Logistic regression 2

Percentage Accurate: 99.1% → 99.1%

Time: 10.1s

Alternatives: 12

Speedup: 1.7×

• could not determine a ground truth

  1. x = 1.0514640750446848e+188
  2. y = -4.899190107231248e+38

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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 97.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := -x \cdot y\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 1:\\ \;\;\;\;-\log \left(\mathsf{fma}\left(x, -0.25, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (- (* x y))))
   (if (<= t_0 0.01) t_1 (if (<= t_0 1.0) (- (log (fma x -0.25 0.5))) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = t_1;
	} else if (t_0 <= 1.0) {
		tmp = -log(fma(x, -0.25, 0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(-Float64(x * y))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = t_1;
	elseif (t_0 <= 1.0)
		tmp = Float64(-log(fma(x, -0.25, 0.5)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * y), $MachinePrecision])}, If[LessEqual[t$95$0, 0.01], t$95$1, If[LessEqual[t$95$0, 1.0], (-N[Log[N[(x * -0.25 + 0.5), $MachinePrecision]], $MachinePrecision]), t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 0.0100000000000000002 or 1 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if 0.0100000000000000002 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 1

   1. Initial program 100.0%

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

      1. lift-exp.f64 N/A

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

      2. +-commutative N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. lower-expm1.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{\color{blue}{\mathsf{expm1}\left(x\right)}}{e^{x} \cdot e^{x} - 1 \cdot 1}\right)\right)\right) - x \cdot y \]

      11. lift-exp.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{\mathsf{expm1}\left(x\right)}{\color{blue}{e^{x}} \cdot e^{x} - 1 \cdot 1}\right)\right)\right) - x \cdot y \]

      12. lift-exp.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{\mathsf{expm1}\left(x\right)}{e^{x} \cdot \color{blue}{e^{x}} - 1 \cdot 1}\right)\right)\right) - x \cdot y \]

      13. prod-exp N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{\mathsf{expm1}\left(x\right)}{\color{blue}{e^{x + x}} - 1 \cdot 1}\right)\right)\right) - x \cdot y \]

      14. metadata-eval N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{\mathsf{expm1}\left(x\right)}{e^{x + x} - \color{blue}{1}}\right)\right)\right) - x \cdot y \]

      15. lower-expm1.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\frac{\mathsf{expm1}\left(x\right)}{\color{blue}{\mathsf{expm1}\left(x + x\right)}}\right)\right)\right) - x \cdot y \]

      16. lower-+.f64 99.9

          \[\leadsto \left(-\log \left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(\color{blue}{x + x}\right)}\right)\right) - x \cdot y \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(-\log \left(\frac{\mathsf{expm1}\left(x\right)}{\mathsf{expm1}\left(x + x\right)}\right)\right)} - x \cdot y \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\mathsf{neg}\left(\log \color{blue}{\left(\frac{1}{2} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right)\right)}\right)\right) - x \cdot y \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)}\right)\right) - x \cdot y \]

      2. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \color{blue}{\left(\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}, \frac{1}{2}\right)\right)}\right)\right) - x \cdot y \]

      3. sub-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      5. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      6. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      7. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, x \cdot \left(\left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) \cdot x, \frac{-1}{4}\right)}, \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      9. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      11. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{480} \cdot {x}^{2} + \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      12. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{480}} + \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      13. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{480}, \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      14. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{480}, \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right)\right) - x \cdot y \]

      15. lower-*.f64 99.7

          \[\leadsto \left(-\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0020833333333333333, 0.020833333333333332\right), -0.25\right), 0.5\right)\right)\right) - x \cdot y \]

   7. Applied rewrites 99.7%

      \[\leadsto \left(-\log \color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.0020833333333333333, 0.020833333333333332\right), -0.25\right), 0.5\right)\right)}\right) - x \cdot y \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \log \left(\frac{1}{2} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{2} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right)\right)\right)} \]

      2. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\log \left(\frac{1}{2} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right)\right)\right)} \]

