symmetry log of sum of exp

Percentage Accurate: 54.3% → 98.8%

Time: 11.0s

Alternatives: 15

Speedup: 2.9×

• could not determine a ground truth

  1. a = 6.397621113039284e+291
  2. b = -5.276060817328637e-295

Specification

Math

\[\begin{array}{l} \\ \log \left(e^{a} + e^{b}\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return log((exp(a) + exp(b)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log((exp(a) + exp(b)))
end function

Java

public static double code(double a, double b) {
	return Math.log((Math.exp(a) + Math.exp(b)));
}

Python

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))

Julia

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = log((exp(a) + exp(b)));
end

Wolfram

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 54.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{a} + e^{b}\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))

C

double code(double a, double b) {
	return log((exp(a) + exp(b)));
}

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log((exp(a) + exp(b)))
end function

Java

public static double code(double a, double b) {
	return Math.log((Math.exp(a) + Math.exp(b)));
}

Python

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))

Julia

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = log((exp(a) + exp(b)));
end

Wolfram

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;0 - \log \left(\frac{1}{e^{a} + e^{b}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (- 0.0 (log (/ 1.0 (+ (exp a) (exp b)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = 0.0 - log((1.0 / (exp(a) + exp(b))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = 0.0d0 - log((1.0d0 / (exp(a) + exp(b))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = 0.0 - Math.log((1.0 / (Math.exp(a) + Math.exp(b))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = 0.0 - math.log((1.0 / (math.exp(a) + math.exp(b))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(0.0 - log(Float64(1.0 / Float64(exp(a) + exp(b)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = 0.0 - log((1.0 / (exp(a) + exp(b))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(0.0 - N[Log[N[(1.0 / N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \log \left(\frac{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}\right) \]

      2. clear-num N/A

         \[\leadsto \log \left(\frac{1}{\frac{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}}\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{neg}\left(\log \left(\frac{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\log \left(\frac{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}\right)\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{\frac{{\left(e^{a}\right)}^{3} + {\left(e^{b}\right)}^{3}}{e^{a} \cdot e^{a} + \left(e^{b} \cdot e^{b} - e^{a} \cdot e^{b}\right)}}\right)\right)\right) \]

      7. flip3-+ N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1}{e^{a} + e^{b}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(1, \left(e^{a} + e^{b}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(\left(e^{a}\right), \left(e^{b}\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 63.7%

      \[\leadsto \color{blue}{-\log \left(\frac{1}{e^{a} + e^{b}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;0 - \log \left(\frac{1}{e^{a} + e^{b}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + exp(b)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((exp(a) + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := e^{a} + 1\\ \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\log \left(t\_0 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (if (<= (exp a) 0.0)
     (/ b t_0)
     (log (+ t_0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / t_0;
	} else {
		tmp = log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(a) + 1.0d0
    if (exp(a) <= 0.0d0) then
        tmp = b / t_0
    else
        tmp = log((t_0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.exp(a) + 1.0;
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / t_0;
	} else {
		tmp = Math.log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.exp(a) + 1.0
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / t_0
	else:
		tmp = math.log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / t_0);
	else
		tmp = log(Float64(t_0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = exp(a) + 1.0;
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / t_0;
	else
		tmp = log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / t$95$0), $MachinePrecision], N[Log[N[(t$95$0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \mathsf{log.f64}\left(\left(\left(1 + e^{a}\right) + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(1 + e^{a}\right), \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \left(e^{a}\right)\right), \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(\frac{1}{6} \cdot b\right)\right)\right)\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

      10. *-lowering-*.f64 60.2%

          \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 60.2%

      \[\leadsto \log \color{blue}{\left(\left(1 + e^{a}\right) + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(e^{a} + 1\right) + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(1 + b \cdot \left(1 + b \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (log (+ (exp a) (+ 1.0 (* b (+ 1.0 (* b 0.5))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + (1.0 + (b * (1.0 + (b * 0.5))))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + (1.0d0 + (b * (1.0d0 + (b * 0.5d0))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + (1.0 + (b * (1.0 + (b * 0.5))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + (1.0 + (b * (1.0 + (b * 0.5))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + Float64(1.0 + Float64(b * Float64(1.0 + Float64(b * 0.5))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((exp(a) + (1.0 + (b * (1.0 + (b * 0.5))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(1.0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \left(b \cdot \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(1 + \frac{1}{2} \cdot b\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot b\right)\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 61.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \frac{1}{2}\right)\right)\right)\right)\right)\right) \]

