symmetry log of sum of exp

Percentage Accurate: 54.1% → 98.8%

Time: 16.0s

Alternatives: 13

Speedup: 1.5×

• could not determine a ground truth

  1. a = 2.4681162903496214e+22
  2. b = -2.3844274572458948e+272

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% 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: 98.8% accurate, 0.7× speedup

Math

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

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ 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) <= 0.0) {
		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) <= 0.0d0) 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) <= 0.0) {
		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) <= 0.0:
		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) <= 0.0)
		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) <= 0.0)
		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], 0.0], 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) < 0.0

   1. Initial program 6.0%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.4%

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

4. Final simplification 73.1%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (log (+ 1.0 (+ (exp a) (+ b (* 0.5 (* b b))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((1.0 + (exp(a) + (b + (0.5 * (b * 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 / (exp(a) + 1.0d0)
    else
        tmp = log((1.0d0 + (exp(a) + (b + (0.5d0 * (b * 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 / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((1.0 + (Math.exp(a) + (b + (0.5 * (b * b))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((1.0 + (math.exp(a) + (b + (0.5 * (b * b))))))
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 6.0%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.4%

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

   2. Taylor expanded in b around 0 61.9%

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

      1. unpow2 61.9%

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

   4. Simplified 61.9%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(1 + \left(e^{a} + \left(b + 0.5 \cdot \left(b \cdot b\right)\right)\right)\right)\\ \end{array} \]

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + (b + 1.0)));
	}
	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 (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + (b + 1.0d0)))
    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 / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((Math.exp(a) + (b + 1.0)));
	}
	return tmp;
}

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 6.0%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.4%

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

   2. Taylor expanded in b around 0 60.6%

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

      1. associate-+r+ 60.6%

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

      2. +-commutative 60.6%

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

      3. associate-+l+ 60.6%

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

   4. Simplified 60.6%

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

4. Final simplification 69.4%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 6.0%

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

   2. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.4%

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

   2. Taylor expanded in b around 0 61.6%

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

      1. log1p-def 61.8%

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

   4. Simplified 61.8%

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

4. Final simplification 70.3%

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

Alternative 5: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(exp(a) + expm1(b)))
end

Wolfram

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

Derivation

1. Initial program 52.2%

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

   1. add-sqr-sqrt 51.1%

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

   2. log-prod 51.5%

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

3. Applied egg-rr 51.5%

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

   1. log-prod 51.1%

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

   2. rem-square-sqrt 52.2%

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

   3. log1p-expm1 52.2%

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

   4. expm1-def 52.2%

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

   5. rem-exp-log 52.2%

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

   6. associate--l+ 52.3%

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

   7. expm1-def 72.9%

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

5. Simplified 72.9%

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

6. Final simplification 72.9%

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

Alternative 6: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 0.5 + \log 2\right) + \frac{b}{a + 2}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.01)
   (/ b (+ (exp a) 1.0))
   (+ (+ (* a 0.5) (log 2.0)) (/ b (+ a 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.01) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = ((a * 0.5) + log(2.0)) + (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 (exp(a) <= 0.01d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = ((a * 0.5d0) + log(2.0d0)) + (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 (Math.exp(a) <= 0.01) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = ((a * 0.5) + Math.log(2.0)) + (b / (a + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.01:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = ((a * 0.5) + math.log(2.0)) + (b / (a + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.01)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(Float64(a * 0.5) + log(2.0)) + 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 (exp(a) <= 0.01)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = ((a * 0.5) + log(2.0)) + (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[N[Exp[a], $MachinePrecision], 0.01], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision] + N[(b / N[(a + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0100000000000000002

   1. Initial program 8.7%

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

   2. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 96.8%

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

   if 0.0100000000000000002 < (exp.f64 a)

   1. Initial program 65.2%

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

   2. Taylor expanded in a around 0 62.6%

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

      1. associate-+r+ 62.6%

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

      2. +-commutative 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in b around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. +-commutative 59.9%

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

   7. Simplified 59.9%

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

   8. Taylor expanded in a around 0 60.0%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot 0.5 + \log 2\right) + \frac{b}{a + 2}\\ \end{array} \]

Alternative 7: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a + 2} + \log \left(a + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.01)
   (/ b (+ (exp a) 1.0))
   (+ (/ b (+ a 2.0)) (log (+ a 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.01) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (b / (a + 2.0)) + 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 (exp(a) <= 0.01d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (b / (a + 2.0d0)) + 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 (Math.exp(a) <= 0.01) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (b / (a + 2.0)) + Math.log((a + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.01:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (b / (a + 2.0)) + math.log((a + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.01)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(b / Float64(a + 2.0)) + 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 (exp(a) <= 0.01)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = (b / (a + 2.0)) + 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[N[Exp[a], $MachinePrecision], 0.01], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b / N[(a + 2.0), $MachinePrecision]), $MachinePrecision] + N[Log[N[(a + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0100000000000000002

   1. Initial program 8.7%

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

   2. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 96.8%

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

   if 0.0100000000000000002 < (exp.f64 a)

   1. Initial program 65.2%

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

   2. Taylor expanded in a around 0 62.6%

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

      1. associate-+r+ 62.6%

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

      2. +-commutative 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in b around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. +-commutative 59.9%

