symmetry log of sum of exp

Percentage Accurate: 54.2% → 98.9%

Time: 15.1s

Alternatives: 15

Speedup: 2.9×

• could not determine a ground truth

  1. a = 1.9183161336615734e+218
  2. b = 1.007723669148924e-309

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: 54.2% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -36:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -36.0) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -36.0) {
		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 (a <= (-36.0d0)) 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 (a <= -36.0) {
		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 a <= -36.0:
		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 (a <= -36.0)
		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 (a <= -36.0)
		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[a, -36.0], 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 a < -36

   1. Initial program 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. exp-lowering-exp.f64 98.4%

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

   5. Simplified 98.4%

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

   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 98.4%

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

   8. Simplified 98.4%

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

   if -36 < a

   1. Initial program 72.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -36:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := e^{a} + 1\\ \mathsf{log1p}\left(e^{a}\right) + b \cdot \left(\left(1 + b \cdot 0.5\right) \cdot \frac{1}{t\_0} + \frac{b \cdot -0.5}{{t\_0}^{2}}\right) \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (+
    (log1p (exp a))
    (* b (+ (* (+ 1.0 (* b 0.5)) (/ 1.0 t_0)) (/ (* b -0.5) (pow t_0 2.0)))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	return log1p(exp(a)) + (b * (((1.0 + (b * 0.5)) * (1.0 / t_0)) + ((b * -0.5) / pow(t_0, 2.0))));
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.exp(a) + 1.0;
	return Math.log1p(Math.exp(a)) + (b * (((1.0 + (b * 0.5)) * (1.0 / t_0)) + ((b * -0.5) / Math.pow(t_0, 2.0))));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.exp(a) + 1.0
	return math.log1p(math.exp(a)) + (b * (((1.0 + (b * 0.5)) * (1.0 / t_0)) + ((b * -0.5) / math.pow(t_0, 2.0))))

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	return Float64(log1p(exp(a)) + Float64(b * Float64(Float64(Float64(1.0 + Float64(b * 0.5)) * Float64(1.0 / t_0)) + Float64(Float64(b * -0.5) / (t_0 ^ 2.0)))))
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]}, N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(b * N[(N[(N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(b * -0.5), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 58.4%

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

3. Taylor expanded in b around 0

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

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

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

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\left(e^{a}\right)\right), \left(\color{blue}{b} \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \left(\frac{1}{2} \cdot \left(b \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \left(\left(\frac{1}{2} \cdot b\right) \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) + \frac{\color{blue}{1}}{1 + e^{a}}\right)\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \left(\left(b \cdot \frac{1}{2}\right) \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right) + \frac{1}{1 + e^{a}}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \left(b \cdot \left(b \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{\color{blue}{1}}{1 + e^{a}}\right)\right)\right) \]

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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \color{blue}{\left(b \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{log1p.f64}\left(\mathsf{exp.f64}\left(a\right)\right), \mathsf{*.f64}\left(b, \left(\frac{1}{1 + e^{a}} + \color{blue}{b \cdot \left(\frac{1}{2} \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right)}\right)\right)\right) \]

5. Simplified 75.9%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(e^{a}\right) + b \cdot \left(\left(1 + b \cdot 0.5\right) \cdot \frac{1}{1 + e^{a}} + \frac{b \cdot -0.5}{{\left(1 + e^{a}\right)}^{2}}\right)} \]

6. Final simplification 75.9%

   \[\leadsto \mathsf{log1p}\left(e^{a}\right) + b \cdot \left(\left(1 + b \cdot 0.5\right) \cdot \frac{1}{e^{a} + 1} + \frac{b \cdot -0.5}{{\left(e^{a} + 1\right)}^{2}}\right) \]
7. Add Preprocessing

Alternative 3: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (log1p (+ (exp a) (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666)))))))))

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(a) + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	}
	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(a) + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	}
	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(a) + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))))
	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(Float64(exp(a) + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	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[(N[Exp[a], $MachinePrecision] + 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 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. exp-lowering-exp.f64 98.4%

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

   5. Simplified 98.4%

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

   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 98.4%

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

   8. Simplified 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. *-commutative N/A

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

      12. *-lowering-*.f64 68.4%

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

   5. Simplified 68.4%

      \[\leadsto \log \color{blue}{\left(e^{a} + \left(1 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

         \[\leadsto \mathsf{log1p.f64}\left(\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right) + e^{a}\right)\right) \]

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

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right), \left(e^{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \left(1 + b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right), \left(e^{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \left(b \cdot \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right)\right), \left(e^{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \left(\frac{1}{2} + b \cdot \frac{1}{6}\right)\right)\right)\right), \left(e^{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \left(b \cdot \frac{1}{6}\right)\right)\right)\right)\right), \left(e^{a}\right)\right)\right) \]

