Logistic regression 2

Percentage Accurate: 99.1% → 98.9%

Time: 13.4s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. x = 3.965558043119397e+63
  2. y = -2.3039404847152117e-144

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: 99.1% 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: 98.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -236000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot \left(y + \left(x \cdot \left(-0.125 + x \cdot \left(x \cdot 0.005208333333333333\right)\right) + -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -236000.0)
   (- 0.0 (* x y))
   (-
    (log 2.0)
    (* x (+ y (+ (* x (+ -0.125 (* x (* x 0.005208333333333333)))) -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -236000.0) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = log(2.0) - (x * (y + ((x * (-0.125 + (x * (x * 0.005208333333333333)))) + -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-236000.0d0)) then
        tmp = 0.0d0 - (x * y)
    else
        tmp = log(2.0d0) - (x * (y + ((x * ((-0.125d0) + (x * (x * 0.005208333333333333d0)))) + (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -236000.0) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = Math.log(2.0) - (x * (y + ((x * (-0.125 + (x * (x * 0.005208333333333333)))) + -0.5)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -236000.0:
		tmp = 0.0 - (x * y)
	else:
		tmp = math.log(2.0) - (x * (y + ((x * (-0.125 + (x * (x * 0.005208333333333333)))) + -0.5)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -236000.0)
		tmp = Float64(0.0 - Float64(x * y));
	else
		tmp = Float64(log(2.0) - Float64(x * Float64(y + Float64(Float64(x * Float64(-0.125 + Float64(x * Float64(x * 0.005208333333333333)))) + -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -236000.0)
		tmp = 0.0 - (x * y);
	else
		tmp = log(2.0) - (x * (y + ((x * (-0.125 + (x * (x * 0.005208333333333333)))) + -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -236000.0], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * N[(y + N[(N[(x * N[(-0.125 + N[(x * N[(x * 0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -236000

   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 -236000 < x

   1. Initial program 98.3%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 98.4%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -236000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot \left(y + \left(x \cdot \left(-0.125 + x \cdot \left(x \cdot 0.005208333333333333\right)\right) + -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% 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 98.8%

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

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

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

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

   7. neg-sub0 N/A

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

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

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

   9. log1p-define N/A

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

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

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

   11. exp-lowering-exp.f64 98.8%

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

4. Applied egg-rr 98.8%

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

Alternative 3: 99.2% 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 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. Add Preprocessing

Alternative 4: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -236000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -236000.0)
   (- 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 <= -236000.0) {
		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 <= (-236000.0d0)) 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 <= -236000.0) {
		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 <= -236000.0:
		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 <= -236000.0)
		tmp = Float64(0.0 - Float64(x * y));
	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 <= -236000.0)
		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, -236000.0], N[(0.0 - N[(x * y), $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 < -236000

   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 -236000 < x

   1. Initial program 98.3%

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

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

      2. remove-double-neg N/A

         \[\leadsto \log 2 + \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \log 2 + \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \log 2 + \left(\mathsf{neg}\left(\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(-1 \cdot x\right)\right)\right) \]

      5. unsub-neg N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-out N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. sub-neg N/A

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

      20. --lowering--.f64 N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -236000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(\left(0.5 + x \cdot 0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{log1p}\left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e-7)
   (- 0.0 (* x y))
   (if (<= x 2.4e-126) (log1p (+ x 1.0)) (* x (+ 0.5 (- (* x 0.125) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-7) {
		tmp = 0.0 - (x * y);
	} else if (x <= 2.4e-126) {
		tmp = log1p((x + 1.0));
	} else {
		tmp = x * (0.5 + ((x * 0.125) - y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-7) {
		tmp = 0.0 - (x * y);
	} else if (x <= 2.4e-126) {
		tmp = Math.log1p((x + 1.0));
	} else {
		tmp = x * (0.5 + ((x * 0.125) - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.1e-7:
		tmp = 0.0 - (x * y)
	elif x <= 2.4e-126:
		tmp = math.log1p((x + 1.0))
	else:
		tmp = x * (0.5 + ((x * 0.125) - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-7)
		tmp = Float64(0.0 - Float64(x * y));
	elseif (x <= 2.4e-126)
		tmp = log1p(Float64(x + 1.0));
	else
		tmp = Float64(x * Float64(0.5 + Float64(Float64(x * 0.125) - y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e-7], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-126], N[Log[1 + N[(x + 1.0), $MachinePrecision]], $MachinePrecision], N[(x * N[(0.5 + N[(N[(x * 0.125), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.1000000000000001e-7

   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 98.9%

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

   5. Simplified 98.9%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   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 98.9%

