Logistic regression 2

Percentage Accurate: 99.4% → 99.5%

Time: 11.8s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. x = 3.974837101315998e+56
  2. y = -1.4569465225392064e-155

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: 99.4% 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: 99.5% 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 100.0%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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 100.0%

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

3. Simplified 100.0%

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

Alternative 2: 99.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -180.0)
   (* x (- 0.0 y))
   (+
    (log 2.0)
    (* x (+ 0.5 (- (* x (+ 0.125 (* -0.005208333333333333 (* x x)))) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -180.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-180.0d0)) then
        tmp = x * (0.0d0 - y)
    else
        tmp = log(2.0d0) + (x * (0.5d0 + ((x * (0.125d0 + ((-0.005208333333333333d0) * (x * x)))) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -180.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -180.0:
		tmp = x * (0.0 - y)
	else:
		tmp = math.log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -180.0)
		tmp = Float64(x * Float64(0.0 - y));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(0.5 + Float64(Float64(x * Float64(0.125 + Float64(-0.005208333333333333 * Float64(x * x)))) - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -180.0)
		tmp = x * (0.0 - y);
	else
		tmp = log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -180.0], N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 + N[(N[(x * N[(0.125 + N[(-0.005208333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -180

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. 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%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   8. 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%

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

   9. Applied egg-rr 100.0%

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

   if -180 < x

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot \left(\left(y - x \cdot 0.125\right) - 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -180.0)
   (* x (- 0.0 y))
   (- (log 2.0) (* x (- (- y (* x 0.125)) 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -180.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = log(2.0) - (x * ((y - (x * 0.125)) - 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 <= (-180.0d0)) then
        tmp = x * (0.0d0 - y)
    else
        tmp = log(2.0d0) - (x * ((y - (x * 0.125d0)) - 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -180.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = Math.log(2.0) - (x * ((y - (x * 0.125)) - 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -180.0:
		tmp = x * (0.0 - y)
	else:
		tmp = math.log(2.0) - (x * ((y - (x * 0.125)) - 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -180.0)
		tmp = Float64(x * Float64(0.0 - y));
	else
		tmp = Float64(log(2.0) - Float64(x * Float64(Float64(y - Float64(x * 0.125)) - 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -180.0)
		tmp = x * (0.0 - y);
	else
		tmp = log(2.0) - (x * ((y - (x * 0.125)) - 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -180.0], N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] - N[(x * N[(N[(y - N[(x * 0.125), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -180

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. 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%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   8. 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%

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

   9. Applied egg-rr 100.0%

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

   if -180 < x

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot x\right) - y\right)} \]
   6. 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. +-commutative N/A

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

      5. associate--l+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l- N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot \left(\left(y - x \cdot 0.125\right) - 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -180.0) (* x (- 0.0 y)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -180.0) {
		tmp = x * (0.0 - 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 <= (-180.0d0)) then
        tmp = x * (0.0d0 - 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 <= -180.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -180.0:
		tmp = x * (0.0 - y)
	else:
		tmp = math.log(2.0) + (x * (0.5 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -180.0)
		tmp = Float64(x * Float64(0.0 - 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 <= -180.0)
		tmp = x * (0.0 - y);
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -180.0], N[(x * N[(0.0 - 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 < -180

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. 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%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   8. 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%

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

   9. Applied egg-rr 100.0%

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

   if -180 < x

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(\frac{1}{2} - y\right)} \]
   6. 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 99.6%

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

   7. Simplified 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -180.0) (* x (- 0.0 y)) (- (log1p 1.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -180.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = log1p(1.0) - (x * y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -180.0) {
		tmp = x * (0.0 - y);
	} else {
		tmp = Math.log1p(1.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -180.0:
		tmp = x * (0.0 - y)
	else:
		tmp = math.log1p(1.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -180.0)
		tmp = Float64(x * Float64(0.0 - y));
	else
		tmp = Float64(log1p(1.0) - Float64(x * y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -180.0], N[(x * N[(0.0 - 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 < -180

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. 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%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   8. 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%