      3. lower-log.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{1}{2} + x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right)\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(x \cdot \left({x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}\right) + \frac{1}{2}\right)}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) - \frac{1}{4}, \frac{1}{2}\right)\right)}\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}, \frac{1}{2}\right)\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, {x}^{2} \cdot \left(\frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}\right) + \color{blue}{\frac{-1}{4}}, \frac{1}{2}\right)\right)\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}, \frac{-1}{4}\right)}, \frac{1}{2}\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{48} + \frac{-1}{480} \cdot {x}^{2}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{-1}{480} \cdot {x}^{2} + \frac{1}{48}}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{-1}{480}} + \frac{1}{48}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{480}, \frac{1}{48}\right)}, \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{neg}\left(\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{480}, \frac{1}{48}\right), \frac{-1}{4}\right), \frac{1}{2}\right)\right)\right) \]

      15. lower-*.f64 98.1

          \[\leadsto -\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.0020833333333333333, 0.020833333333333332\right), -0.25\right), 0.5\right)\right) \]

   10. Applied rewrites 98.1%

       \[\leadsto \color{blue}{-\log \left(\mathsf{fma}\left(x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.0020833333333333333, 0.020833333333333332\right), -0.25\right), 0.5\right)\right)} \]

   11. Taylor expanded in x around 0

       \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{1}{2} + \frac{-1}{4} \cdot x\right)}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{neg}\left(\log \color{blue}{\left(\frac{-1}{4} \cdot x + \frac{1}{2}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{neg}\left(\log \left(\color{blue}{x \cdot \frac{-1}{4}} + \frac{1}{2}\right)\right) \]

       3. lower-fma.f64 97.8

          \[\leadsto -\log \color{blue}{\left(\mathsf{fma}\left(x, -0.25, 0.5\right)\right)} \]

   13. Applied rewrites 97.8%

       \[\leadsto -\log \color{blue}{\left(\mathsf{fma}\left(x, -0.25, 0.5\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.01:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 1:\\ \;\;\;\;-\log \left(\mathsf{fma}\left(x, -0.25, 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := -x \cdot y\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (- (* x y))))
   (if (<= t_0 0.01) t_1 (if (<= t_0 5.0) (fma x 0.5 (log 2.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = t_1;
	} else if (t_0 <= 5.0) {
		tmp = fma(x, 0.5, log(2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(-Float64(x * y))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = t_1;
	elseif (t_0 <= 5.0)
		tmp = fma(x, 0.5, log(2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * y), $MachinePrecision])}, If[LessEqual[t$95$0, 0.01], t$95$1, If[LessEqual[t$95$0, 5.0], N[(x * 0.5 + N[Log[2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 0.0100000000000000002 or 5 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if 0.0100000000000000002 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 98.7

         \[\leadsto \mathsf{fma}\left(x, 0.5 - y, \color{blue}{\log 2}\right) \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5 - y, \log 2\right)} \]

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 97.0

         \[\leadsto \mathsf{fma}\left(x, 0.5, \color{blue}{\log 2}\right) \]

   8. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \log 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.01:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 5:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5, \log 2\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := -x \cdot y\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (- (* x y))))
   (if (<= t_0 0.01) t_1 (if (<= t_0 5.0) (log (+ x 2.0)) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = t_1;
	} else if (t_0 <= 5.0) {
		tmp = log((x + 2.0));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = log((1.0d0 + exp(x))) - (x * y)
    t_1 = -(x * y)
    if (t_0 <= 0.01d0) then
        tmp = t_1
    else if (t_0 <= 5.0d0) then
        tmp = log((x + 2.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = t_1;
	} else if (t_0 <= 5.0) {
		tmp = Math.log((x + 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (x * y)
	t_1 = -(x * y)
	tmp = 0
	if t_0 <= 0.01:
		tmp = t_1
	elif t_0 <= 5.0:
		tmp = math.log((x + 2.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(-Float64(x * y))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = t_1;
	elseif (t_0 <= 5.0)
		tmp = log(Float64(x + 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = log((1.0 + exp(x))) - (x * y);
	t_1 = -(x * y);
	tmp = 0.0;
	if (t_0 <= 0.01)
		tmp = t_1;
	elseif (t_0 <= 5.0)
		tmp = log((x + 2.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * y), $MachinePrecision])}, If[LessEqual[t$95$0, 0.01], t$95$1, If[LessEqual[t$95$0, 5.0], N[Log[N[(x + 2.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 0.0100000000000000002 or 5 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if 0.0100000000000000002 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\log \left(2 + x\right)} \]
   7. Step-by-step derivation