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

   7. Applied egg-rr 98.9%

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

   if -1.1000000000000001e-7 < x < 2.40000000000000007e-126

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. log1p-define N/A

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

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

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{x}\right)\right) \]

      3. exp-lowering-exp.f64 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{\left(1 + x\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 83.3%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(1, x\right)\right) \]

   8. Simplified 83.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x}\right) \]

   if 2.40000000000000007e-126 < x

   1. Initial program 93.1%

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

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

      2. remove-double-neg N/A

         \[\leadsto \log 2 + \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \log 2 + \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \log 2 + \left(\mathsf{neg}\left(\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(-1 \cdot x\right)\right)\right) \]

      5. unsub-neg N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-out N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. sub-neg N/A

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

      20. --lowering--.f64 N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 92.6%

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

   8. Simplified 83.5%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. associate-+r+ N/A

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

       7. +-commutative N/A

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

       8. sub-neg N/A

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

       9. associate--r- N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. div-sub N/A

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

       13. unsub-neg N/A

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

       14. mul-1-neg N/A

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

       15. distribute-rgt-in N/A

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

   11. Simplified 65.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{log1p}\left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-126}:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e-9)
   (- 0.0 (* x y))
   (if (<= x 2.4e-126) (log (+ x 2.0)) (* x (+ 0.5 (- (* x 0.125) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-9) {
		tmp = 0.0 - (x * y);
	} else if (x <= 2.4e-126) {
		tmp = log((x + 2.0));
	} else {
		tmp = 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 <= (-1.15d-9)) then
        tmp = 0.0d0 - (x * y)
    else if (x <= 2.4d-126) then
        tmp = log((x + 2.0d0))
    else
        tmp = 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 <= -1.15e-9) {
		tmp = 0.0 - (x * y);
	} else if (x <= 2.4e-126) {
		tmp = Math.log((x + 2.0));
	} else {
		tmp = x * (0.5 + ((x * 0.125) - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15e-9:
		tmp = 0.0 - (x * y)
	elif x <= 2.4e-126:
		tmp = math.log((x + 2.0))
	else:
		tmp = x * (0.5 + ((x * 0.125) - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15e-9)
		tmp = Float64(0.0 - Float64(x * y));
	elseif (x <= 2.4e-126)
		tmp = log(Float64(x + 2.0));
	else
		tmp = 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 <= -1.15e-9)
		tmp = 0.0 - (x * y);
	elseif (x <= 2.4e-126)
		tmp = log((x + 2.0));
	else
		tmp = x * (0.5 + ((x * 0.125) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15e-9], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-126], N[Log[N[(x + 2.0), $MachinePrecision]], $MachinePrecision], N[(x * N[(0.5 + N[(N[(x * 0.125), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15e-9

   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 98.9%

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

   5. Simplified 98.9%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   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 98.9%

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

   7. Applied egg-rr 98.9%

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

   if -1.15e-9 < x < 2.40000000000000007e-126

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. log1p-define N/A

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

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

         \[\leadsto \mathsf{log1p.f64}\left(\left(e^{x}\right)\right) \]

      3. exp-lowering-exp.f64 83.3%

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

   5. Simplified 83.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{log1p.f64}\left(\color{blue}{\left(1 + x\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 83.3%

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(1, x\right)\right) \]

   8. Simplified 83.3%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x}\right) \]
   9. Step-by-step derivation

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

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

      2. associate-+r+ N/A

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

      3. metadata-eval N/A

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

      4. +-lowering-+.f64 83.3%

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

   10. Applied egg-rr 83.3%

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

   if 2.40000000000000007e-126 < x

   1. Initial program 93.1%

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

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

      2. remove-double-neg N/A

         \[\leadsto \log 2 + \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \log 2 + \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \log 2 + \left(\mathsf{neg}\left(\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(-1 \cdot x\right)\right)\right) \]

      5. unsub-neg N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-out N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. sub-neg N/A

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

      20. --lowering--.f64 N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 92.6%

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

   8. Simplified 83.5%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. associate-+r+ N/A

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

       7. +-commutative N/A

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

       8. sub-neg N/A

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

       9. associate--r- N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. div-sub N/A

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

       13. unsub-neg N/A

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

       14. mul-1-neg N/A

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

       15. distribute-rgt-in N/A

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

   11. Simplified 65.0%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-126}:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -236000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -236000.0) (- 0.0 (* x y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -236000.0) {
		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 <= (-236000.0d0)) 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 <= -236000.0) {
		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 <= -236000.0:
		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 <= -236000.0)
		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 <= -236000.0)
		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, -236000.0], 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 < -236000

