symmetry log of sum of exp

Percentage Accurate: 53.3% → 98.9%

Time: 17.2s

Alternatives: 11

Speedup: 2.9×

• could not determine a ground truth

  1. a = -0.003074037684853584
  2. b = 8.744295791413106e+247

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: 53.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-29) (/ b (+ (exp a) 1.0)) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-29) {
		tmp = b / (exp(a) + 1.0);
	} 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) <= 5d-29) then
        tmp = b / (exp(a) + 1.0d0)
    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) <= 5e-29) {
		tmp = b / (Math.exp(a) + 1.0);
	} 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) <= 5e-29:
		tmp = b / (math.exp(a) + 1.0)
	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) <= 5e-29)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	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) <= 5e-29)
		tmp = b / (exp(a) + 1.0);
	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], 5e-29], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], 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) < 4.99999999999999986e-29

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 99.5%

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

   if 4.99999999999999986e-29 < (exp.f64 a)

   1. Initial program 65.4%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-29) (/ b (+ (exp a) 1.0)) (log (+ 1.0 (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-29) {
		tmp = b / (exp(a) + 1.0);
	} 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) <= 5d-29) then
        tmp = b / (exp(a) + 1.0d0)
    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) <= 5e-29) {
		tmp = b / (Math.exp(a) + 1.0);
	} 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) <= 5e-29:
		tmp = b / (math.exp(a) + 1.0)
	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) <= 5e-29)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	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) <= 5e-29)
		tmp = b / (exp(a) + 1.0);
	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], 5e-29], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(1.0 + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999986e-29

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 99.5%

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

   if 4.99999999999999986e-29 < (exp.f64 a)

   1. Initial program 65.4%

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

   3. Taylor expanded in a around 0 63.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% 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 51.2%

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

3. Taylor expanded in b around 0 72.2%

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

   1. log1p-def 72.3%

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

5. Simplified 72.3%

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

6. Final simplification 72.3%

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

Alternative 4: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-29) (/ b (+ (exp a) 1.0)) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-29) {
		tmp = b / (exp(a) + 1.0);
	} 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) <= 5e-29) {
		tmp = b / (Math.exp(a) + 1.0);
	} 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) <= 5e-29:
		tmp = b / (math.exp(a) + 1.0)
	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) <= 5e-29)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	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], 5e-29], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999986e-29

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 99.5%

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

   if 4.99999999999999986e-29 < (exp.f64 a)

   1. Initial program 65.4%

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

   3. Taylor expanded in a around 0 63.7%

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

      1. log1p-def 63.7%

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

   5. Simplified 63.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot \left(0.5 - b \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-29)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (+ (* b 0.5) (* a (- 0.5 (* b 0.25)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-29) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + ((b * 0.5) + (a * (0.5 - (b * 0.25))));
	}
	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) <= 5d-29) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + ((b * 0.5d0) + (a * (0.5d0 - (b * 0.25d0))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-29) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + ((b * 0.5) + (a * (0.5 - (b * 0.25))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-29:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + ((b * 0.5) + (a * (0.5 - (b * 0.25))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-29)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(Float64(b * 0.5) + Float64(a * Float64(0.5 - Float64(b * 0.25)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-29)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + ((b * 0.5) + (a * (0.5 - (b * 0.25))));
	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], 5e-29], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(N[(b * 0.5), $MachinePrecision] + N[(a * N[(0.5 - N[(b * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999986e-29

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 99.5%

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

   if 4.99999999999999986e-29 < (exp.f64 a)

   1. Initial program 65.4%

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

   3. Taylor expanded in b around 0 63.6%

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

      1. log1p-def 63.6%

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

   5. Simplified 63.6%

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

   6. Taylor expanded in a around 0 62.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot \left(0.5 - b \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + \left(b + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-29) (/ b (+ (exp a) 1.0)) (log1p (+ a (+ b 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-29) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log1p((a + (b + 1.0)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 5e-29) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p((a + (b + 1.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-29:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p((a + (b + 1.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-29)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(Float64(a + 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[N[Exp[a], $MachinePrecision], 5e-29], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(a + N[(b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999986e-29

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in b around inf 99.5%

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

   if 4.99999999999999986e-29 < (exp.f64 a)

