symmetry log of sum of exp

Percentage Accurate: 87.4% → 97.5%

Time: 17.6s

Alternatives: 8

Speedup: 1.4×

• could not determine a ground truth

  1. a = 1.7646876937244838e+131
  2. b = -4.20873821247005e-185

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: 87.4% 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: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := \mathsf{log1p}\left(e^{a}\right)\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{-84} \lor \neg \left(b \leq 8.2 \cdot 10^{-84}\right):\\ \;\;\;\;t_0 + \frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (log1p (exp a))))
   (if (or (<= b -4.2e-84) (not (<= b 8.2e-84)))
     (+ t_0 (/ b (+ (exp a) 1.0)))
     t_0)))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = log1p(exp(a));
	double tmp;
	if ((b <= -4.2e-84) || !(b <= 8.2e-84)) {
		tmp = t_0 + (b / (exp(a) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.log1p(Math.exp(a));
	double tmp;
	if ((b <= -4.2e-84) || !(b <= 8.2e-84)) {
		tmp = t_0 + (b / (Math.exp(a) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.log1p(math.exp(a))
	tmp = 0
	if (b <= -4.2e-84) or not (b <= 8.2e-84):
		tmp = t_0 + (b / (math.exp(a) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = log1p(exp(a))
	tmp = 0.0
	if ((b <= -4.2e-84) || !(b <= 8.2e-84))
		tmp = Float64(t_0 + Float64(b / Float64(exp(a) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[b, -4.2e-84], N[Not[LessEqual[b, 8.2e-84]], $MachinePrecision]], N[(t$95$0 + N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if b < -4.19999999999999996e-84 or 8.2000000000000001e-84 < b

   1. Initial program 71.5%

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

   3. Taylor expanded in b around 0 45.3%

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

      1. log1p-def 45.3%

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

   5. Simplified 45.3%

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

   if -4.19999999999999996e-84 < b < 8.2000000000000001e-84

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0 99.0%

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

      1. log1p-def 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-84} \lor \neg \left(b \leq 8.2 \cdot 10^{-84}\right):\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right) + \frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-84} \lor \neg \left(b \leq 8.2 \cdot 10^{-84}\right):\\ \;\;\;\;\mathsf{log1p}\left(b + e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (or (<= b -4.2e-84) (not (<= b 8.2e-84)))
   (log1p (+ b (exp a)))
   (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((b <= -4.2e-84) || !(b <= 8.2e-84)) {
		tmp = log1p((b + exp(a)));
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if ((b <= -4.2e-84) || !(b <= 8.2e-84)) {
		tmp = Math.log1p((b + Math.exp(a)));
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if (b <= -4.2e-84) or not (b <= 8.2e-84):
		tmp = math.log1p((b + math.exp(a)))
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if ((b <= -4.2e-84) || !(b <= 8.2e-84))
		tmp = log1p(Float64(b + exp(a)));
	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[Or[LessEqual[b, -4.2e-84], N[Not[LessEqual[b, 8.2e-84]], $MachinePrecision]], N[Log[1 + N[(b + N[Exp[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.19999999999999996e-84 or 8.2000000000000001e-84 < b

   1. Initial program 71.5%

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

   3. Taylor expanded in b around 0 30.9%

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

      1. associate-+r+ 30.9%

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

      2. +-commutative 30.9%

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

   5. Simplified 30.9%

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

   6. Taylor expanded in a around inf 30.9%

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

      1. log1p-def 42.6%

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

   8. Simplified 42.6%

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

   if -4.19999999999999996e-84 < b < 8.2000000000000001e-84

   1. Initial program 99.0%

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

   3. Taylor expanded in b around 0 99.0%

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

      1. log1p-def 99.2%

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

   5. Simplified 99.2%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-84} \lor \neg \left(b \leq 8.2 \cdot 10^{-84}\right):\\ \;\;\;\;\mathsf{log1p}\left(b + e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.9%

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

3. Taylor expanded in b around 0 66.5%

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

   1. log1p-def 66.7%

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

5. Simplified 66.7%

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

6. Final simplification 66.7%

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

Alternative 4: 53.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right) + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.4) (* b 0.5) (+ (log (+ b 2.0)) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.4) {
		tmp = b * 0.5;
	} else {
		tmp = log((b + 2.0)) + (a * 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 <= (-1.4d0)) then
        tmp = b * 0.5d0
    else
        tmp = log((b + 2.0d0)) + (a * 0.5d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -1.4) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((b + 2.0)) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.4:
		tmp = b * 0.5
	else:
		tmp = math.log((b + 2.0)) + (a * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.4)
		tmp = Float64(b * 0.5);
	else
		tmp = Float64(log(Float64(b + 2.0)) + Float64(a * 0.5));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.4)
		tmp = b * 0.5;
	else
		tmp = log((b + 2.0)) + (a * 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, -1.4], N[(b * 0.5), $MachinePrecision], N[(N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3999999999999999

