symmetry log of sum of exp

Percentage Accurate: 52.6% → 98.7%

Time: 10.2s

Alternatives: 11

Speedup: 1.5×

• could not determine a ground truth

  1. a = -1.0697485635018318e-233
  2. b = 5.583307549055951e+86

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: 52.6% 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, 0.7× speedup

Math

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

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log((exp(b) + exp(a)))
    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 / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log((Math.exp(b) + 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 / (1.0 + math.exp(a))
	else:
		tmp = math.log((math.exp(b) + 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(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(b) + exp(a)));
	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 / (1.0 + exp(a));
	else
		tmp = log((exp(b) + exp(a)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 15.2%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 73.9%

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

4. Final simplification 80.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(log1p(exp(a)) + Float64(b / Float64(1.0 + exp(a))))
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[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

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

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ 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 79.0

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

5. Applied rewrites 79.0%

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

6. Final simplification 79.0%

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

Alternative 3: 55.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{b} + e^{a} \leq 1.5:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp b) (exp a)) 1.5) (* 0.5 b) (fma 0.5 b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((exp(b) + exp(a)) <= 1.5) {
		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 (Float64(exp(b) + exp(a)) <= 1.5)
		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[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision], 1.5], N[(0.5 * b), $MachinePrecision], N[(0.5 * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.5

   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. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 56.5

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

   5. Applied rewrites 56.5%

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

   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 3.7%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 11.5%

       \[\leadsto 0.5 \cdot b \]

   if 1.5 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 99.3%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 98.5

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

   5. Applied rewrites 98.5%

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

   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 96.0%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} + e^{a} \leq 1.5:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		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 (Math.exp(a) <= 0.0) {
		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 math.exp(a) <= 0.0:
		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 (exp(a) <= 0.0)
		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[N[Exp[a], $MachinePrecision], 0.0], 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 (exp.f64 a) < 0.0

   1. Initial program 15.2%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 73.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

4. Final simplification 79.4%

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

Alternative 5: 97.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 15.2%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 73.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 70.8%

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

4. Final simplification 78.6%

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

Alternative 6: 97.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 15.2%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 73.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 70.6%

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

4. Final simplification 78.4%

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

Alternative 7: 56.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 15.2%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 5.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if 0.0 < (exp.f64 a)

   1. Initial program 73.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 70.6%

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

Alternative 8: 55.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log1p((1.0 + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p((1.0 + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = 0.5 * b
	else:
		tmp = math.log1p((1.0 + b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(0.5 * b);
	else
		tmp = log1p(Float64(1.0 + b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + N[(1.0 + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 15.2%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 5.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if 0.0 < (exp.f64 a)

   1. Initial program 73.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 69.8%

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

4. Final simplification 56.2%

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

Alternative 9: 55.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		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 (Math.exp(a) <= 0.0) {
		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 math.exp(a) <= 0.0:
		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 (exp(a) <= 0.0)
		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[N[Exp[a], $MachinePrecision], 0.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + N[(1.0 + a), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 15.2%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 5.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if 0.0 < (exp.f64 a)

   1. Initial program 73.9%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 71.5

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.1%

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

4. Final simplification 57.2%

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

Alternative 10: 55.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		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 (Math.exp(a) <= 0.0) {
		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 math.exp(a) <= 0.0:
		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 (exp(a) <= 0.0)
		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[N[Exp[a], $MachinePrecision], 0.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 15.2%

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

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ 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 100.0

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

   5. Applied rewrites 100.0%

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

   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 5.1%

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

   9. Taylor expanded in b around inf

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

   11. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if 0.0 < (exp.f64 a)

   1. Initial program 73.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 72.0

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

   5. Applied rewrites 72.0%

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

   6. Taylor expanded in b around 0

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

   8. Applied rewrites 70.7%

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

Alternative 11: 12.1% 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 58.3%

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

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ 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 79.0

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

5. Applied rewrites 79.0%

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

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 53.1%

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

9. Taylor expanded in b around inf

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

11. Applied rewrites 7.3%

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

Reproduce

herbie shell --seed 2024278 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))