   1. Initial program 65.4%

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

   3. Taylor expanded in b around 0 62.6%

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

      1. associate-+r+ 62.6%

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

      2. +-commutative 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in a around inf 62.6%

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

      1. log1p-def 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in a around 0 61.7%

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

       1. +-commutative 61.7%

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

       2. associate-+l+ 61.7%

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

       3. +-commutative 61.7%

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

   11. Simplified 61.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-29}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + \left(b + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 6.3%

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

      1. associate-+r+ 6.3%

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

      2. +-commutative 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in a around inf 6.3%

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

      1. log1p-def 95.1%

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

   8. Simplified 95.1%

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

   9. Taylor expanded in b around inf 94.6%

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

   if -1 < a

   1. Initial program 65.4%

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

   3. Taylor expanded in b around 0 62.6%

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

      1. associate-+r+ 62.6%

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

      2. +-commutative 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in a around inf 62.6%

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

      1. log1p-def 62.6%

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

   8. Simplified 62.6%

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

   9. Taylor expanded in a around 0 61.7%

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

       1. +-commutative 61.7%

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

       2. associate-+l+ 61.7%

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

       3. +-commutative 61.7%

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

   11. Simplified 61.7%

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

4. Final simplification 69.5%

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

Alternative 8: 94.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -57:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -57.0) (log1p b) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -57.0) {
		tmp = log1p(b);
	} else {
		tmp = log((b + 2.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -57.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -57.0:
		tmp = math.log1p(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 <= -57.0)
		tmp = log1p(b);
	else
		tmp = log(Float64(b + 2.0));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -57.0], N[Log[1 + b], $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -57

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 6.3%

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

      1. associate-+r+ 6.3%

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

      2. +-commutative 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in a around inf 6.3%

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

      1. log1p-def 95.1%

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

   8. Simplified 95.1%

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

   9. Taylor expanded in b around inf 94.6%

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

   if -57 < a

   1. Initial program 65.4%

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

   3. Taylor expanded in a around 0 63.7%

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

   4. Taylor expanded in b around 0 61.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -57:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -65:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -65.0) (* b 0.5) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -65.0) {
		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 <= (-65.0d0)) 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 <= -65.0) {
		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 <= -65.0:
		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 <= -65.0)
		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 <= -65.0)
		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, -65.0], N[(b * 0.5), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -65

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in a around 0 4.5%

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

   7. Taylor expanded in b around inf 18.7%

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

      1. *-commutative 18.7%

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

   9. Simplified 18.7%

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

   if -65 < a

   1. Initial program 65.4%

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

   3. Taylor expanded in a around 0 63.7%

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

      1. log1p-def 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in b around 0 62.1%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -65:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -65:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -65.0) (log1p b) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -65.0) {
		tmp = log1p(b);
	} else {
		tmp = log(2.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -65.0) {
		tmp = Math.log1p(b);
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -65.0:
		tmp = math.log1p(b)
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -65.0)
		tmp = log1p(b);
	else
		tmp = log(2.0);
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -65.0], N[Log[1 + b], $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -65

   1. Initial program 6.1%

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

   3. Taylor expanded in b around 0 6.3%

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

      1. associate-+r+ 6.3%

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

      2. +-commutative 6.3%

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

   5. Simplified 6.3%

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

   6. Taylor expanded in a around inf 6.3%

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

      1. log1p-def 95.1%

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

   8. Simplified 95.1%

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

   9. Taylor expanded in b around inf 94.6%

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

   if -65 < a

   1. Initial program 65.4%

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

   3. Taylor expanded in a around 0 63.7%

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

      1. log1p-def 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in b around 0 62.1%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -65:\\ \;\;\;\;\mathsf{log1p}\left(b\right)\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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 51.2%

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

3. Taylor expanded in b around 0 72.2%

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

   1. log1p-def 72.3%

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

5. Simplified 72.3%

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

6. Taylor expanded in a around 0 48.5%

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

7. Taylor expanded in b around inf 7.0%

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

   1. *-commutative 7.0%

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

9. Simplified 7.0%

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

10. Final simplification 7.0%

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

Reproduce

herbie shell --seed 2024017 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))