7. Applied egg-rr 100.0%

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

   if -31 < x

   1. Initial program 99.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\log 2}, \mathsf{*.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

      1. log-lowering-log.f64 97.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(\color{blue}{x}, y\right)\right) \]

   5. Simplified 97.8%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -31:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.7% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ 0 - y \cdot x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 0.0 (* y x)))

C

double code(double x, double y) {
	return 0.0 - (y * x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0 - (y * x)
end function

Java

public static double code(double x, double y) {
	return 0.0 - (y * x);
}

Python

def code(x, y):
	return 0.0 - (y * x)

Julia

function code(x, y)
	return Float64(0.0 - Float64(y * x))
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0 - (y * x);
end

Wolfram

code[x_, y_] := N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(x \cdot y\right) \]

   2. distribute-rgt-neg-in N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(0 - \color{blue}{y}\right)\right) \]

   5. --lowering--.f64 52.1%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(0, \color{blue}{y}\right)\right) \]

5. Simplified 52.1%

   \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
6. Step-by-step derivation

   1. sub0-neg N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\mathsf{neg}\left(y\right)\right)\right) \]

   2. neg-lowering-neg.f64 52.1%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{neg.f64}\left(y\right)\right) \]

7. Applied egg-rr 52.1%

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

8. Final simplification 52.1%

   \[\leadsto 0 - y \cdot x \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0:\\ \;\;\;\;\log \left(1 + e^{x}\right) - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{-x}\right) - \left(-x\right) \cdot \left(1 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0)
   (- (log (+ 1.0 (exp x))) (* x y))
   (- (log (+ 1.0 (exp (- x)))) (* (- x) (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = log((1.0 + exp(x))) - (x * y);
	} else {
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0d0) then
        tmp = log((1.0d0 + exp(x))) - (x * y)
    else
        tmp = log((1.0d0 + exp(-x))) - (-x * (1.0d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0) {
		tmp = Math.log((1.0 + Math.exp(x))) - (x * y);
	} else {
		tmp = Math.log((1.0 + Math.exp(-x))) - (-x * (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0:
		tmp = math.log((1.0 + math.exp(x))) - (x * y)
	else:
		tmp = math.log((1.0 + math.exp(-x))) - (-x * (1.0 - y))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0)
		tmp = log((1.0 + exp(x))) - (x * y);
	else
		tmp = log((1.0 + exp(-x))) - (-x * (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0], N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(1.0 + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[((-x) * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024192 
(FPCore (x y)
  :name "Logistic regression 2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (<= x 0) (- (log (+ 1 (exp x))) (* x y)) (- (log (+ 1 (exp (- x)))) (* (- x) (- 1 y)))))

  (- (log (+ 1.0 (exp x))) (* x y)))