symmetry log of sum of exp

Percentage Accurate: 54.3% → 98.7%

Time: 10.8s

Alternatives: 14

Speedup: 2.8×

• could not determine a ground truth

  1. a = -2.4576640232955314e-123
  2. b = 1.1795379718390992e+134

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 1.0× speedup

Math

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

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 (+ 1.0 (exp a))) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -36.0) {
		tmp = b / (1.0 + exp(a));
	} 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 / (1.0d0 + exp(a))
    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 / (1.0 + Math.exp(a));
	} 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 / (1.0 + math.exp(a))
	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(1.0 + exp(a)));
	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 / (1.0 + exp(a));
	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[(1.0 + N[Exp[a], $MachinePrecision]), $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 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.8%

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

   if -36 < a

   1. Initial program 66.3%

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

Alternative 2: 57.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 0.035:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (log (+ (exp a) (exp b))) 0.035) (* 0.5 b) (fma 0.5 b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (log((exp(a) + exp(b))) <= 0.035) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(0.5, b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (log(Float64(exp(a) + exp(b))) <= 0.035)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(0.5, b, log(2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.035], N[(0.5 * b), $MachinePrecision], N[(0.5 * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (exp.f64 a) (exp.f64 b))) < 0.035000000000000003

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 54.8

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

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.7%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 54.7%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 11.2%

       \[\leadsto 0.5 \cdot b \]

   if 0.035000000000000003 < (log.f64 (+.f64 (exp.f64 a) (exp.f64 b)))

   1. Initial program 96.6%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 90.9%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log \left(e^{a} + e^{b}\right) \leq 0.035:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-exp.f64 72.9

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

5. Applied rewrites 72.9%

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

Alternative 4: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -370:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -370.0) (/ b (+ 1.0 (exp a))) (+ (* 0.5 b) (log1p (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -370.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = (0.5 * b) + log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -370.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = (0.5 * b) + Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -370.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = (0.5 * b) + math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -370.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = Float64(Float64(0.5 * b) + 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, -370.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * b), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -370

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.8%

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

   if -370 < a

   1. Initial program 66.3%

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

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 61.9

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

   5. Applied rewrites 61.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto \frac{1}{2} \cdot b + \mathsf{log1p}\left(\color{blue}{e^{a}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 61.9%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -370:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -36.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(a) + (1.0 + 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 / (1.0d0 + exp(a))
    else
        tmp = log((exp(a) + (1.0d0 + 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 / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log((Math.exp(a) + (1.0 + b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -36.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log((math.exp(a) + (1.0 + b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -36.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(a) + Float64(1.0 + 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 / (1.0 + exp(a));
	else
		tmp = log((exp(a) + (1.0 + 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[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -36

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.8%

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

   if -36 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 60.9

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

   5. Applied rewrites 60.9%

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

Alternative 6: 97.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -29.5:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -29.5) (/ b (+ 1.0 (exp a))) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -29.5) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -29.5) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -29.5:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -29.5)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log1p(exp(b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -29.5], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -29.5

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.8%

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

   if -29.5 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in a around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 63.1

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

   5. Applied rewrites 63.1%

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

Alternative 7: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -370:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -370.0) (/ b (+ 1.0 (exp a))) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -370.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -370.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -370.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -370.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	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, -370.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -370

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.8%

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

   if -370 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 61.7

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

   5. Applied rewrites 61.7%

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

Alternative 8: 97.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -2.6)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = fma(fma(fma(Float64(a * a), -0.005208333333333333, 0.125), a, 0.5), a, log(2.0));
	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, -2.6], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * -0.005208333333333333 + 0.125), $MachinePrecision] * a + 0.5), $MachinePrecision] * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.60000000000000009

   1. Initial program 9.9%

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

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 97.6

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

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 97.6%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 97.6%

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

   if -2.60000000000000009 < a

   1. Initial program 66.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.8%

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

Alternative 9: 97.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -29.5:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -29.5) (/ b (+ 1.0 (exp a))) (fma (fma 0.125 a 0.5) a (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -29.5) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = fma(fma(0.125, a, 0.5), a, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -29.5)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = fma(fma(0.125, a, 0.5), a, log(2.0));
	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, -29.5], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * a + 0.5), $MachinePrecision] * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -29.5

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]
   11. Step-by-step derivation

   12. Applied rewrites 98.8%

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

   if -29.5 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), \color{blue}{a}, \log 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -110:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -110.0) (* 0.5 b) (fma (fma 0.125 a 0.5) a (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -110.0) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(fma(0.125, a, 0.5), a, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -110.0)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(fma(0.125, a, 0.5), a, log(2.0));
	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, -110.0], N[(0.5 * b), $MachinePrecision], N[(N[(0.125 * a + 0.5), $MachinePrecision] * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -110

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 18.6%

       \[\leadsto 0.5 \cdot b \]

   if -110 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), \color{blue}{a}, \log 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 57.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.4) (* 0.5 b) (fma 0.5 a (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.4) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(0.5, a, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.4)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(0.5, a, log(2.0));
	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, -1.4], N[(0.5 * b), $MachinePrecision], N[(0.5 * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3999999999999999

   1. Initial program 9.9%

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

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 97.6

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

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 97.6%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 18.4%

       \[\leadsto 0.5 \cdot b \]

   if -1.3999999999999999 < a

   1. Initial program 66.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.9%

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

Alternative 12: 57.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 9.9%

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

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 97.6

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

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 97.6%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 18.4%

       \[\leadsto 0.5 \cdot b \]

   if -1 < a

   1. Initial program 66.1%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 62.0

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.8%

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

Alternative 13: 56.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -110:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -110.0) (* 0.5 b) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -110.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -110.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -110.0:
		tmp = 0.5 * b
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -110.0)
		tmp = Float64(0.5 * b);
	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, -110.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -110

   1. Initial program 8.7%

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

   3. Taylor expanded in b around 0

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

      1. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 98.8

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

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

      \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 98.8%

       \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]

   11. Taylor expanded in a around 0

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

   13. Applied rewrites 18.6%

       \[\leadsto 0.5 \cdot b \]

   if -110 < a

   1. Initial program 66.3%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 60.1%

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

Alternative 14: 11.9% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

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 = 0.5d0 * b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

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

   2. associate-*r/ N/A

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

   3. +-commutative N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-exp.f64 72.9

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

5. Applied rewrites 72.9%

   \[\leadsto \color{blue}{\frac{b}{e^{a} + 1} + \mathsf{log1p}\left(e^{a}\right)} \]
6. Step-by-step derivation

7. Applied rewrites 72.8%

   \[\leadsto \mathsf{fma}\left(\frac{b}{\mathsf{expm1}\left(a + a\right)}, \color{blue}{\mathsf{expm1}\left(a\right)}, \mathsf{log1p}\left(e^{a}\right)\right) \]

8. Taylor expanded in b around inf

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

10. Applied rewrites 31.6%

    \[\leadsto \mathsf{expm1}\left(a\right) \cdot \color{blue}{\frac{b}{\mathsf{expm1}\left(2 \cdot a\right)}} \]

11. Taylor expanded in a around 0

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

13. Applied rewrites 7.9%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))