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

         \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \frac{1}{6}\right)\right)\right)\right)\right), \left(e^{a}\right)\right)\right) \]

      10. exp-lowering-exp.f64 68.5%

          \[\leadsto \mathsf{log1p.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(b, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(b, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(b, \frac{1}{6}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(a\right)\right)\right) \]

   7. Applied egg-rr 68.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right) + e^{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% 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 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. exp-lowering-exp.f64 98.4%

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

   5. Simplified 98.4%

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

   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 98.4%

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

   8. Simplified 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 69.3%

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

   5. Simplified 69.3%

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

4. Final simplification 76.0%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

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 a) 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(a) + 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(a) + 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(a) + 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(Float64(exp(a) + 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[(N[Exp[a], $MachinePrecision] + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. exp-lowering-exp.f64 98.4%

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

   5. Simplified 98.4%

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

   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 98.4%

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

   8. Simplified 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 68.1%

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

   5. Simplified 68.1%

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

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

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

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

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

      6. exp-lowering-exp.f64 68.2%

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

   7. Applied egg-rr 68.2%

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

4. Final simplification 75.1%

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

Alternative 6: 98.5% 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 58.4%

   \[\log \left(e^{a} + e^{b}\right) \]
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. accelerator-lowering-log1p.f64 N/A

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

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

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

   6. associate-*r/ N/A

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

   7. *-rgt-identity N/A

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

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

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

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

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

   10. exp-lowering-exp.f64 75.7%

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

5. Simplified 75.7%

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

6. Final simplification 75.7%

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

Alternative 7: 96.9% accurate, 1.4× speedup

Math

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

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) (* a 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) + (a * 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) + (a * 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) + (a * 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) + (a * 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(a * 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) + (a * 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[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. exp-lowering-exp.f64 98.4%

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

   5. Simplified 98.4%

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

   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 98.4%

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

   8. Simplified 98.4%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

      2. exp-lowering-exp.f64 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 67.9%

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

   8. Simplified 67.9%

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

4. Final simplification 75.0%

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

Alternative 8: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -380:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -380.0) (/ b (+ (exp a) 1.0)) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -380.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -380.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -380.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -380.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(a));
	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, -380.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -380

   1. Initial program 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. exp-lowering-exp.f64 98.4%

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

   5. Simplified 98.4%

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

   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 98.4%

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

   8. Simplified 98.4%

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

   if -380 < a

   1. Initial program 72.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

      2. exp-lowering-exp.f64 68.1%

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

   5. Simplified 68.1%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -380:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.5:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.5)
   (/ b (+ (exp a) 1.0))
   (+
    (log 2.0)
    (* a (+ 0.5 (* a (+ 0.125 (* -0.005208333333333333 (* a a)))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.5) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * (0.5 + (a * (0.125 + (-0.005208333333333333 * (a * a))))));
	}
	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 <= (-2.5d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * (0.5d0 + (a * (0.125d0 + ((-0.005208333333333333d0) * (a * a))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.5) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * (0.5 + (a * (0.125 + (-0.005208333333333333 * (a * a))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.5:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * (0.5 + (a * (0.125 + (-0.005208333333333333 * (a * a))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.5)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(a * Float64(0.5 + Float64(a * Float64(0.125 + Float64(-0.005208333333333333 * Float64(a * a)))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.5)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * (0.5 + (a * (0.125 + (-0.005208333333333333 * (a * a))))));
	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, -2.5], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * N[(0.5 + N[(a * N[(0.125 + N[(-0.005208333333333333 * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.5

   1. Initial program 14.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. exp-lowering-exp.f64 96.8%

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

   5. Simplified 96.8%

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

   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 96.8%

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

   8. Simplified 96.8%

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

   if -2.5 < a

   1. Initial program 71.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

      2. exp-lowering-exp.f64 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 68.3%

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

   8. Simplified 68.3%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.5:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 + -0.005208333333333333 \cdot \left(a \cdot a\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -80:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -80.0)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* a (+ 0.5 (* a 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -80.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * (0.5 + (a * 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 (a <= (-80.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * (0.5d0 + (a * 0.125d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -80.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -80.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * (0.5 + (a * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -80.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(a * Float64(0.5 + Float64(a * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -80.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * (0.5 + (a * 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[a, -80.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * N[(0.5 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -80

   1. Initial program 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

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

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

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

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

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

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

      10. exp-lowering-exp.f64 98.4%

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

   5. Simplified 98.4%

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

   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 98.4%

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

   8. Simplified 98.4%

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

   if -80 < a

   1. Initial program 72.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

      2. exp-lowering-exp.f64 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 68.0%

