Logistic regression 2

Percentage Accurate: 99.1% → 99.1%

Time: 16.0s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. x = 8.895839788415986e+122
  2. y = 1370.9605161725012

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: 99.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right) - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6)
   (- 0.0 (* y x))
   (-
    (log1p (+ 1.0 (* x (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666)))))))
    (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log1p((1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))) - (y * x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.6) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log1p((1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))) - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.6:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log1p((1.0 + (x * (1.0 + (x * (0.5 + (x * 0.16666666666666666))))))) - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(log1p(Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * Float64(0.5 + Float64(x * 0.16666666666666666))))))) - Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(1.0 + N[(x * N[(1.0 + N[(x * N[(0.5 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001

   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0 - x \cdot y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -1.6000000000000001 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-lowering-*.f64 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

      7. *-lowering-*.f64 99.5%

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

   7. Simplified 99.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)}\right) - x \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.6:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + x \cdot \left(1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\right)\right) - y \cdot x\\ \end{array} \]
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 99.6%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. accelerator-lowering-log1p.f64 N/A

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

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

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

   4. *-lowering-*.f64 99.6%

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

3. Simplified 99.6%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

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

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

   6. neg-sub0 N/A

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

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

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

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

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

   9. exp-lowering-exp.f64 99.6%

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

6. Applied egg-rr 99.6%

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

Alternative 3: 99.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (- (log1p (exp x)) (* y x)))

C

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

Java

public static double code(double x, double y) {
	return Math.log1p(Math.exp(x)) - (y * x);
}

Python

def code(x, y):
	return math.log1p(math.exp(x)) - (y * x)

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. accelerator-lowering-log1p.f64 N/A

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

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

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

   4. *-lowering-*.f64 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+15}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + x \cdot 0.5\right) + x \cdot \left(x \cdot 0.125 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+15)
   (- 0.0 (* y x))
   (+ (+ (log 2.0) (* x 0.5)) (* x (- (* x 0.125) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+15) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = (log(2.0) + (x * 0.5)) + (x * ((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 <= (-5d+15)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = (log(2.0d0) + (x * 0.5d0)) + (x * ((x * 0.125d0) - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+15) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = (Math.log(2.0) + (x * 0.5)) + (x * ((x * 0.125) - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+15:
		tmp = 0.0 - (y * x)
	else:
		tmp = (math.log(2.0) + (x * 0.5)) + (x * ((x * 0.125) - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+15)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(Float64(log(2.0) + Float64(x * 0.5)) + Float64(x * Float64(Float64(x * 0.125) - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+15)
		tmp = 0.0 - (y * x);
	else
		tmp = (log(2.0) + (x * 0.5)) + (x * ((x * 0.125) - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+15], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(x * 0.125), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e15

   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0 - x \cdot y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -5e15 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-lowering-*.f64 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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. associate--l+ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.5%

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

   7. Simplified 99.5%

      \[\leadsto \color{blue}{\log 2 + x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

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

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

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

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

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

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

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

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

      10. *-commutative N/A

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

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

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

      12. --lowering--.f64 N/A

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

      13. *-lowering-*.f64 99.5%

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

   9. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+15}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(\log 2 + x \cdot 0.5\right) + x \cdot \left(x \cdot 0.125 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+15}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5e+15)
   (- 0.0 (* y x))
   (+ (log 2.0) (* x (+ 0.5 (- (* x 0.125) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5e+15) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log(2.0) + (x * (0.5 + ((x * 0.125) - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5d+15)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = log(2.0d0) + (x * (0.5d0 + ((x * 0.125d0) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5e+15) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 + ((x * 0.125) - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5e+15:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0) + (x * (0.5 + ((x * 0.125) - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5e+15)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(0.5 + Float64(Float64(x * 0.125) - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5e+15)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0) + (x * (0.5 + ((x * 0.125) - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5e+15], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 + N[(N[(x * 0.125), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e15

   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0 - x \cdot y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -5e15 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-lowering-*.f64 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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. associate--l+ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 99.5%

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

   7. Simplified 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+15}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - y \cdot x\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-98}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 280:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 0.0 (* y x))))
   (if (<= x -1.25e-98) t_0 (if (<= x 280.0) (+ (log 2.0) (* x 0.5)) t_0))))