      1. lower-log.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 97.0

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

   8. Applied rewrites 97.0%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.01:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 5:\\ \;\;\;\;\log \left(x + 2\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(1 + e^{x}\right) - x \cdot y\\ t_1 := -x \cdot y\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (log (+ 1.0 (exp x))) (* x y))) (t_1 (- (* x y))))
   (if (<= t_0 0.01) t_1 (if (<= t_0 5.0) (log 2.0) t_1))))

C

double code(double x, double y) {
	double t_0 = log((1.0 + exp(x))) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = t_1;
	} else if (t_0 <= 5.0) {
		tmp = log(2.0);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = log((1.0d0 + exp(x))) - (x * y)
    t_1 = -(x * y)
    if (t_0 <= 0.01d0) then
        tmp = t_1
    else if (t_0 <= 5.0d0) then
        tmp = log(2.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.log((1.0 + Math.exp(x))) - (x * y);
	double t_1 = -(x * y);
	double tmp;
	if (t_0 <= 0.01) {
		tmp = t_1;
	} else if (t_0 <= 5.0) {
		tmp = Math.log(2.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((1.0 + math.exp(x))) - (x * y)
	t_1 = -(x * y)
	tmp = 0
	if t_0 <= 0.01:
		tmp = t_1
	elif t_0 <= 5.0:
		tmp = math.log(2.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(log(Float64(1.0 + exp(x))) - Float64(x * y))
	t_1 = Float64(-Float64(x * y))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = t_1;
	elseif (t_0 <= 5.0)
		tmp = log(2.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = log((1.0 + exp(x))) - (x * y);
	t_1 = -(x * y);
	tmp = 0.0;
	if (t_0 <= 0.01)
		tmp = t_1;
	elseif (t_0 <= 5.0)
		tmp = log(2.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Log[N[(1.0 + N[Exp[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * y), $MachinePrecision])}, If[LessEqual[t$95$0, 0.01], t$95$1, If[LessEqual[t$95$0, 5.0], N[Log[2.0], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 0.0100000000000000002 or 5 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if 0.0100000000000000002 < (-.f64 (log.f64 (+.f64 #s(literal 1 binary64) (exp.f64 x))) (*.f64 x y)) < 5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-log.f64 96.1

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

   5. Applied rewrites 96.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 0.01:\\ \;\;\;\;-x \cdot y\\ \mathbf{elif}\;\log \left(1 + e^{x}\right) - x \cdot y \leq 5:\\ \;\;\;\;\log 2\\ \mathbf{else}:\\ \;\;\;\;-x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), 0.5 - y\right), \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -48.0)
   (- (* x y))
   (fma
    x
    (fma x (fma -0.005208333333333333 (* x x) 0.125) (- 0.5 y))
    (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -48.0) {
		tmp = -(x * y);
	} else {
		tmp = fma(x, fma(x, fma(-0.005208333333333333, (x * x), 0.125), (0.5 - y)), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -48.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = fma(x, fma(x, fma(-0.005208333333333333, Float64(x * x), 0.125), Float64(0.5 - y)), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -48.0], (-N[(x * y), $MachinePrecision]), N[(x * N[(x * N[(-0.005208333333333333 * N[(x * x), $MachinePrecision] + 0.125), $MachinePrecision] + N[(0.5 - y), $MachinePrecision]), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -48

