symmetry log of sum of exp

Percentage Accurate: 53.9% → 98.9%

Time: 14.1s

Alternatives: 11

Speedup: 2.9×

• could not determine a ground truth

  1. a = -3.140793630900253e-289
  2. b = 2.1648905614463254e+298

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: 53.9% 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: 98.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 67.6%

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

4. Final simplification 76.1%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + b \cdot \left(0.5 + a \cdot -0.25\right)\\ \end{array} \end{array} \]

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 (+ (log1p (exp a)) (* b (+ 0.5 (* a -0.25))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b;
	} else {
		tmp = log1p(exp(a)) + (b * (0.5 + (a * -0.25)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b;
	} else {
		tmp = Math.log1p(Math.exp(a)) + (b * (0.5 + (a * -0.25)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b
	else:
		tmp = math.log1p(math.exp(a)) + (b * (0.5 + (a * -0.25)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = b;
	else
		tmp = Float64(log1p(exp(a)) + Float64(b * Float64(0.5 + Float64(a * -0.25))));
	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], b, N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(b * N[(0.5 + N[(a * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 67.6%

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

   2. Taylor expanded in b around 0 64.1%

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

      1. log1p-def 64.1%

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

   4. Simplified 64.1%

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

   5. Taylor expanded in a around 0 64.1%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{\left(-0.25 \cdot \left(a \cdot b\right) + 0.5 \cdot b\right)} \]
   6. Step-by-step derivation

      1. +-commutative 64.1%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{\left(0.5 \cdot b + -0.25 \cdot \left(a \cdot b\right)\right)} \]

      2. associate-*r* 64.1%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \left(0.5 \cdot b + \color{blue}{\left(-0.25 \cdot a\right) \cdot b}\right) \]

      3. distribute-rgt-out 64.1%

         \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{b \cdot \left(0.5 + -0.25 \cdot a\right)} \]

   7. Simplified 64.1%

      \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \color{blue}{b \cdot \left(0.5 + -0.25 \cdot a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + b \cdot \left(0.5 + a \cdot -0.25\right)\\ \end{array} \]

Alternative 3: 98.5% 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 53.2%

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

2. Taylor expanded in b around 0 73.5%

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

   1. log1p-def 73.5%

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

4. Simplified 73.5%

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

5. Final simplification 73.5%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

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

C

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

Fortran

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

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = b;
	else
		tmp = log(Float64(1.0 + exp(b)));
	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;
	else
		tmp = log((1.0 + exp(b)));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 67.6%

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

   2. Taylor expanded in a around 0 65.0%

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

4. Final simplification 74.2%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

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 (log1p (+ (exp a) b))))

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 67.6%

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

   2. Taylor expanded in b around 0 63.2%

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

      1. associate-+r+ 63.2%

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

      2. +-commutative 63.2%

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

   4. Simplified 63.2%

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

   5. Taylor expanded in a around inf 63.2%

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

      1. log1p-def 63.2%

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

   7. Simplified 63.2%

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

4. Final simplification 72.9%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

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

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 (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b;
	} 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;
	} 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
	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 = b;
	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], b, 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 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 67.6%

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

   2. Taylor expanded in a around 0 65.0%

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

      1. log1p-def 65.0%

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

   4. Simplified 65.0%

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

4. Final simplification 74.2%

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

Alternative 7: 97.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 67.6%

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

   2. Taylor expanded in b around 0 63.2%

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

      1. associate-+r+ 63.2%

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

      2. +-commutative 63.2%

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

   4. Simplified 63.2%

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

   5. Taylor expanded in a around inf 63.2%

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

      1. log1p-def 63.2%

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

   7. Simplified 63.2%

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

   8. Taylor expanded in a around 0 62.4%

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

      1. associate-+r+ 62.4%

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

   10. Simplified 62.4%

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

4. Final simplification 72.2%

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

Alternative 8: 97.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b;
	} else {
		tmp = log((a + 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 <= (-1.0d0)) then
        tmp = b
    else
        tmp = log((a + 2.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.0)
		tmp = b;
	else
		tmp = log((a + 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, -1.0], b, N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if -1 < a

   1. Initial program 67.6%

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

   2. Taylor expanded in b around 0 63.1%

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

   3. Taylor expanded in a around 0 62.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1:\\ \;\;\;\;b\\ \mathbf{else}:\\ \;\;\;\;\log \left(a + 2\right)\\ \end{array} \]

Alternative 9: 97.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -15.5:\\ \;\;\;\;b\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -15.5) b (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -15.5) {
		tmp = b;
	} 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 <= (-15.5d0)) then
        tmp = b
    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 <= -15.5) {
		tmp = b;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -15.5:
		tmp = b
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -15.5)
		tmp = b;
	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 <= -15.5)
		tmp = b;
	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, -15.5], b, N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -15.5

   1. Initial program 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if -15.5 < a

   1. Initial program 67.6%

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

   2. Taylor expanded in b around 0 63.2%

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

      1. associate-+r+ 63.2%

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

      2. +-commutative 63.2%

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

   4. Simplified 63.2%

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

   5. Taylor expanded in a around 0 62.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -15.5:\\ \;\;\;\;b\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]

Alternative 10: 96.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -17:\\ \;\;\;\;b\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -17.0) b (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -17.0) {
		tmp = b;
	} 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 <= (-17.0d0)) then
        tmp = b
    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 <= -17.0) {
		tmp = b;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -17.0:
		tmp = b
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -17.0)
		tmp = b;
	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 <= -17.0)
		tmp = b;
	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, -17.0], b, N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -17

   1. Initial program 12.6%

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

   2. Taylor expanded in b around 0 8.7%

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

      1. associate-+r+ 8.7%

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

      2. +-commutative 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in a around inf 8.7%

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

      1. log1p-def 91.9%

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

   7. Simplified 91.9%

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

   8. Taylor expanded in b around inf 91.9%

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

   9. Taylor expanded in b around 0 100.0%

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

   if -17 < a

   1. Initial program 67.6%

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

   2. Taylor expanded in b around 0 63.1%

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

   3. Taylor expanded in a around 0 62.1%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -17:\\ \;\;\;\;b\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]

Alternative 11: 52.9% accurate, 303.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return 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 = b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

2. Taylor expanded in b around 0 49.0%

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

   1. associate-+r+ 49.0%

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

   2. +-commutative 49.0%

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

4. Simplified 49.0%

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

5. Taylor expanded in a around inf 49.0%

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

   1. log1p-def 70.7%

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

7. Simplified 70.7%

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

8. Taylor expanded in b around inf 26.0%

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

9. Taylor expanded in b around 0 28.8%

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

10. Final simplification 28.8%

    \[\leadsto b \]

Reproduce

herbie shell --seed 2023318 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))