C

double code(double x, double y) {
	double t_0 = 0.0 - (y * x);
	double tmp;
	if (x <= -1.25e-98) {
		tmp = t_0;
	} else if (x <= 280.0) {
		tmp = log(2.0) + (x * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (y * x)
    if (x <= (-1.25d-98)) then
        tmp = t_0
    else if (x <= 280.0d0) then
        tmp = log(2.0d0) + (x * 0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.0 - (y * x);
	double tmp;
	if (x <= -1.25e-98) {
		tmp = t_0;
	} else if (x <= 280.0) {
		tmp = Math.log(2.0) + (x * 0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.0 - (y * x)
	tmp = 0
	if x <= -1.25e-98:
		tmp = t_0
	elif x <= 280.0:
		tmp = math.log(2.0) + (x * 0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.0 - Float64(y * x))
	tmp = 0.0
	if (x <= -1.25e-98)
		tmp = t_0;
	elseif (x <= 280.0)
		tmp = Float64(log(2.0) + Float64(x * 0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.0 - (y * x);
	tmp = 0.0;
	if (x <= -1.25e-98)
		tmp = t_0;
	elseif (x <= 280.0)
		tmp = log(2.0) + (x * 0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e-98], t$95$0, If[LessEqual[x, 280.0], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.25000000000000005e-98 or 280 < x

   1. Initial program 99.1%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-lowering-*.f64 99.1%

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

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

      4. *-lowering-*.f64 92.2%

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

   7. Simplified 92.2%

      \[\leadsto \color{blue}{0 - x \cdot y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 92.2%

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

   9. Applied egg-rr 92.2%

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

   if -1.25000000000000005e-98 < x < 280

   1. Initial program 99.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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 y around 0

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

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

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

      2. exp-lowering-exp.f64 84.3%

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

   7. Simplified 84.3%

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

   8. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 83.5%

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

   10. Simplified 83.5%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-98}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{elif}\;x \leq 280:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0 - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-99}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{elif}\;x \leq 0.00016:\\ \;\;\;\;\mathsf{log1p}\left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - x \cdot -0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e-99)
   (- 0.0 (* y x))
   (if (<= x 0.00016) (log1p (+ x 1.0)) (* x (- (- 0.5 (* x -0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e-99) {
		tmp = 0.0 - (y * x);
	} else if (x <= 0.00016) {
		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 <= -6.8e-99) {
		tmp = 0.0 - (y * x);
	} else if (x <= 0.00016) {
		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 <= -6.8e-99:
		tmp = 0.0 - (y * x)
	elif x <= 0.00016:
		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 <= -6.8e-99)
		tmp = Float64(0.0 - Float64(y * x));
	elseif (x <= 0.00016)
		tmp = log1p(Float64(x + 1.0));
	else
		tmp = Float64(x * Float64(Float64(0.5 - Float64(x * -0.125)) - y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e-99], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00016], N[Log[1 + N[(x + 1.0), $MachinePrecision]], $MachinePrecision], N[(x * N[(N[(0.5 - N[(x * -0.125), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.80000000000000014e-99

   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

      4. *-lowering-*.f64 91.2%

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

   7. Simplified 91.2%

      \[\leadsto \color{blue}{0 - x \cdot y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 91.2%

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

   9. Applied egg-rr 91.2%

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

   if -6.80000000000000014e-99 < x < 1.60000000000000013e-4

   1. Initial program 99.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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 y around 0

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

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

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

      2. exp-lowering-exp.f64 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 83.9%

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

   10. Simplified 83.9%

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

   if 1.60000000000000013e-4 < x

   1. Initial program 92.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-lowering-*.f64 92.9%

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

   3. Simplified 92.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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. associate--l+ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 93.9%

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

   7. Simplified 93.9%

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

   8. 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. Simplified 93.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-99}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{elif}\;x \leq 0.00016:\\ \;\;\;\;\mathsf{log1p}\left(x + 1\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - x \cdot -0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+29}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e+29) (- 0.0 (* y x)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+29) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.9d+29)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = log(2.0d0) + (x * (0.5d0 - y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+29) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.9e+29:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0) + (x * (0.5 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e+29)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(0.5 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.9e+29)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0) + (x * (0.5 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e+29], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9000000000000001e29

   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0 - x \cdot y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -4.9000000000000001e29 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-lowering-*.f64 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+29}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-98}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - x \cdot -0.125\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e-98)
   (- 0.0 (* y x))
   (if (<= x 4.6e-5) (log 2.0) (* x (- (- 0.5 (* x -0.125)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15e-98) {
		tmp = 0.0 - (y * x);
	} else if (x <= 4.6e-5) {
		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 <= (-1.15d-98)) then
        tmp = 0.0d0 - (y * x)
    else if (x <= 4.6d-5) 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 <= -1.15e-98) {
		tmp = 0.0 - (y * x);
	} else if (x <= 4.6e-5) {
		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 <= -1.15e-98:
		tmp = 0.0 - (y * x)
	elif x <= 4.6e-5:
		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 <= -1.15e-98)
		tmp = Float64(0.0 - Float64(y * x));
	elseif (x <= 4.6e-5)
		tmp = log(2.0);
	else
		tmp = 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 <= -1.15e-98)
		tmp = 0.0 - (y * x);
	elseif (x <= 4.6e-5)
		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, -1.15e-98], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-5], N[Log[2.0], $MachinePrecision], N[(x * N[(N[(0.5 - N[(x * -0.125), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.15e-98