   5. Simplified 61.1%

      \[\leadsto \log \left(e^{a} + \color{blue}{\left(1 + b \cdot \left(1 + b \cdot 0.5\right)\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(1 + b \cdot \left(1 + b \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (+ b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + (b + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + (b + 1.0d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + (b + 1.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + (b + 1.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + Float64(b + 1.0)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((exp(a) + (b + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \color{blue}{\left(1 + b\right)}\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 60.1%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{exp.f64}\left(a\right), \mathsf{+.f64}\left(1, b\right)\right)\right) \]

   5. Simplified 60.1%

      \[\leadsto \log \left(e^{a} + \color{blue}{\left(1 + b\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (+ (log1p (exp a)) (/ b (+ (exp a) 1.0))))

C

assert(a < b);
double code(double a, double b) {
	return log1p(exp(a)) + (b / (exp(a) + 1.0));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p(Math.exp(a)) + (b / (Math.exp(a) + 1.0));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p(math.exp(a)) + (b / (math.exp(a) + 1.0))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(log1p(exp(a)) + Float64(b / Float64(exp(a) + 1.0)))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. *-rgt-identity N/A

      \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

   2. associate-*r/ N/A

      \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

   4. log1p-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 70.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

5. Simplified 70.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

6. Final simplification 70.3%

   \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1} \]
7. Add Preprocessing

Alternative 7: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ (exp a) 1.0)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. log1p-define N/A

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

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

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

      3. exp-lowering-exp.f64 60.9%

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

   5. Simplified 60.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot \left(0.125 + -0.005208333333333333 \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (+
    (log 2.0)
    (* b (+ 0.5 (* b (+ 0.125 (* -0.005208333333333333 (* b b)))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * (0.5 + (b * (0.125 + (-0.005208333333333333 * (b * b))))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * (0.5d0 + (b * (0.125d0 + ((-0.005208333333333333d0) * (b * b))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * (0.5 + (b * (0.125 + (-0.005208333333333333 * (b * b))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * (0.5 + (b * (0.125 + (-0.005208333333333333 * (b * b))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * Float64(0.5 + Float64(b * Float64(0.125 + Float64(-0.005208333333333333 * Float64(b * b)))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * (0.5 + (b * (0.125 + (-0.005208333333333333 * (b * b))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * N[(0.5 + N[(b * N[(0.125 + N[(-0.005208333333333333 * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

      2. exp-lowering-exp.f64 60.9%

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

   5. Simplified 60.9%

      \[\leadsto \log \color{blue}{\left(1 + e^{b}\right)} \]

   6. Taylor expanded in b around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 59.7%

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

   8. Simplified 59.7%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot \left(0.125 + -0.005208333333333333 \cdot \left(b \cdot b\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* b (+ 0.5 (* b 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * (0.5d0 + (b * 0.125d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * (0.5 + (b * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * Float64(0.5 + Float64(b * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * (0.5 + (b * 0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

      2. exp-lowering-exp.f64 60.9%

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

   5. Simplified 60.9%

      \[\leadsto \log \color{blue}{\left(1 + e^{b}\right)} \]

   6. Taylor expanded in b around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 59.6%

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

   8. Simplified 59.6%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ (exp a) 1.0)) (+ (log 2.0) (* b 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * 0.5);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   if 0.0 < (exp.f64 a)

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 61.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 61.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. *-lowering-*.f64 59.9%

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

   8. Simplified 59.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -112:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -112.0) (/ b 2.0) (+ (log 2.0) (* b 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -112.0) {
		tmp = b / 2.0;
	} else {
		tmp = log(2.0) + (b * 0.5);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-112.0d0)) then
        tmp = b / 2.0d0
    else
        tmp = log(2.0d0) + (b * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -112.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -112.0:
		tmp = b / 2.0
	else:
		tmp = math.log(2.0) + (b * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -112.0)
		tmp = Float64(b / 2.0);
	else
		tmp = Float64(log(2.0) + Float64(b * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -112.0)
		tmp = b / 2.0;
	else
		tmp = log(2.0) + (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -112.0], N[(b / 2.0), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -112

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
   10. Step-by-step derivation

   11. Simplified 18.8%

       \[\leadsto \frac{b}{\color{blue}{2}} \]

   if -112 < a

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 61.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 61.2%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. *-lowering-*.f64 59.9%

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

   8. Simplified 59.9%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -112:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -146:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -146.0) (/ b 2.0) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -146.0) {
		tmp = b / 2.0;
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-146.0d0)) then
        tmp = b / 2.0d0
    else
        tmp = log((b + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -146.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -146.0:
		tmp = b / 2.0
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -146.0)
		tmp = Float64(b / 2.0);
	else
		tmp = log(Float64(b + 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -146.0)
		tmp = b / 2.0;
	else
		tmp = log((b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -146.0], N[(b / 2.0), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -146

