Logistic regression 2

Percentage Accurate: 99.6% → 99.6%

Time: 14.0s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. x = 4.025490589719136e+83
  2. y = 6.885745082131581e+119

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: 99.6% 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: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -11.2:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -11.2)
   (- 0.0 (* x y))
   (+ (log 2.0) (* x (+ 0.5 (- (* x 0.125) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -11.2) {
		tmp = 0.0 - (x * y);
	} 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 <= (-11.2d0)) then
        tmp = 0.0d0 - (x * y)
    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 <= -11.2) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 + ((x * 0.125) - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -11.2:
		tmp = 0.0 - (x * y)
	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 <= -11.2)
		tmp = Float64(0.0 - Float64(x * y));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(0.5 + Float64(Float64(x * 0.125) - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -11.2)
		tmp = 0.0 - (x * y);
	else
		tmp = log(2.0) + (x * (0.5 + ((x * 0.125) - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -11.2], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 + N[(N[(x * 0.125), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -11.199999999999999

   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 -11.199999999999999 < x

   1. Initial program 98.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. associate--l+ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11.2:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% 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]

Derivation

1. Initial program 99.2%

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

Alternative 3: 99.7% 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. --lowering--.f64 N/A

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

   2. log1p-define N/A

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

   3. log1p-lowering-log1p.f64 N/A

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

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

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

   5. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

3. Simplified 99.2%

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

Alternative 4: 97.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\frac{1}{1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}\right) - x \cdot y \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (-
  (log1p
   (/ 1.0 (+ 1.0 (* x (+ -1.0 (* x (+ 0.5 (* x -0.16666666666666666))))))))
  (* x y)))

C

double code(double x, double y) {
	return log1p((1.0 / (1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))))) - (x * y);
}

Java

public static double code(double x, double y) {
	return Math.log1p((1.0 / (1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))))) - (x * y);
}

Python

def code(x, y):
	return math.log1p((1.0 / (1.0 + (x * (-1.0 + (x * (0.5 + (x * -0.16666666666666666)))))))) - (x * y)

Julia

function code(x, y)
	return Float64(log1p(Float64(1.0 / Float64(1.0 + Float64(x * Float64(-1.0 + Float64(x * Float64(0.5 + Float64(x * -0.16666666666666666)))))))) - Float64(x * y))
end

Wolfram

code[x_, y_] := N[(N[Log[1 + N[(1.0 / N[(1.0 + N[(x * N[(-1.0 + N[(x * N[(0.5 + N[(x * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. --lowering--.f64 N/A

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

   2. log1p-define N/A

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

   3. log1p-lowering-log1p.f64 N/A

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

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

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

   5. *-lowering-*.f64 99.2%

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

3. Simplified 99.2%

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

   1. remove-double-div N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\left(\frac{1}{\frac{1}{e^{x}}}\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{e^{x}}\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   3. rec-exp N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \left(e^{\mathsf{neg}\left(x\right)}\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   5. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(0 - x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   6. +-inverses N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\log \left(1 + e^{x}\right) - \log \left(1 + e^{x}\right)\right) - x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   7. remove-double-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right)\right)\right)\right) - \log \left(1 + e^{x}\right)\right) - x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(-1 \cdot \left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right)\right) - \log \left(1 + e^{x}\right)\right) - x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\left(\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right)\right) \cdot -1 - \log \left(1 + e^{x}\right)\right) - x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   10. fmm-undef N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right), -1, \mathsf{neg}\left(\log \left(1 + e^{x}\right)\right)\right) - x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   11. neg-mul-1 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right), -1, -1 \cdot \log \left(1 + e^{x}\right)\right) - x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right), -1, \log \left(1 + e^{x}\right) \cdot -1\right) - x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right), -1, \log \left(1 + e^{x}\right) \cdot -1\right)\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right), -1, -1 \cdot \log \left(1 + e^{x}\right)\right)\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   15. neg-mul-1 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{fma}\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right), -1, \mathsf{neg}\left(\log \left(1 + e^{x}\right)\right)\right)\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   16. fmm-undef N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right)\right) \cdot -1 - \log \left(1 + e^{x}\right)\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right)\right) - \log \left(1 + e^{x}\right)\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   18. neg-mul-1 N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\log \left(1 + e^{x}\right)\right)\right)\right)\right) - \log \left(1 + e^{x}\right)\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   19. remove-double-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \left(1 + e^{x}\right) - \log \left(1 + e^{x}\right)\right), x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   20. +-inverses 99.2%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, x\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

6. Applied egg-rr 99.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\frac{1}{e^{0 - x}}}\right) - x \cdot y \]

7. Taylor expanded in x around 0

   \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(1 + x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)}\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]
8. Step-by-step derivation

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \left(x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) - 1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right) + -1\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \left(-1 + x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \left(x \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \left(\frac{1}{2} + \frac{-1}{6} \cdot x\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

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

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{-1}{6} \cdot x\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \left(x \cdot \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

   10. *-lowering-*.f64 98.8%

       \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(-1, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x, \frac{-1}{6}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(x, y\right)\right) \]

9. Simplified 98.8%

   \[\leadsto \mathsf{log1p}\left(\frac{1}{\color{blue}{1 + x \cdot \left(-1 + x \cdot \left(0.5 + x \cdot -0.16666666666666666\right)\right)}}\right) - x \cdot y \]
10. Add Preprocessing