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

   7. Simplified 59.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a + 2} + \log \left(a + 2\right)\\ \end{array} \]

Alternative 8: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a + 2} + \mathsf{log1p}\left(a + 1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.01)
   (/ b (+ (exp a) 1.0))
   (+ (/ b (+ a 2.0)) (log1p (+ a 1.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.01) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (b / (a + 2.0)) + log1p((a + 1.0));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.01) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (b / (a + 2.0)) + Math.log1p((a + 1.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.01:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (b / (a + 2.0)) + math.log1p((a + 1.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.01)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(b / Float64(a + 2.0)) + log1p(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], 0.01], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(b / N[(a + 2.0), $MachinePrecision]), $MachinePrecision] + N[Log[1 + N[(a + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0100000000000000002

   1. Initial program 8.7%

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

   2. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 96.8%

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

   if 0.0100000000000000002 < (exp.f64 a)

   1. Initial program 65.2%

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

   2. Taylor expanded in a around 0 62.6%

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

      1. associate-+r+ 62.6%

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

      2. +-commutative 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in b around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. +-commutative 59.9%

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

   7. Simplified 59.9%

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

      1. log1p-expm1-u 59.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(a + 2\right)\right)\right)} + \frac{b}{a + 2} \]

      2. expm1-udef 59.9%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(a + 2\right)} - 1}\right) + \frac{b}{a + 2} \]

      3. add-exp-log 59.9%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(a + 2\right)} - 1\right) + \frac{b}{a + 2} \]

   9. Applied egg-rr 59.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(a + 2\right) - 1\right)} + \frac{b}{a + 2} \]
   10. Step-by-step derivation

       1. associate--l+ 59.9%

          \[\leadsto \mathsf{log1p}\left(\color{blue}{a + \left(2 - 1\right)}\right) + \frac{b}{a + 2} \]

       2. metadata-eval 59.9%

          \[\leadsto \mathsf{log1p}\left(a + \color{blue}{1}\right) + \frac{b}{a + 2} \]

   11. Simplified 59.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{a + 2} + \mathsf{log1p}\left(a + 1\right)\\ \end{array} \]

Alternative 9: 96.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.01) (/ b (+ (exp a) 1.0)) (+ (* a 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.01) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = (a * 0.5) + log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.01d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = (a * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.01) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = (a * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.01:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = (a * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.01)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(Float64(a * 0.5) + log(2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.01)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = (a * 0.5) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.01], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0100000000000000002

   1. Initial program 8.7%

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

   2. Taylor expanded in b around 0 99.5%

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

      1. log1p-def 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in b around inf 96.8%

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

   if 0.0100000000000000002 < (exp.f64 a)

   1. Initial program 65.2%

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

   2. Taylor expanded in a around 0 62.6%

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

      1. associate-+r+ 62.6%

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

      2. +-commutative 62.6%

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

   4. Simplified 62.6%

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

   5. Taylor expanded in b around 0 60.1%

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

      1. +-commutative 60.1%

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

   7. Simplified 60.1%

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

   8. Taylor expanded in a around 0 60.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.01:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 0.5 + \log 2\\ \end{array} \]

Alternative 10: 50.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return log(2.0) + (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 = log(2.0d0) + (b * 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in a around 0 48.0%

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

3. Taylor expanded in b around 0 46.7%

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

4. Final simplification 46.7%

   \[\leadsto \log 2 + b \cdot 0.5 \]

Alternative 11: 49.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in a around 0 48.0%

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

3. Taylor expanded in b around 0 45.9%

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

   1. +-commutative 45.9%

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

5. Simplified 45.9%

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

6. Final simplification 45.9%

   \[\leadsto \log \left(b + 2\right) \]

Alternative 12: 49.2% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in a around 0 48.0%

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

3. Taylor expanded in b around 0 46.8%

   \[\leadsto \color{blue}{\log 2} \]

4. Final simplification 46.8%

   \[\leadsto \log 2 \]

Alternative 13: 5.1% accurate, 60.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return b / (a + 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

2. Taylor expanded in a around 0 48.2%

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

   1. associate-+r+ 48.2%

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

   2. +-commutative 48.2%

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

4. Simplified 48.2%

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

5. Taylor expanded in b around 0 46.1%

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

   1. +-commutative 46.1%

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

   2. +-commutative 46.1%

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

7. Simplified 46.1%

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

   1. log1p-expm1-u 46.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(a + 2\right)\right)\right)} + \frac{b}{a + 2} \]

   2. expm1-udef 46.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\log \left(a + 2\right)} - 1}\right) + \frac{b}{a + 2} \]

   3. add-exp-log 46.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(a + 2\right)} - 1\right) + \frac{b}{a + 2} \]

9. Applied egg-rr 46.1%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(a + 2\right) - 1\right)} + \frac{b}{a + 2} \]
10. Step-by-step derivation

    1. associate--l+ 46.1%

       \[\leadsto \mathsf{log1p}\left(\color{blue}{a + \left(2 - 1\right)}\right) + \frac{b}{a + 2} \]

    2. metadata-eval 46.1%

       \[\leadsto \mathsf{log1p}\left(a + \color{blue}{1}\right) + \frac{b}{a + 2} \]

11. Simplified 46.1%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(a + 1\right)} + \frac{b}{a + 2} \]

12. Taylor expanded in b around inf 3.8%

    \[\leadsto \color{blue}{\frac{b}{2 + a}} \]
13. Step-by-step derivation

    1. +-commutative 3.8%

       \[\leadsto \frac{b}{\color{blue}{a + 2}} \]

14. Simplified 3.8%

    \[\leadsto \color{blue}{\frac{b}{a + 2}} \]

15. Final simplification 3.8%

    \[\leadsto \frac{b}{a + 2} \]

Reproduce

herbie shell --seed 2023252 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))