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -48 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. associate--l+ N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-log.f64 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(-0.005208333333333333, x \cdot x, 0.125\right), 0.5 - y\right), \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -53:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -53.0) (- (* x y)) (fma x (- (fma x 0.125 0.5) y) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -53.0) {
		tmp = -(x * y);
	} else {
		tmp = fma(x, (fma(x, 0.125, 0.5) - y), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -53.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = fma(x, Float64(fma(x, 0.125, 0.5) - y), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -53.0], (-N[(x * y), $MachinePrecision]), N[(x * N[(N[(x * 0.125 + 0.5), $MachinePrecision] - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -53 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-log.f64 99.3

         \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \color{blue}{\log 2}\right) \]

   5. Applied rewrites 99.3%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -53:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.125, 0.5\right) - y, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -48.0) (- (* x y)) (fma x (- 0.5 y) (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -48.0) {
		tmp = -(x * y);
	} else {
		tmp = fma(x, (0.5 - y), log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -48.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = fma(x, Float64(0.5 - y), log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -48.0], (-N[(x * y), $MachinePrecision]), N[(x * N[(0.5 - y), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -48

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -48 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-log.f64 99.0

         \[\leadsto \mathsf{fma}\left(x, 0.5 - y, \color{blue}{\log 2}\right) \]

   5. Applied rewrites 99.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -48:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.5 - y, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -53:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, \log 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -53.0) (- (* x y)) (fma (- y) x (log 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -53.0) {
		tmp = -(x * y);
	} else {
		tmp = fma(-y, x, log(2.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -53.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = fma(Float64(-y), x, log(2.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -53.0], (-N[(x * y), $MachinePrecision]), N[((-y) * x + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -53 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2} - x \cdot y \]
   4. Step-by-step derivation

      1. lower-log.f64 98.3

         \[\leadsto \color{blue}{\log 2} - x \cdot y \]

   5. Applied rewrites 98.3%

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

      1. lift-log.f64 N/A

         \[\leadsto \color{blue}{\log 2} - x \cdot y \]

      2. lift-*.f64 N/A

         \[\leadsto \log 2 - \color{blue}{x \cdot y} \]

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 98.3

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

   7. Applied rewrites 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -53:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, x, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 98.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -53:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -53.0) (- (* x y)) (- (log 2.0) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -53.0) {
		tmp = -(x * y);
	} else {
		tmp = log(2.0) - (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 <= (-53.0d0)) then
        tmp = -(x * y)
    else
        tmp = log(2.0d0) - (x * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -53.0) {
		tmp = -(x * y);
	} else {
		tmp = Math.log(2.0) - (x * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -53.0:
		tmp = -(x * y)
	else:
		tmp = math.log(2.0) - (x * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -53.0)
		tmp = Float64(-Float64(x * y));
	else
		tmp = Float64(log(2.0) - Float64(x * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -53.0)
		tmp = -(x * y);
	else
		tmp = log(2.0) - (x * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -53.0], (-N[(x * y), $MachinePrecision]), N[(N[Log[2.0], $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -53

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -53 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log 2} - x \cdot y \]
   4. Step-by-step derivation

      1. lower-log.f64 98.3

         \[\leadsto \color{blue}{\log 2} - x \cdot y \]

   5. Applied rewrites 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -53:\\ \;\;\;\;-x \cdot y\\ \mathbf{else}:\\ \;\;\;\;\log 2 - x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.7% accurate, 26.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around inf

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

   1. mul-1-neg N/A

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

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

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 49.5

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

5. Applied rewrites 49.5%

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

6. Final simplification 49.5%

   \[\leadsto -x \cdot y \]
7. Add Preprocessing

Alternative 12: 3.6% accurate, 35.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. associate--l+ N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-log.f64 75.8

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

5. Applied rewrites 75.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 12.3%

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

9. Taylor expanded in x around 0

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

    1. lower-*.f64 N/A

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

    2. lower--.f64 32.9

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

11. Applied rewrites 32.9%

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

12. Taylor expanded in y around 0

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

14. Applied rewrites 3.5%

    \[\leadsto x \cdot \color{blue}{0.5} \]
15. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× 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 2024214 
(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)))