   1. Initial program 68.6%

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

   3. Taylor expanded in a around 0 3.7%

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

      1. log1p-def 3.7%

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

   5. Simplified 3.7%

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

   6. Taylor expanded in b around 0 4.0%

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

      1. *-commutative 4.0%

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

   8. Simplified 4.0%

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

   9. Taylor expanded in b around inf 9.6%

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

       1. *-commutative 9.6%

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

   11. Simplified 9.6%

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

   if -1.3999999999999999 < a

   1. Initial program 91.4%

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

   3. Taylor expanded in b around 0 66.0%

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

      1. associate-+r+ 66.0%

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

      2. +-commutative 66.0%

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

   5. Simplified 66.0%

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

   6. Taylor expanded in a around 0 64.9%

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

      1. +-commutative 64.9%

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

      2. +-commutative 64.9%

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

   8. Simplified 64.9%

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

   9. Taylor expanded in b around 0 64.9%

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

       1. *-commutative 64.9%

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

   11. Simplified 64.9%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right) + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b * 0.5;
	} else {
		tmp = log((b + (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 * 0.5d0
    else
        tmp = log((b + (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 * 0.5;
	} else {
		tmp = Math.log((b + (a + 2.0)));
	}
	return tmp;
}

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 68.6%

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

   3. Taylor expanded in a around 0 3.7%

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

      1. log1p-def 3.7%

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

   5. Simplified 3.7%

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

   6. Taylor expanded in b around 0 4.0%

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

      1. *-commutative 4.0%

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

   8. Simplified 4.0%

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

   9. Taylor expanded in b around inf 9.6%

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

       1. *-commutative 9.6%

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

   11. Simplified 9.6%

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

   if -1 < a

   1. Initial program 91.4%

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

   3. Taylor expanded in b around 0 66.0%

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

      1. associate-+r+ 66.0%

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

      2. +-commutative 66.0%

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

   5. Simplified 66.0%

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

   6. Taylor expanded in a around 0 64.8%

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

      1. +-commutative 64.8%

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

      2. +-commutative 64.8%

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

      3. associate-+l+ 64.8%

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

   8. Simplified 64.8%

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

4. Final simplification 51.4%

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

Alternative 6: 53.1% accurate, 2.8× speedup

Math

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

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 0.5) (log (+ a 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		tmp = b * 0.5;
	} 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 * 0.5d0
    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 * 0.5;
	} 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 * 0.5
	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 = Float64(b * 0.5);
	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 * 0.5;
	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], N[(b * 0.5), $MachinePrecision], N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 68.6%

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

   3. Taylor expanded in a around 0 3.7%

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

      1. log1p-def 3.7%

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

   5. Simplified 3.7%

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

   6. Taylor expanded in b around 0 4.0%

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

      1. *-commutative 4.0%

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

   8. Simplified 4.0%

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

   9. Taylor expanded in b around inf 9.6%

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

       1. *-commutative 9.6%

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

   11. Simplified 9.6%

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

   if -1 < a

   1. Initial program 91.4%

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

   3. Taylor expanded in b around 0 66.0%

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

      1. associate-+r+ 66.0%

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

      2. +-commutative 66.0%

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

   5. Simplified 66.0%

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

   6. Taylor expanded in a around 0 64.8%

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

      1. +-commutative 64.8%

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

      2. +-commutative 64.8%

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

      3. associate-+l+ 64.8%

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

   8. Simplified 64.8%

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

   9. Taylor expanded in b around 0 65.4%

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

       1. +-commutative 65.4%

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

   11. Simplified 65.4%

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

4. Final simplification 51.8%

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

Alternative 7: 52.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -80:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -80.0) (* b 0.5) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -80.0) {
		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 <= (-80.0d0)) 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 <= -80.0) {
		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 <= -80.0:
		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 <= -80.0)
		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 <= -80.0)
		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, -80.0], N[(b * 0.5), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -80

   1. Initial program 68.7%

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

   3. Taylor expanded in a around 0 3.6%

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

      1. log1p-def 3.6%

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

   5. Simplified 3.6%

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

   6. Taylor expanded in b around 0 3.9%

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

      1. *-commutative 3.9%

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

   8. Simplified 3.9%

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

   9. Taylor expanded in b around inf 9.7%

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

       1. *-commutative 9.7%

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

   11. Simplified 9.7%

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

   if -80 < a

   1. Initial program 91.2%

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

   3. Taylor expanded in b around 0 66.5%

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

      1. log1p-def 66.7%

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

   5. Simplified 66.7%

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

   6. Taylor expanded in a around 0 64.2%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -80:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 7.5% 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 85.9%

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

3. Taylor expanded in a around 0 67.5%

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

   1. log1p-def 67.5%

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

5. Simplified 67.5%

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

6. Taylor expanded in b around 0 50.0%

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

   1. *-commutative 50.0%

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

8. Simplified 50.0%

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

9. Taylor expanded in b around inf 4.8%

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

    1. *-commutative 4.8%

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

11. Simplified 4.8%

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

12. Final simplification 4.8%

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

Reproduce

herbie shell --seed 2024018 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))