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

   8. Simplified 68.0%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -80:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.35) (* b 0.5) (+ (log 2.0) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.35) {
		tmp = b * 0.5;
	} else {
		tmp = log(2.0) + (a * 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 <= (-1.35d0)) then
        tmp = b * 0.5d0
    else
        tmp = log(2.0d0) + (a * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.35) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log(2.0) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.35:
		tmp = b * 0.5
	else:
		tmp = math.log(2.0) + (a * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.35)
		tmp = Float64(b * 0.5);
	else
		tmp = Float64(log(2.0) + Float64(a * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.35)
		tmp = b * 0.5;
	else
		tmp = log(2.0) + (a * 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, -1.35], N[(b * 0.5), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3500000000000001

   1. Initial program 14.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. log-lowering-log.f64 3.9%

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

   8. Simplified 3.9%

      \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]

   9. Taylor expanded in b around inf

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

       1. *-lowering-*.f64 18.3%

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

   11. Simplified 18.3%

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

   if -1.3500000000000001 < a

   1. Initial program 71.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

      2. exp-lowering-exp.f64 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 68.2%

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

   8. Simplified 68.2%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -1.0) (* b 0.5) (log1p (+ a 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b * 0.5;
	} else {
		tmp = log1p((a + 1.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log1p((a + 1.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.0:
		tmp = b * 0.5
	else:
		tmp = math.log1p((a + 1.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log1p(Float64(a + 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, -1.0], N[(b * 0.5), $MachinePrecision], N[Log[1 + N[(a + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 14.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. log-lowering-log.f64 3.9%

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

   8. Simplified 3.9%

      \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]

   9. Taylor expanded in b around inf

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

       1. *-lowering-*.f64 18.3%

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

   11. Simplified 18.3%

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

   if -1 < a

   1. Initial program 71.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

      2. exp-lowering-exp.f64 68.4%

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

   5. Simplified 68.4%

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

   6. Taylor expanded in a around 0

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

      1. +-lowering-+.f64 68.1%

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

   8. Simplified 68.1%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -160:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -160.0) (* b 0.5) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -160.0) {
		tmp = b * 0.5;
	} 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 <= (-160.0d0)) then
        tmp = b * 0.5d0
    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 <= -160.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -160.0:
		tmp = b * 0.5
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -160.0)
		tmp = Float64(b * 0.5);
	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 <= -160.0)
		tmp = b * 0.5;
	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, -160.0], N[(b * 0.5), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -160

   1. Initial program 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. log-lowering-log.f64 3.9%

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

   8. Simplified 3.9%

      \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]

   9. Taylor expanded in b around inf

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

       1. *-lowering-*.f64 18.5%

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

   11. Simplified 18.5%

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

   if -160 < a

   1. Initial program 72.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 69.9%

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

   5. Simplified 69.9%

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

   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. +-commutative N/A

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

      2. +-lowering-+.f64 67.5%

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

   8. Simplified 67.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -160:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -145:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -145.0) (* b 0.5) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -145.0) {
		tmp = b * 0.5;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -145.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -145.0:
		tmp = b * 0.5
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -145.0)
		tmp = Float64(b * 0.5);
	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, -145.0], N[(b * 0.5), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -145

   1. Initial program 13.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. log-lowering-log.f64 3.9%

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

   8. Simplified 3.9%

      \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]

   9. Taylor expanded in b around inf

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

       1. *-lowering-*.f64 18.5%

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

   11. Simplified 18.5%

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

   if -145 < a

   1. Initial program 72.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. accelerator-lowering-log1p.f64 N/A

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

      2. exp-lowering-exp.f64 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 67.5%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -145:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 11.9% accurate, 101.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot 0.5 \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* b 0.5))

C

assert(a < b);
double code(double a, double b) {
	return b * 0.5;
}

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 * 0.5d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b * 0.5;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b * 0.5

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(b * 0.5)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = b * 0.5;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(b * 0.5), $MachinePrecision]

Derivation

1. Initial program 58.4%

   \[\log \left(e^{a} + e^{b}\right) \]
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 54.6%

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

5. Simplified 54.6%

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

6. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

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

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

   3. *-commutative N/A

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

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

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

   5. log-lowering-log.f64 53.4%

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

8. Simplified 53.4%

   \[\leadsto \color{blue}{b \cdot 0.5 + \log 2} \]

9. Taylor expanded in b around inf

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

    1. *-lowering-*.f64 6.8%

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

11. Simplified 6.8%

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

12. Final simplification 6.8%

    \[\leadsto b \cdot 0.5 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))