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

   9. Applied egg-rr 100.0%

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

   if -180 < x

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 98.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e-108) (* x (- 0.0 y)) (+ (log 2.0) (* x 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.2e-108) {
		tmp = x * (0.0 - y);
	} else {
		tmp = log(2.0) + (x * 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 <= (-4.2d-108)) then
        tmp = x * (0.0d0 - y)
    else
        tmp = log(2.0d0) + (x * 0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.2e-108) {
		tmp = x * (0.0 - y);
	} else {
		tmp = Math.log(2.0) + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.2e-108:
		tmp = x * (0.0 - y)
	else:
		tmp = math.log(2.0) + (x * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.2e-108)
		tmp = Float64(x * Float64(0.0 - y));
	else
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.2e-108)
		tmp = x * (0.0 - y);
	else
		tmp = log(2.0) + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.2e-108], N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999998e-108

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. 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 86.9%

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

   7. Simplified 86.9%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   8. 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 86.9%

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

   9. Applied egg-rr 86.9%

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

   if -4.1999999999999998e-108 < x

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

      4. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 99.6%

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

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)} \]

   8. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

      12. log-lowering-log.f64 79.7%

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

   10. Simplified 79.7%

       \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \left(0.125 + x \cdot \left(x \cdot -0.005208333333333333\right)\right)\right) + \log 2} \]

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 79.5%

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

   13. Simplified 79.5%

       \[\leadsto \color{blue}{x \cdot 0.5} + \log 2 \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.14 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.14e-110) (* x (- 0.0 y)) (log (+ x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.14e-110) {
		tmp = x * (0.0 - 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 <= (-1.14d-110)) then
        tmp = x * (0.0d0 - 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 <= -1.14e-110) {
		tmp = x * (0.0 - y);
	} else {
		tmp = Math.log((x + 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.14e-110:
		tmp = x * (0.0 - y)
	else:
		tmp = math.log((x + 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.14e-110)
		tmp = Float64(x * Float64(0.0 - y));
	else
		tmp = log(Float64(x + 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.14e-110)
		tmp = x * (0.0 - y);
	else
		tmp = log((x + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.14e-110], N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], N[Log[N[(x + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.14e-110

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. 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 86.9%

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

   7. Simplified 86.9%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   8. 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 86.9%

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

   9. Applied egg-rr 86.9%

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

   if -1.14e-110 < x

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log \left(2 + x\right)} \]
   9. 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 79.5%

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

   10. Simplified 79.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.14 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-108) (* x (- 0.0 y)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-108) {
		tmp = x * (0.0 - 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 <= (-3.5d-108)) then
        tmp = x * (0.0d0 - 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 <= -3.5e-108) {
		tmp = x * (0.0 - y);
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.5e-108:
		tmp = x * (0.0 - y)
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-108)
		tmp = Float64(x * Float64(0.0 - y));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-108)
		tmp = x * (0.0 - y);
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e-108], N[(x * N[(0.0 - y), $MachinePrecision]), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4999999999999999e-108

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
   6. 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 86.9%

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

   7. Simplified 86.9%

      \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
   8. 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 86.9%

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

   9. Applied egg-rr 86.9%

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

   if -3.4999999999999999e-108 < x

   1. Initial program 100.0%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. log-lowering-log.f64 79.0%

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

   7. Simplified 79.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-108}:\\ \;\;\;\;x \cdot \left(0 - y\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.0% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0 - y\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot y\right)} \]
6. 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 49.5%

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

7. Simplified 49.5%

   \[\leadsto \color{blue}{x \cdot \left(0 - y\right)} \]
8. 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 49.5%

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

9. Applied egg-rr 49.5%

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

10. Final simplification 49.5%

    \[\leadsto x \cdot \left(0 - y\right) \]
11. Add Preprocessing

Alternative 10: 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 100.0%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in x around 0

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

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

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

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

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

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

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

   4. associate--l+ N/A

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

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

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

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

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

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

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

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

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

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

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

   10. unpow2 N/A

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

   11. *-lowering-*.f64 75.8%

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

7. Simplified 75.8%

   \[\leadsto \color{blue}{\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)} \]

8. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

   9. associate-*l* N/A

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

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

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

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

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

   12. log-lowering-log.f64 55.6%

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

10. Simplified 55.6%

    \[\leadsto \color{blue}{x \cdot \left(0.5 + x \cdot \left(0.125 + x \cdot \left(x \cdot -0.005208333333333333\right)\right)\right) + \log 2} \]

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{{x}^{4} \cdot \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{192}\right)} \]
12. Step-by-step derivation

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

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{4}\right), \color{blue}{\left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right) - \frac{1}{192}\right)}\right) \]

    2. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(3 + 1\right)}\right), \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}\right) - \frac{1}{192}\right)\right) \]

    3. pow-plus N/A

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{3} \cdot x\right), \left(\color{blue}{\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)} - \frac{1}{192}\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\left(x \cdot {x}^{3}\right), \left(\color{blue}{\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)} - \frac{1}{192}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left({x}^{3}\right)\right), \left(\color{blue}{\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)} - \frac{1}{192}\right)\right) \]

    6. cube-mult N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot \left(x \cdot x\right)\right)\right), \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}\right) - \frac{1}{192}\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot {x}^{2}\right)\right), \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right) - \frac{1}{192}\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left({x}^{2}\right)\right)\right), \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \color{blue}{\frac{1}{8} \cdot \frac{1}{{x}^{2}}}\right) - \frac{1}{192}\right)\right) \]

    9. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \left(x \cdot x\right)\right)\right), \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right) - \frac{1}{192}\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \color{blue}{\frac{1}{{x}^{2}}}\right) - \frac{1}{192}\right)\right) \]

    11. sub-neg N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{192}\right)\right)}\right)\right) \]

    12. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right) + \frac{-1}{192}\right)\right) \]

    13. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \left(\frac{-1}{192} + \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{3}} + \frac{1}{8} \cdot \frac{1}{{x}^{2}}\right)}\right)\right) \]

    15. +-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \left(\frac{1}{8} \cdot \frac{1}{{x}^{2}} + \color{blue}{\frac{\frac{1}{2}}{{x}^{3}}}\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{\left(\frac{\frac{1}{2}}{{x}^{3}}\right)}\right)\right)\right) \]

    17. associate-*r/ N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \mathsf{+.f64}\left(\left(\frac{\frac{1}{8} \cdot 1}{{x}^{2}}\right), \left(\frac{\color{blue}{\frac{1}{2}}}{{x}^{3}}\right)\right)\right)\right) \]

    18. metadata-eval N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \mathsf{+.f64}\left(\left(\frac{\frac{1}{8}}{{x}^{2}}\right), \left(\frac{\frac{1}{2}}{{x}^{3}}\right)\right)\right)\right) \]

    19. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \left({x}^{2}\right)\right), \left(\frac{\color{blue}{\frac{1}{2}}}{{x}^{3}}\right)\right)\right)\right) \]

    20. unpow2 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \left(x \cdot x\right)\right), \left(\frac{\frac{1}{2}}{{x}^{3}}\right)\right)\right)\right) \]

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

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \left(\frac{\frac{1}{2}}{{x}^{3}}\right)\right)\right)\right) \]

    22. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, x\right)\right)\right), \mathsf{+.f64}\left(\frac{-1}{192}, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{8}, \mathsf{*.f64}\left(x, x\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{3}\right)}\right)\right)\right)\right) \]

    23. cube-mult N/A

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

    24. unpow2 N/A

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

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

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

    26. unpow2 N/A

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

    27. *-lowering-*.f64 5.1%

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

13. Simplified 5.1%

    \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(-0.005208333333333333 + \left(\frac{0.125}{x \cdot x} + \frac{0.5}{x \cdot \left(x \cdot x\right)}\right)\right)} \]

14. Taylor expanded in x around 0

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

    1. *-lowering-*.f64 3.5%

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

16. Simplified 3.5%

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

17. Final simplification 3.5%

    \[\leadsto x \cdot 0.5 \]
18. 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 2024145 
(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)))