   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 -236000 < x

   1. Initial program 98.3%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 97.9%

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

   5. Simplified 97.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -236000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-11}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-126}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.7e-11)
   (- 0.0 (* x y))
   (if (<= x 1.05e-126) (log 2.0) (* x (+ 0.5 (- (* x 0.125) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.7e-11) {
		tmp = 0.0 - (x * y);
	} else if (x <= 1.05e-126) {
		tmp = log(2.0);
	} else {
		tmp = 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 <= (-5.7d-11)) then
        tmp = 0.0d0 - (x * y)
    else if (x <= 1.05d-126) then
        tmp = log(2.0d0)
    else
        tmp = 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 <= -5.7e-11) {
		tmp = 0.0 - (x * y);
	} else if (x <= 1.05e-126) {
		tmp = Math.log(2.0);
	} else {
		tmp = x * (0.5 + ((x * 0.125) - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.7e-11:
		tmp = 0.0 - (x * y)
	elif x <= 1.05e-126:
		tmp = math.log(2.0)
	else:
		tmp = x * (0.5 + ((x * 0.125) - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.7e-11)
		tmp = Float64(0.0 - Float64(x * y));
	elseif (x <= 1.05e-126)
		tmp = log(2.0);
	else
		tmp = 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 <= -5.7e-11)
		tmp = 0.0 - (x * y);
	elseif (x <= 1.05e-126)
		tmp = log(2.0);
	else
		tmp = x * (0.5 + ((x * 0.125) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.7e-11], N[(0.0 - N[(x * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e-126], N[Log[2.0], $MachinePrecision], N[(x * N[(0.5 + N[(N[(x * 0.125), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6999999999999997e-11

   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 98.9%

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

   5. Simplified 98.9%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   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 98.9%

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

   7. Applied egg-rr 98.9%

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

   if -5.6999999999999997e-11 < x < 1.0499999999999999e-126

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 83.1%

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

   5. Simplified 83.1%

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

   if 1.0499999999999999e-126 < x

   1. Initial program 93.1%

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

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

      2. remove-double-neg N/A

         \[\leadsto \log 2 + \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \log 2 + \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(\mathsf{neg}\left(-1 \cdot x\right)\right) \]

      4. distribute-rgt-neg-out N/A

         \[\leadsto \log 2 + \left(\mathsf{neg}\left(\left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right) \cdot \left(-1 \cdot x\right)\right)\right) \]

      5. unsub-neg N/A

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

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

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

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

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

      8. *-commutative N/A

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

      9. mul-1-neg N/A

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

      10. distribute-lft-neg-out N/A

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

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

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

      12. mul-1-neg N/A

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

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

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

      14. mul-1-neg N/A

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

      15. sub-neg N/A

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

      16. +-commutative N/A

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

      17. distribute-neg-in N/A

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

      18. remove-double-neg N/A

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

      19. sub-neg N/A

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

      20. --lowering--.f64 N/A

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

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

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

      22. *-commutative N/A

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

      23. *-lowering-*.f64 92.6%

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

   5. Simplified 92.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. remove-double-neg N/A

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

      8. mul-1-neg N/A

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

      9. distribute-neg-in N/A

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

      10. +-commutative N/A

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

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

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

      12. unpow2 N/A

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

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

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. mul-1-neg N/A

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

      17. remove-double-neg N/A

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

   8. Simplified 83.5%

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

   9. Taylor expanded in x around inf

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

       1. unpow2 N/A

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

       2. associate-*l* N/A

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

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

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

       4. sub-neg N/A

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

       5. +-commutative N/A

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

       6. associate-+r+ N/A

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

       7. +-commutative N/A

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

       8. sub-neg N/A

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

       9. associate--r- N/A

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

       10. associate-*r/ N/A

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

       11. metadata-eval N/A

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

       12. div-sub N/A

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

       13. unsub-neg N/A

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

       14. mul-1-neg N/A

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

       15. distribute-rgt-in N/A

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

   11. Simplified 65.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.7 \cdot 10^{-11}:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-126}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -236000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -236000.0) (- 0.0 (* x y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -236000.0) {
		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 <= -236000.0) {
		tmp = 0.0 - (x * y);
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -236000.0:
		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 <= -236000.0)
		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, -236000.0], 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 < -236000

   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 -236000 < x

   1. Initial program 98.3%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 98.3%

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

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 97.3%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -236000:\\ \;\;\;\;0 - x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.9% 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 98.8%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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 52.6%

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

5. Simplified 52.6%

   \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
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 52.6%

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

7. Applied egg-rr 52.6%

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

8. Final simplification 52.6%

   \[\leadsto 0 - x \cdot y \]
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 2024155 
(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)))