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
   10. Step-by-step derivation

   11. Simplified 18.8%

       \[\leadsto \frac{b}{\color{blue}{2}} \]

   if -146 < a

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(1 + e^{b}\right)}\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(1, \left(e^{b}\right)\right)\right) \]

      2. exp-lowering-exp.f64 60.9%

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

   5. Simplified 60.9%

      \[\leadsto \log \color{blue}{\left(1 + e^{b}\right)} \]

   6. Taylor expanded in b around 0

      \[\leadsto \mathsf{log.f64}\left(\color{blue}{\left(2 + b\right)}\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 58.8%

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

   8. Simplified 58.8%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -146:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -1.0) (/ b 2.0) (log (+ a 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b / 2.0;
	} else {
		tmp = log((a + 2.0));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.0d0)) then
        tmp = b / 2.0d0
    else
        tmp = log((a + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log((a + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.0:
		tmp = b / 2.0
	else:
		tmp = math.log((a + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.0)
		tmp = Float64(b / 2.0);
	else
		tmp = log(Float64(a + 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.0)
		tmp = b / 2.0;
	else
		tmp = log((a + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.0], N[(b / 2.0), $MachinePrecision], N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
   10. Step-by-step derivation

   11. Simplified 18.8%

       \[\leadsto \frac{b}{\color{blue}{2}} \]

   if -1 < a

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(1 + a\right)}, \mathsf{exp.f64}\left(b\right)\right)\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 62.6%

         \[\leadsto \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(1, a\right), \mathsf{exp.f64}\left(b\right)\right)\right) \]

   5. Simplified 62.6%

      \[\leadsto \log \left(\color{blue}{\left(1 + a\right)} + e^{b}\right) \]

   6. Taylor expanded in b around 0

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

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

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

      2. +-lowering-+.f64 59.9%

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

   8. Simplified 59.9%

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

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -105:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -105.0) (/ b 2.0) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -105.0) {
		tmp = b / 2.0;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -105.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -105.0:
		tmp = b / 2.0
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -105.0)
		tmp = Float64(b / 2.0);
	else
		tmp = log1p(1.0);
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -105.0], N[(b / 2.0), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -105

   1. Initial program 5.6%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

         \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

      4. log1p-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

      8. *-rgt-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

      11. exp-lowering-exp.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

   6. Taylor expanded in b around inf

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

      3. exp-lowering-exp.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

   9. Taylor expanded in a around 0

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
   10. Step-by-step derivation

   11. Simplified 18.8%

       \[\leadsto \frac{b}{\color{blue}{2}} \]

   if -105 < a

   1. Initial program 63.7%

      \[\log \left(e^{a} + e^{b}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

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

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

      3. exp-lowering-exp.f64 60.6%

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

   5. Simplified 60.6%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 59.2%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 11.9% accurate, 101.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \frac{b}{2} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (/ b 2.0))

C

assert(a < b);
double code(double a, double b) {
	return b / 2.0;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b / 2.0d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b / 2.0;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b / 2.0

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b / 2.0)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b / 2.0;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b / 2.0), $MachinePrecision]

Derivation

1. Initial program 50.1%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Add Preprocessing

3. Taylor expanded in b around 0

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

   1. *-rgt-identity N/A

      \[\leadsto \log \left(1 + e^{a}\right) + \frac{b \cdot 1}{\color{blue}{1} + e^{a}} \]

   2. associate-*r/ N/A

      \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \color{blue}{\frac{1}{1 + e^{a}}} \]

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

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

   4. log1p-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{log1p}\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \frac{1}{1 + e^{a}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \frac{1}{1 + e^{a}}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b \cdot 1}{\color{blue}{1 + e^{a}}}\right)\right) \]

   8. *-rgt-identity N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(\frac{b}{\color{blue}{1} + e^{a}}\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right)\right) \]

   11. exp-lowering-exp.f64 70.3%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right)\right) \]

5. Simplified 70.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}}} \]

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
7. Step-by-step derivation

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

      \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{\left(1 + e^{a}\right)}\right) \]

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

      \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \color{blue}{\left(e^{a}\right)}\right)\right) \]

   3. exp-lowering-exp.f64 26.0%

      \[\leadsto \mathsf{/.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{exp.f64}\left(a\right)\right)\right) \]

8. Simplified 26.0%

   \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{/.f64}\left(b, \color{blue}{2}\right) \]
10. Step-by-step derivation

11. Simplified 7.0%

    \[\leadsto \frac{b}{\color{blue}{2}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))