symmetry log of sum of exp

Percentage Accurate: 53.7% → 98.6%

Time: 19.5s

Alternatives: 17

Speedup: 2.9×

• could not determine a ground truth

  1. a = 7.914789803951773e+302
  2. b = -4.311015526308106e+65

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: 53.7% 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: 98.6% 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 56.1%

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

3. Taylor expanded in b around 0 74.1%

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

   1. log1p-define 74.1%

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

5. Simplified 74.1%

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

6. Final simplification 74.1%

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

Alternative 2: 99.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-187) (/ 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-187) {
		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-187) 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-187) {
		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-187:
		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-187)
		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-187)
		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-187], 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) < 4.9999999999999996e-187

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.9999999999999996e-187 < (exp.f64 a)

   1. Initial program 68.9%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := e^{a} + 1\\ \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\log \left(t\_0 + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (if (<= (exp a) 5e-187)
     (/ b t_0)
     (log (+ t_0 (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	double tmp;
	if (exp(a) <= 5e-187) {
		tmp = b / t_0;
	} else {
		tmp = log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = exp(a) + 1.0d0
    if (exp(a) <= 5d-187) then
        tmp = b / t_0
    else
        tmp = log((t_0 + (b * (1.0d0 + (b * (0.5d0 + (b * 0.16666666666666666d0)))))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.exp(a) + 1.0;
	double tmp;
	if (Math.exp(a) <= 5e-187) {
		tmp = b / t_0;
	} else {
		tmp = Math.log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.exp(a) + 1.0
	tmp = 0
	if math.exp(a) <= 5e-187:
		tmp = b / t_0
	else:
		tmp = math.log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	tmp = 0.0
	if (exp(a) <= 5e-187)
		tmp = Float64(b / t_0);
	else
		tmp = log(Float64(t_0 + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = exp(a) + 1.0;
	tmp = 0.0;
	if (exp(a) <= 5e-187)
		tmp = b / t_0;
	else
		tmp = log((t_0 + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
	end
	tmp_2 = 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[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 5e-187], N[(b / t$95$0), $MachinePrecision], N[Log[N[(t$95$0 + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999996e-187

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.9999999999999996e-187 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.4%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{a} + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 65.4%

         \[\leadsto \log \color{blue}{\left(\left(1 + e^{a}\right) + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)\right)} \]

      2. *-commutative 65.4%

         \[\leadsto \log \left(\left(1 + e^{a}\right) + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)\right) \]

   5. Simplified 65.4%

      \[\leadsto \log \color{blue}{\left(\left(1 + e^{a}\right) + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(e^{a} + 1\right) + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := e^{a} + 1\\ \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\log \left(t\_0 + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ (exp a) 1.0)))
   (if (<= (exp a) 5e-187) (/ b t_0) (log (+ t_0 (* b (+ 1.0 (* b 0.5))))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = exp(a) + 1.0;
	double tmp;
	if (exp(a) <= 5e-187) {
		tmp = b / t_0;
	} else {
		tmp = log((t_0 + (b * (1.0 + (b * 0.5)))));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(a) + 1.0d0
    if (exp(a) <= 5d-187) then
        tmp = b / t_0
    else
        tmp = log((t_0 + (b * (1.0d0 + (b * 0.5d0)))))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = Math.exp(a) + 1.0;
	double tmp;
	if (Math.exp(a) <= 5e-187) {
		tmp = b / t_0;
	} else {
		tmp = Math.log((t_0 + (b * (1.0 + (b * 0.5)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = math.exp(a) + 1.0
	tmp = 0
	if math.exp(a) <= 5e-187:
		tmp = b / t_0
	else:
		tmp = math.log((t_0 + (b * (1.0 + (b * 0.5)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(exp(a) + 1.0)
	tmp = 0.0
	if (exp(a) <= 5e-187)
		tmp = Float64(b / t_0);
	else
		tmp = log(Float64(t_0 + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	t_0 = exp(a) + 1.0;
	tmp = 0.0;
	if (exp(a) <= 5e-187)
		tmp = b / t_0;
	else
		tmp = log((t_0 + (b * (1.0 + (b * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Exp[a], $MachinePrecision], 5e-187], N[(b / t$95$0), $MachinePrecision], N[Log[N[(t$95$0 + N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999996e-187

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.9999999999999996e-187 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 66.4%

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

      1. associate-+r+ 66.4%

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

      2. *-commutative 66.4%

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

   5. Simplified 66.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(e^{a} + 1\right) + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-187) (/ 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) <= 5e-187) {
		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) <= 5d-187) 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) <= 5e-187) {
		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) <= 5e-187:
		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) <= 5e-187)
		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) <= 5e-187)
		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], 5e-187], 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) < 4.9999999999999996e-187

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.9999999999999996e-187 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.0%

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

      1. associate-+r+ 65.0%

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

      2. +-commutative 65.0%

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

   5. Simplified 65.0%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.0% 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 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.1%

