symmetry log of sum of exp

Percentage Accurate: 54.0% → 98.1%

Time: 14.0s

Alternatives: 9

Speedup: 2.9×

• could not determine a ground truth

  1. a = -3.38148714912964e-123
  2. b = 1.8623127264239894e+182

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: 54.0% 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: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (log1p (+ (exp a) (expm1 b))))

C

assert(a < b);
double code(double a, double b) {
	return log1p((exp(a) + expm1(b)));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p((Math.exp(a) + Math.expm1(b)));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p((math.exp(a) + math.expm1(b)))

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(exp(a) + expm1(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] + N[(Exp[b] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.5%

   \[\log \left(e^{a} + e^{b}\right) \]
2. Step-by-step derivation

   1. add-sqr-sqrt 56.2%

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

   2. log-prod 56.6%

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

3. Applied egg-rr 56.6%

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

   1. log-prod 56.2%

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

   2. rem-square-sqrt 57.5%

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

   3. log1p-expm1 57.3%

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

   4. expm1-def 57.3%

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

   5. rem-exp-log 57.3%

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

   6. associate--l+ 57.4%

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

   7. expm1-def 79.7%

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

5. Simplified 79.7%

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

6. Final simplification 79.7%

   \[\leadsto \mathsf{log1p}\left(e^{a} + \mathsf{expm1}\left(b\right)\right) \]

Alternative 2: 57.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 12.1%

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

   2. Taylor expanded in a around 0 3.7%

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

   3. Taylor expanded in b around 0 3.7%

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

   4. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   6. Simplified 18.8%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0 68.1%

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

      1. log1p-def 68.1%

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

   4. Simplified 68.1%

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

4. Final simplification 55.6%

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

Alternative 3: 96.1% 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. Taylor expanded in b around 0 52.6%

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

   1. associate-+r+ 52.6%

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

   2. +-commutative 52.6%

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

   3. associate-+l+ 52.6%

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

4. Simplified 52.6%

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

5. Taylor expanded in a around inf 52.6%

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

   1. log1p-def 74.7%

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

   2. +-commutative 74.7%

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

7. Simplified 74.7%

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

8. Final simplification 74.7%

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

Alternative 4: 56.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -118:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(b + b \cdot \left(b \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -118.0) (* b 0.5) (log (+ 2.0 (+ b (* b (* b 0.5)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -118.0) {
		tmp = b * 0.5;
	} else {
		tmp = log((2.0 + (b + (b * (b * 0.5)))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-118.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log((2.0d0 + (b + (b * (b * 0.5d0)))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -118.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((2.0 + (b + (b * (b * 0.5)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -118.0:
		tmp = b * 0.5
	else:
		tmp = math.log((2.0 + (b + (b * (b * 0.5)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -118.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(Float64(2.0 + Float64(b + Float64(b * Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -118.0)
		tmp = b * 0.5;
	else
		tmp = log((2.0 + (b + (b * (b * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -118.0], N[(b * 0.5), $MachinePrecision], N[Log[N[(2.0 + N[(b + N[(b * N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -118

   1. Initial program 12.1%

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

   2. Taylor expanded in a around 0 3.7%

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

   3. Taylor expanded in b around 0 3.7%

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

   4. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   6. Simplified 18.8%

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

   if -118 < a

   1. Initial program 72.9%

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

   2. Taylor expanded in a around 0 69.0%

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

   3. Taylor expanded in b around 0 67.7%

      \[\leadsto \log \left(1 + \color{blue}{\left(1 + \left(b + 0.5 \cdot {b}^{2}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. associate-+r+ 67.7%

         \[\leadsto \log \left(1 + \color{blue}{\left(\left(1 + b\right) + 0.5 \cdot {b}^{2}\right)}\right) \]

      2. unpow2 67.7%

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

   5. Simplified 67.7%

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

   6. Taylor expanded in b around 0 67.7%

      \[\leadsto \log \color{blue}{\left(2 + \left(b + 0.5 \cdot {b}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. unpow2 67.7%

         \[\leadsto \log \left(2 + \left(b + 0.5 \cdot \color{blue}{\left(b \cdot b\right)}\right)\right) \]

      2. *-commutative 67.7%

         \[\leadsto \log \left(2 + \left(b + \color{blue}{\left(b \cdot b\right) \cdot 0.5}\right)\right) \]

