Logistic regression 2

Percentage Accurate: 99.1% → 99.2%

Time: 13.0s

Alternatives: 10

Speedup: 1.8×

• could not determine a ground truth

  1. x = 1.0419357803721727e+49
  2. y = -2.218013038881438e+119

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.1% 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: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.55:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.55)
   (- 0.0 (* y x))
   (+
    (log 2.0)
    (* x (+ 0.5 (- (* x (+ 0.125 (* -0.005208333333333333 (* x x)))) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.55) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.55d0)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = log(2.0d0) + (x * (0.5d0 + ((x * (0.125d0 + ((-0.005208333333333333d0) * (x * x)))) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.55) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.55:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.55)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(log(2.0) + Float64(x * Float64(0.5 + Float64(Float64(x * Float64(0.125 + Float64(-0.005208333333333333 * Float64(x * x)))) - y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.55)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0) + (x * (0.5 + ((x * (0.125 + (-0.005208333333333333 * (x * x)))) - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.55], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * N[(0.5 + N[(N[(x * N[(0.125 + N[(-0.005208333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5499999999999998

   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 -3.5499999999999998 < x

   1. Initial program 98.2%

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

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

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

      2. log1p-define N/A

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

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

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

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

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

      5. *-lowering-*.f64 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.55:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot \left(0.125 + -0.005208333333333333 \cdot \left(x \cdot x\right)\right) - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

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

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

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

   2. log1p-define N/A

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

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

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

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

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

   5. *-lowering-*.f64 98.8%

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

3. Simplified 98.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

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

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

   5. fma-define N/A

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

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

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

   7. neg-sub0 N/A

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

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

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

   9. log1p-define N/A

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

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

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

   11. exp-lowering-exp.f64 98.8%

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

6. Applied egg-rr 98.8%

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

Alternative 3: 99.1% 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 98.8%

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

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

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

   2. log1p-define N/A

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

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

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

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

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

   5. *-lowering-*.f64 98.8%

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

3. Simplified 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -80:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125 - y\right) + \left(\log 2 + x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -80.0)
   (- 0.0 (* y x))
   (+ (* x (- (* x 0.125) y)) (+ (log 2.0) (* x 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -80.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = (x * ((x * 0.125) - y)) + (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 <= (-80.0d0)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = (x * ((x * 0.125d0) - y)) + (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 <= -80.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = (x * ((x * 0.125) - y)) + (Math.log(2.0) + (x * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -80.0:
		tmp = 0.0 - (y * x)
	else:
		tmp = (x * ((x * 0.125) - y)) + (math.log(2.0) + (x * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -80.0)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = Float64(Float64(x * Float64(Float64(x * 0.125) - y)) + Float64(log(2.0) + Float64(x * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -80.0)
		tmp = 0.0 - (y * x);
	else
		tmp = (x * ((x * 0.125) - y)) + (log(2.0) + (x * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -80.0], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(N[(x * 0.125), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] + N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -80

   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 -80 < x

   1. Initial program 98.2%

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

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

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

      2. log1p-define N/A

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

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

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

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

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

      5. *-lowering-*.f64 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 99.9%

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

   7. Simplified 99.9%

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

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

      3. associate-+r+ N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

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

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

      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 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -80:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot 0.125 - y\right) + \left(\log 2 + x \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -102:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -102.0)
   (- 0.0 (* y x))
   (+ (log 2.0) (* x (+ 0.5 (- (* x 0.125) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -102.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = log(2.0) + (x * (0.5 + ((x * 0.125) - y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-102.0d0)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = log(2.0d0) + (x * (0.5d0 + ((x * 0.125d0) - y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -102.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) + (x * (0.5 + ((x * 0.125) - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -102.0:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0) + (x * (0.5 + ((x * 0.125) - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -102.0)
		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 <= -102.0)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0) + (x * (0.5 + ((x * 0.125) - y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -102.0], 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 < -102