   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

      4. *-lowering-*.f64 91.2%

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

   7. Simplified 91.2%

      \[\leadsto \color{blue}{0 - x \cdot y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 91.2%

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

   9. Applied egg-rr 91.2%

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

   if -1.15e-98 < x < 4.6e-5

   1. Initial program 99.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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 83.7%

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

   7. Simplified 83.7%

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

   if 4.6e-5 < x

   1. Initial program 92.9%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-lowering-*.f64 92.9%

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

   3. Simplified 92.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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. associate--l+ N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 93.9%

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

   7. Simplified 93.9%

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

   8. 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. Simplified 93.9%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-98}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-5}:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\left(0.5 - x \cdot -0.125\right) - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+29}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e+29) (- 0.0 (* y x)) (- (log1p 1.0) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+29) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log1p(1.0) - (y * x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+29) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log1p(1.0) - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.9e+29:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log1p(1.0) - (y * x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e+29)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(log1p(1.0) - Float64(y * x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e+29], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + 1.0], $MachinePrecision] - N[(y * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9000000000000001e29

   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-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. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

      4. *-lowering-*.f64 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0 - x \cdot y} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

   if -4.9000000000000001e29 < x

   1. Initial program 99.4%

      \[\log \left(1 + e^{x}\right) - x \cdot y \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      4. *-lowering-*.f64 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+29}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.5% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
2. 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. accelerator-lowering-log1p.f64 N/A

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

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

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

   4. *-lowering-*.f64 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
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. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

   4. *-lowering-*.f64 48.7%

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

7. Simplified 48.7%

   \[\leadsto \color{blue}{0 - x \cdot y} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

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

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 48.7%

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

9. Applied egg-rr 48.7%

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

10. Final simplification 48.7%

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

Alternative 12: 2.2% accurate, 69.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[\log \left(1 + e^{x}\right) - x \cdot y \]
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. accelerator-lowering-log1p.f64 N/A

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

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

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

   4. *-lowering-*.f64 99.6%

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

3. Simplified 99.6%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
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. neg-sub0 N/A

      \[\leadsto 0 - \color{blue}{x \cdot y} \]

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

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

   4. *-lowering-*.f64 48.7%

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

7. Simplified 48.7%

   \[\leadsto \color{blue}{0 - x \cdot y} \]
8. Step-by-step derivation

   1. sub0-neg N/A

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

   2. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\left(0 + x \cdot y\right)\right) \]

   3. flip3-+ N/A

      \[\leadsto \mathsf{neg}\left(\frac{{0}^{3} + {\left(x \cdot y\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{neg}\left(\frac{0 + {\left(x \cdot y\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)}\right) \]

   5. +-lft-identity N/A

      \[\leadsto \mathsf{neg}\left(\frac{{\left(x \cdot y\right)}^{3}}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)}\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left({\left(x \cdot y\right)}^{3}\right)}{\color{blue}{0 \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)}} \]

   7. cube-neg N/A

      \[\leadsto \frac{{\left(\mathsf{neg}\left(x \cdot y\right)\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)} \]

   8. sqr-pow N/A

      \[\leadsto \frac{{\left(\mathsf{neg}\left(x \cdot y\right)\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\mathsf{neg}\left(x \cdot y\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)} \]

   9. pow-prod-down N/A

      \[\leadsto \frac{{\left(\left(\mathsf{neg}\left(x \cdot y\right)\right) \cdot \left(\mathsf{neg}\left(x \cdot y\right)\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)} \]

   10. sqr-neg N/A

       \[\leadsto \frac{{\left(\left(x \cdot y\right) \cdot \left(x \cdot y\right)\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0} \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)} \]

   11. pow-prod-down N/A

       \[\leadsto \frac{{\left(x \cdot y\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(x \cdot y\right)}^{\left(\frac{3}{2}\right)}}{\color{blue}{0 \cdot 0} + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)} \]

   12. sqr-pow N/A

       \[\leadsto \frac{{\left(x \cdot y\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)} \]

   13. +-lft-identity N/A

       \[\leadsto \frac{0 + {\left(x \cdot y\right)}^{3}}{\color{blue}{0 \cdot 0} + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)} \]

   14. metadata-eval N/A

       \[\leadsto \frac{{0}^{3} + {\left(x \cdot y\right)}^{3}}{\color{blue}{0} \cdot 0 + \left(\left(x \cdot y\right) \cdot \left(x \cdot y\right) - 0 \cdot \left(x \cdot y\right)\right)} \]

   15. flip3-+ N/A

       \[\leadsto 0 + \color{blue}{x \cdot y} \]

   16. +-lft-identity N/A

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 2.2%

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

9. Applied egg-rr 2.2%

   \[\leadsto \color{blue}{y \cdot x} \]
10. 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 2024184 
(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)))