symmetry log of sum of exp

Percentage Accurate: 54.6% → 98.3%

Time: 19.0s

Alternatives: 8

Speedup: 1.5×

• could not determine a ground truth

  1. a = 3.678449014109417e+40
  2. b = 4.767520378324896e-186

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: 54.6% 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: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in b around 0 76.8%

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

   1. log1p-define 76.9%

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

5. Simplified 76.9%

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

6. Final simplification 76.9%

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

Alternative 2: 53.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq 1.75 \cdot 10^{-154}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= b 1.75e-154) (log1p (exp a)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (b <= 1.75e-154) {
		tmp = log1p(exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (b <= 1.75e-154) {
		tmp = Math.log1p(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 b <= 1.75e-154:
		tmp = math.log1p(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 (b <= 1.75e-154)
		tmp = log1p(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[b, 1.75e-154], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.75e-154

   1. Initial program 53.1%

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

   3. Taylor expanded in b around 0 50.0%

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

      1. log1p-define 50.0%

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

   5. Simplified 50.0%

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

   if 1.75e-154 < b

   1. Initial program 72.3%

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

   3. Taylor expanded in a around 0 69.0%

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

      1. log1p-define 69.0%

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

   5. Simplified 69.0%

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

Alternative 3: 96.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in b around 0 55.3%

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

   1. associate-+r+ 55.3%

      \[\leadsto \log \color{blue}{\left(\left(1 + e^{a}\right) + b \cdot \left(1 + 0.5 \cdot b\right)\right)} \]

   2. *-commutative 55.3%

      \[\leadsto \log \left(\left(1 + e^{a}\right) + b \cdot \left(1 + \color{blue}{b \cdot 0.5}\right)\right) \]

5. Simplified 55.3%

   \[\leadsto \log \color{blue}{\left(\left(1 + e^{a}\right) + b \cdot \left(1 + b \cdot 0.5\right)\right)} \]

6. Taylor expanded in b around 0 54.3%

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

7. Taylor expanded in a around inf 54.3%

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

   1. log1p-define 74.4%

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

   2. +-commutative 74.4%

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

9. Simplified 74.4%

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

Alternative 4: 51.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(exp(a))
end

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in b around 0 54.0%

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

   1. log1p-define 54.1%

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

5. Simplified 54.1%

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

Alternative 5: 50.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ b \cdot 0.5 + \log 2 \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return (b * 0.5) + log(2.0);
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (b * 0.5d0) + log(2.0d0)
end function

Java

assert a < b;
public static double code(double a, double b) {
	return (b * 0.5) + Math.log(2.0);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (b * 0.5) + math.log(2.0)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(b * 0.5) + log(2.0))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (b * 0.5) + log(2.0);
end

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in a around 0 54.0%

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

4. Taylor expanded in b around 0 53.2%

   \[\leadsto \color{blue}{\log 2 + 0.5 \cdot b} \]
5. Step-by-step derivation

   1. +-commutative 53.2%

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

6. Simplified 53.2%

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

7. Final simplification 53.2%

   \[\leadsto b \cdot 0.5 + \log 2 \]
8. Add Preprocessing

Alternative 6: 50.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \log \left(b + 2\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log((b + 2.0));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log((b + 2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in a around 0 54.0%

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

4. Taylor expanded in b around 0 52.4%

   \[\leadsto \log \color{blue}{\left(2 + b\right)} \]

5. Final simplification 52.4%

   \[\leadsto \log \left(b + 2\right) \]
6. Add Preprocessing

Alternative 7: 49.6% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log(2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in b around 0 54.0%

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

   1. log1p-define 54.1%

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

5. Simplified 54.1%

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

6. Taylor expanded in a around 0 52.6%

   \[\leadsto \color{blue}{\log 2} \]
7. Add Preprocessing

Alternative 8: 2.6% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a * 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

3. Taylor expanded in b around 0 54.0%

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

   1. log1p-define 54.1%

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

5. Simplified 54.1%

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

6. Taylor expanded in a around 0 52.8%

   \[\leadsto \color{blue}{\log 2 + 0.5 \cdot a} \]
7. Step-by-step derivation

   1. +-commutative 52.8%

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

   2. *-commutative 52.8%

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

8. Simplified 52.8%

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

9. Taylor expanded in a around inf 7.0%

   \[\leadsto \color{blue}{0.5 \cdot a} \]
10. Step-by-step derivation

    1. *-commutative 7.0%

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

11. Simplified 7.0%

    \[\leadsto \color{blue}{a \cdot 0.5} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024087 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))