   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 -102 < x

   1. Initial program 98.2%

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

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

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

      2. log1p-define N/A

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

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

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

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

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

      5. *-lowering-*.f64 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -102:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 + \left(x \cdot 0.125 - y\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.55:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.55) (- 0.0 (* y x)) (+ (log 2.0) (* x (- 0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.55) {
		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 <= (-3.55d0)) 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 <= -3.55) {
		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 <= -3.55:
		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 <= -3.55)
		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 <= -3.55)
		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, -3.55], 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 < -3.5499999999999998

   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 -3.5499999999999998 < x

   1. Initial program 98.2%

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

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

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

      2. log1p-define N/A

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

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

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

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

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

      5. *-lowering-*.f64 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.55:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot \left(0.5 - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -102:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -102.0) (- 0.0 (* y x)) (- (log1p 1.0) (* y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -102.0) {
		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 <= -102.0) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log1p(1.0) - (y * x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -102.0:
		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 <= -102.0)
		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, -102.0], 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 < -102

   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 -102 < x

   1. Initial program 98.2%

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

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

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

      2. log1p-define N/A

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

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

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

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

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

      5. *-lowering-*.f64 98.2%

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

   3. Simplified 98.2%

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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -102:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right) - y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e-6) (- 0.0 (* y x)) (+ (log 2.0) (* x 0.5))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.2e-6) {
		tmp = 0.0 - (y * x);
	} 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-6)) then
        tmp = 0.0d0 - (y * x)
    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-6) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0) + (x * 0.5);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.2e-6:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0) + (x * 0.5)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.2e-6)
		tmp = Float64(0.0 - Float64(y * x));
	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-6)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0) + (x * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.2e-6], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1999999999999996e-6

   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 -4.1999999999999996e-6 < x

   1. Initial program 98.2%

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

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

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

      2. log1p-define N/A

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

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

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

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

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

      5. *-lowering-*.f64 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 74.2%

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

   10. Simplified 74.2%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. fma-define N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, \log 2\right) \]

       3. *-rgt-identity N/A

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

       4. rgt-mult-inverse N/A

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

       5. associate-*l* N/A

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

       6. unpow2 N/A

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

       7. fma-define N/A

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

       8. *-commutative N/A

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

       9. associate-*r* N/A

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

       10. *-commutative N/A

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

       11. remove-double-neg N/A

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

       12. mul-1-neg N/A

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

       13. mul-1-neg N/A

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

       14. associate-*r* N/A

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

       15. metadata-eval N/A

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

       16. *-commutative N/A

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

       17. *-inverses N/A

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

       18. associate-/l* N/A

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

       19. associate-*l/ N/A

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

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

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

   13. Simplified 74.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2 + x \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e-11) (- 0.0 (* y x)) (log 2.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-11) {
		tmp = 0.0 - (y * x);
	} 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 <= (-4.4d-11)) then
        tmp = 0.0d0 - (y * x)
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.4e-11) {
		tmp = 0.0 - (y * x);
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.4e-11:
		tmp = 0.0 - (y * x)
	else:
		tmp = math.log(2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e-11)
		tmp = Float64(0.0 - Float64(y * x));
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.4e-11)
		tmp = 0.0 - (y * x);
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e-11], N[(0.0 - N[(y * x), $MachinePrecision]), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4000000000000003e-11

   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 -4.4000000000000003e-11 < x

   1. Initial program 98.2%

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

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

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

      2. log1p-define N/A

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

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

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

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

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

      5. *-lowering-*.f64 98.2%

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

   3. Simplified 98.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{x}\right) - x \cdot y} \]
   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 73.2%

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

   7. Simplified 73.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;0 - y \cdot x\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.3% 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 98.8%

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

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

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

   2. log1p-define N/A

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

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

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

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

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

   5. *-lowering-*.f64 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in x around 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 53.0%

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

7. Simplified 53.0%

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

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

9. Applied egg-rr 53.0%

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

10. Final simplification 53.0%

    \[\leadsto 0 - y \cdot x \]
11. 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 2024163 
(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)))