symmetry log of sum of exp

Percentage Accurate: 54.1% → 98.2%

Time: 11.8s

Alternatives: 12

Speedup: 2.9×

• could not determine a ground truth

  1. a = 7.938379745900461e+108
  2. b = -4.29491105546138e-99

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: 54.1% 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: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.5%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 69.4%

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

   5. Simplified 69.4%

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

4. Final simplification 76.2%

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

Alternative 2: 98.1% accurate, 1.0× speedup

Math

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

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 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 70.0%

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

   5. Simplified 70.0%

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

4. Final simplification 76.7%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 69.1%

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

   5. Simplified 69.1%

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

4. Final simplification 76.0%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

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 58.8%

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

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

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

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

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

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

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

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

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

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

   11. exp-lowering-exp.f64 76.6%

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

5. Simplified 76.6%

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

6. Final simplification 76.6%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

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));
	}
	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));
	}
	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))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(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[N[Exp[a], $MachinePrecision], 0.0], 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 (exp.f64 a) < 0.0

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.5%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

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

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

      3. exp-lowering-exp.f64 69.6%

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

   5. Simplified 69.6%

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

4. Final simplification 76.3%

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

Alternative 6: 97.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 + a \cdot \left(a \cdot -0.005208333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.6)
   (/ b (+ (exp a) 1.0))
   (+
    (log 2.0)
    (* a (+ 0.5 (* a (+ 0.125 (* a (* a -0.005208333333333333)))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.6) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * (0.5 + (a * (0.125 + (a * (a * -0.005208333333333333))))));
	}
	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 <= (-2.6d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * (0.5d0 + (a * (0.125d0 + (a * (a * (-0.005208333333333333d0)))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -2.6) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * (0.5 + (a * (0.125 + (a * (a * -0.005208333333333333))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -2.6:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * (0.5 + (a * (0.125 + (a * (a * -0.005208333333333333))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.6)
		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(a * Float64(a * -0.005208333333333333)))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -2.6)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * (0.5 + (a * (0.125 + (a * (a * -0.005208333333333333))))));
	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, -2.6], 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[(a * N[(a * -0.005208333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.60000000000000009

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   if -2.60000000000000009 < a

   1. Initial program 72.5%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

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

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

      3. exp-lowering-exp.f64 69.6%

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

   5. Simplified 69.6%

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

   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. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 69.1%

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

   8. Simplified 69.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 + a \cdot \left(a \cdot -0.005208333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -45:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -45.0)
   (/ 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 <= -45.0) {
		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 <= (-45.0d0)) 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 <= -45.0) {
		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 <= -45.0:
		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 <= -45.0)
		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 <= -45.0)
		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, -45.0], 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 < -45

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   if -45 < a

   1. Initial program 72.5%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

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

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

      3. exp-lowering-exp.f64 69.6%

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

   5. Simplified 69.6%

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

   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. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - 0\right) \cdot a\right)\right)\right) \]

      4. mul0-rgt N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \left(a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot 0\right) \cdot a\right)\right)\right) \]

      5. metadata-eval N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. distribute-rgt-out N/A

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

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot 0\right) \cdot a\right)\right)\right) \]

      12. mul0-rgt N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{log.f64}\left(2\right), \mathsf{*.f64}\left(a, \left(\frac{1}{2} + \left(\frac{1}{8} - 0\right) \cdot a\right)\right)\right) \]

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

      16. *-lowering-*.f64 68.9%

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

   8. Simplified 68.9%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -45:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -75:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -75.0) (/ b (+ (exp a) 1.0)) (+ (log 2.0) (* b 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -75.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * 0.5);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-75.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -75.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -75.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -75.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -75.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -75.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -75

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   if -75 < a

   1. Initial program 72.5%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. *-lowering-*.f64 67.8%

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

   8. Simplified 67.8%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -75:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -150:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -150.0) (/ b 2.0) (+ (log 2.0) (* b 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -150.0) {
		tmp = b / 2.0;
	} else {
		tmp = log(2.0) + (b * 0.5);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-150.0d0)) then
        tmp = b / 2.0d0
    else
        tmp = log(2.0d0) + (b * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -150.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -150.0:
		tmp = b / 2.0
	else:
		tmp = math.log(2.0) + (b * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -150.0)
		tmp = Float64(b / 2.0);
	else
		tmp = Float64(log(2.0) + Float64(b * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -150.0)
		tmp = b / 2.0;
	else
		tmp = log(2.0) + (b * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -150.0], N[(b / 2.0), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -150

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Simplified 18.8%

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

   if -150 < a

   1. Initial program 72.5%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 69.9%

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

   5. Simplified 69.9%

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

   6. Taylor expanded in a around 0

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

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

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

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

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

      3. *-lowering-*.f64 67.8%

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

   8. Simplified 67.8%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -150:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -150:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -150.0) (/ b 2.0) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -150.0) {
		tmp = b / 2.0;
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-150.0d0)) then
        tmp = b / 2.0d0
    else
        tmp = log((b + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -150.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -150.0:
		tmp = b / 2.0
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -150.0)
		tmp = Float64(b / 2.0);
	else
		tmp = log(Float64(b + 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -150.0)
		tmp = b / 2.0;
	else
		tmp = log((b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -150.0], N[(b / 2.0), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -150

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Simplified 18.8%

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

   if -150 < a

   1. Initial program 72.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 68.7%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in b around 0

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

      1. +-lowering-+.f64 67.0%

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

   8. Simplified 67.0%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -150:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -145:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -145.0) (/ b 2.0) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -145.0) {
		tmp = b / 2.0;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -145.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -145.0:
		tmp = b / 2.0
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -145.0)
		tmp = Float64(b / 2.0);
	else
		tmp = log1p(1.0);
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -145.0], N[(b / 2.0), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -145

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

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

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

      4. log1p-define N/A

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

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

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

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

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

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

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

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

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

      11. exp-lowering-exp.f64 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf

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

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

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

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

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

      3. exp-lowering-exp.f64 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in a around 0

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

   11. Simplified 18.8%

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

   if -145 < a

   1. Initial program 72.5%

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

   3. Taylor expanded in b around 0

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

      1. log1p-define N/A

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

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

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

      3. exp-lowering-exp.f64 69.6%

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

   5. Simplified 69.6%

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

   6. Taylor expanded in a around 0

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

   8. Simplified 67.4%

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

Alternative 12: 12.0% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.8%

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

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

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

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

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

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

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

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

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

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

   11. exp-lowering-exp.f64 76.6%

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

5. Simplified 76.6%

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

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 24.9%

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

8. Simplified 24.9%

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

9. Taylor expanded in a around 0

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

11. Simplified 6.8%

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

Reproduce

herbie shell --seed 2024138 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))