      3. associate-*r* 67.7%

         \[\leadsto \log \left(2 + \left(b + \color{blue}{b \cdot \left(b \cdot 0.5\right)}\right)\right) \]

   8. Simplified 67.7%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -118:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + \left(b + b \cdot \left(b \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 5: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;a \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.36) (* b 0.5) (+ (* a 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = b * 0.5;
	} else {
		tmp = (a * 0.5) + log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.36d0)) then
        tmp = b * 0.5d0
    else
        tmp = (a * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = b * 0.5;
	} else {
		tmp = (a * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.36:
		tmp = b * 0.5
	else:
		tmp = (a * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.36)
		tmp = Float64(b * 0.5);
	else
		tmp = Float64(Float64(a * 0.5) + log(2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.36)
		tmp = b * 0.5;
	else
		tmp = (a * 0.5) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.36], N[(b * 0.5), $MachinePrecision], N[(N[(a * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3600000000000001

   1. Initial program 15.9%

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

   2. Taylor expanded in a around 0 3.8%

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

   3. Taylor expanded in b around 0 4.0%

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

   4. Taylor expanded in b around inf 18.3%

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

      1. *-commutative 18.3%

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

   6. Simplified 18.3%

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

   if -1.3600000000000001 < a

   1. Initial program 72.5%

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

   2. Taylor expanded in b around 0 68.7%

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

      1. log1p-def 68.7%

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

   4. Simplified 68.7%

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

   5. Taylor expanded in a around 0 68.4%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;a \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 6: 56.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -125:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -125.0) (* b 0.5) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -125.0) {
		tmp = b * 0.5;
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-125.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log((b + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -125.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -125.0:
		tmp = b * 0.5
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -125.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(Float64(b + 2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -125.0)
		tmp = b * 0.5;
	else
		tmp = log((b + 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -125.0], N[(b * 0.5), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -125

   1. Initial program 12.1%

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

   2. Taylor expanded in a around 0 3.7%

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

   3. Taylor expanded in b around 0 3.7%

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

   4. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   6. Simplified 18.8%

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

   if -125 < a

   1. Initial program 72.9%

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

   2. Taylor expanded in a around 0 69.0%

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

   3. Taylor expanded in b around 0 66.5%

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

      1. +-commutative 66.5%

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

   5. Simplified 66.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -125:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 7: 56.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -112:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -112.0) (* b 0.5) (log1p (+ b 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -112.0) {
		tmp = b * 0.5;
	} else {
		tmp = log1p((b + 1.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -112.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log1p((b + 1.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -112.0:
		tmp = b * 0.5
	else:
		tmp = math.log1p((b + 1.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -112.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log1p(Float64(b + 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, -112.0], N[(b * 0.5), $MachinePrecision], N[Log[1 + N[(b + 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -112

   1. Initial program 12.1%

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

   2. Taylor expanded in a around 0 3.7%

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

   3. Taylor expanded in b around 0 3.7%

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

   4. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   6. Simplified 18.8%

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

   if -112 < a

   1. Initial program 72.9%

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

   2. Taylor expanded in a around 0 69.0%

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

      1. log1p-def 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in b around 0 66.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -112:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(b + 1\right)\\ \end{array} \]

Alternative 8: 56.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -90:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -90.0) (* b 0.5) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -90.0) {
		tmp = b * 0.5;
	} else {
		tmp = log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-90.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -90.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -90.0:
		tmp = b * 0.5
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -90.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -90.0)
		tmp = b * 0.5;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -90.0], N[(b * 0.5), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -90

   1. Initial program 12.1%

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

   2. Taylor expanded in a around 0 3.7%

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

   3. Taylor expanded in b around 0 3.7%

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

   4. Taylor expanded in b around inf 18.8%

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

      1. *-commutative 18.8%

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

   6. Simplified 18.8%

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

   if -90 < a

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0 68.1%

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

      1. log1p-def 68.1%

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

   4. Simplified 68.1%

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

   5. Taylor expanded in a around 0 66.9%

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

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -90:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 9: 12.0% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in a around 0 52.4%

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

3. Taylor expanded in b around 0 51.2%

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

4. Taylor expanded in b around inf 7.4%

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

   1. *-commutative 7.4%

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

6. Simplified 7.4%

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

7. Final simplification 7.4%

   \[\leadsto b \cdot 0.5 \]

Reproduce

herbie shell --seed 2023217 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))