Alternative 5: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + x \cdot 0.5\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.45) (- 0.0 (* x y)) (- (+ (log 2.0) (* x 0.5)) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.45) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = (log(2.0) + (x * 0.5)) - (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 <= (-2.45d0)) then
        tmp = 0.0d0 - (x * y)
    else
        tmp = (log(2.0d0) + (x * 0.5d0)) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.45) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = (Math.log(2.0) + (x * 0.5)) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.45:
		tmp = 0.0 - (x * y)
	else:
		tmp = (math.log(2.0) + (x * 0.5)) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.45)
		tmp = Float64(0.0 - Float64(x * y));
	else
		tmp = Float64(Float64(log(2.0) + Float64(x * 0.5)) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.45)
		tmp = 0.0 - (x * y);
	else
		tmp = (log(2.0) + (x * 0.5)) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.45], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4500000000000002

   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 -2.4500000000000002 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\color{blue}{\left(\log 2 + \frac{1}{2} \cdot x\right)}, \mathsf{*.f64}\left(x, y\right)\right) \]
   4. Step-by-step derivation

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

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

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

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

      3. *-lowering-*.f64 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + x \cdot 0.5\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.45) (- 0.0 (* x y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.45) {
		tmp = 0.0 - (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 <= (-2.45d0)) then
        tmp = 0.0d0 - (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 <= -2.45) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.45:
		tmp = 0.0 - (x * y)
	else:
		tmp = math.log(2.0) + (x * (0.5 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.45)
		tmp = Float64(0.0 - Float64(x * 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 <= -2.45)
		tmp = 0.0 - (x * y);
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.45], N[(0.0 - N[(x * y), $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 < -2.4500000000000002

   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 -2.4500000000000002 < x

   1. Initial program 98.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 99.5%

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

   5. Simplified 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -70.0) (- 0.0 (* x y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = log1p(1.0) - (x * y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -70.0) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -70.0:
		tmp = 0.0 - (x * y)
	else:
		tmp = math.log1p(1.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -70.0)
		tmp = Float64(0.0 - Float64(x * y));
	else
		tmp = Float64(log1p(1.0) - Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -70.0], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + 1.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -70

   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 -70 < x

   1. Initial program 98.8%

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

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

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

      2. log1p-define N/A

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

      3. log1p-lowering-log1p.f64 N/A

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

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

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

      5. *-lowering-*.f64 98.8%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(x\right)\right), \mathsf{*.f64}\left(x, \color{blue}{y}\right)\right) \]

   3. Simplified 98.8%

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

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{\_.f64}\left(\mathsf{log1p.f64}\left(\color{blue}{1}\right), \mathsf{*.f64}\left(x, y\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -70:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-30}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.22e-30) (- 0.0 (* x y)) (log (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.22e-30) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = log((x + 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 <= (-1.22d-30)) then
        tmp = 0.0d0 - (x * y)
    else
        tmp = log((x + 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.22e-30) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = Math.log((x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.22e-30:
		tmp = 0.0 - (x * y)
	else:
		tmp = math.log((x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.22e-30)
		tmp = Float64(0.0 - Float64(x * y));
	else
		tmp = log(Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.22e-30)
		tmp = 0.0 - (x * y);
	else
		tmp = log((x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.22e-30], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.22e-30

   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 97.0%

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

   5. Simplified 97.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   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 97.0%

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

   7. Applied egg-rr 97.0%

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

   if -1.22e-30 < x

   1. Initial program 98.8%

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

   3. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 100.0%

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

   5. Simplified 100.0%

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

   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 76.7%

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

   8. Simplified 76.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-30}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-30}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.02e-30) (- 0.0 (* x y)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-30) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = log(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.02d-30)) then
        tmp = 0.0d0 - (x * y)
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.02e-30) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.02e-30:
		tmp = 0.0 - (x * y)
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.02e-30)
		tmp = Float64(0.0 - Float64(x * y));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.02e-30)
		tmp = 0.0 - (x * y);
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.02e-30], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0199999999999999e-30

   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 97.0%

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

   5. Simplified 97.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   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 97.0%

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

   7. Applied egg-rr 97.0%

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

   if -1.0199999999999999e-30 < x

   1. Initial program 98.8%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.02 \cdot 10^{-30}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.0% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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 53.0%

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

5. Simplified 53.0%

   \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
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 53.0%

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

7. Applied egg-rr 53.0%

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

8. Final simplification 53.0%

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

Alternative 11: 3.5% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. associate--l+ N/A

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

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

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

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

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

   7. *-commutative N/A

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

   8. *-lowering-*.f64 75.5%

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

5. Simplified 75.5%

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

6. Taylor expanded in x around inf

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

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

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

   2. unpow2 N/A

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

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

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

   4. associate--l+ N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. div-sub N/A

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

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

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

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

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

   10. --lowering--.f64 22.1%

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

8. Simplified 22.1%

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

9. Taylor expanded in x around 0

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

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

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

    2. --lowering--.f64 35.3%

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

11. Simplified 35.3%

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

12. Taylor expanded in y around 0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot x} \]
13. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto x \cdot \color{blue}{\frac{1}{2}} \]

    2. *-lowering-*.f64 3.7%

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

14. Simplified 3.7%

    \[\leadsto \color{blue}{x \cdot 0.5} \]
15. 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 2024164 
(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)))