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

      1. log1p-define 65.1%

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

   5. Simplified 65.1%

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

4. Final simplification 73.6%

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

Alternative 7: 98.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b \cdot \left(1 + b \cdot 0.5\right) + \left(2 + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-7)
   (/ b (+ (exp a) 1.0))
   (log
    (+
     (* b (+ 1.0 (* b 0.5)))
     (+ 2.0 (* a (+ 1.0 (* a (+ 0.5 (* a 0.16666666666666666))))))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-7) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(((b * (1.0 + (b * 0.5))) + (2.0 + (a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))))));
	}
	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-7) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(((b * (1.0d0 + (b * 0.5d0))) + (2.0d0 + (a * (1.0d0 + (a * (0.5d0 + (a * 0.16666666666666666d0))))))))
    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-7) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(((b * (1.0 + (b * 0.5))) + (2.0 + (a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-7:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(((b * (1.0 + (b * 0.5))) + (2.0 + (a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-7)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(Float64(b * Float64(1.0 + Float64(b * 0.5))) + Float64(2.0 + Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(a * 0.16666666666666666))))))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-7)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(((b * (1.0 + (b * 0.5))) + (2.0 + (a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))))));
	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-7], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 + N[(a * N[(1.0 + N[(a * N[(0.5 + N[(a * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999977e-7

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.99999999999999977e-7 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 66.4%

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

      1. associate-+r+ 66.4%

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

      2. *-commutative 66.4%

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

   5. Simplified 66.4%

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

   6. Taylor expanded in a around 0 65.9%

      \[\leadsto \log \left(\color{blue}{\left(2 + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)} + b \cdot \left(1 + b \cdot 0.5\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b \cdot \left(1 + b \cdot 0.5\right) + \left(2 + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right) + \left(b + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-7)
   (/ b (+ (exp a) 1.0))
   (log (+ (* a (+ 1.0 (* a (+ 0.5 (* a 0.16666666666666666))))) (+ b 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-7) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(((a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))) + (b + 2.0)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 5d-7) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(((a * (1.0d0 + (a * (0.5d0 + (a * 0.16666666666666666d0))))) + (b + 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) <= 5e-7) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(((a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))) + (b + 2.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-7:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(((a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))) + (b + 2.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-7)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(Float64(a * Float64(1.0 + Float64(a * Float64(0.5 + Float64(a * 0.16666666666666666))))) + Float64(b + 2.0)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-7)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(((a * (1.0 + (a * (0.5 + (a * 0.16666666666666666))))) + (b + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 5e-7], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(a * N[(1.0 + N[(a * N[(0.5 + N[(a * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999977e-7

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.99999999999999977e-7 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in a around 0 67.9%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{b} + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 67.9%

         \[\leadsto \log \color{blue}{\left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)} \]

      2. *-commutative 67.9%

         \[\leadsto \log \left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + \color{blue}{a \cdot 0.16666666666666666}\right)\right)\right) \]

   5. Simplified 67.9%

      \[\leadsto \log \color{blue}{\left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)} \]

   6. Taylor expanded in b around 0 64.6%

      \[\leadsto \log \left(\color{blue}{\left(2 + b\right)} + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right) + \left(b + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot \left(0.5 - b \cdot 0.25\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-7)
   (/ 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-7) {
		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-7) 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-7) {
		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-7:
		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-7)
		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-7)
		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-7], 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) < 4.99999999999999977e-7

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.99999999999999977e-7 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.8%

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

      1. log1p-define 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in a around 0 64.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot \left(0.5 - b \cdot 0.25\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-187)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* a (+ 0.5 (* a 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-187) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	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-187) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (a * (0.5d0 + (a * 0.125d0)))
    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-187) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * (0.5 + (a * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-187:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * (0.5 + (a * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-187)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(a * Float64(0.5 + Float64(a * 0.125))));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 5e-187)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (a * (0.5 + (a * 0.125)));
	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-187], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * N[(0.5 + N[(a * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.9999999999999996e-187

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.9999999999999996e-187 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.1%

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

      1. log1p-define 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in a around 0 64.7%

      \[\leadsto \color{blue}{\log 2 + a \cdot \left(0.5 + 0.125 \cdot a\right)} \]
   7. Step-by-step derivation

      1. metadata-eval 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \color{blue}{\left(0.125 - 0\right)} \cdot a\right) \]

      2. mul0-rgt 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \left(0.125 - \color{blue}{b \cdot 0}\right) \cdot a\right) \]

      3. metadata-eval 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \left(0.125 - b \cdot \color{blue}{\left(-0.125 + 0.125\right)}\right) \cdot a\right) \]

      4. distribute-rgt-out 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \left(0.125 - \color{blue}{\left(-0.125 \cdot b + 0.125 \cdot b\right)}\right) \cdot a\right) \]

      5. *-commutative 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + \color{blue}{a \cdot \left(0.125 - \left(-0.125 \cdot b + 0.125 \cdot b\right)\right)}\right) \]

      6. distribute-rgt-out 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 - \color{blue}{b \cdot \left(-0.125 + 0.125\right)}\right)\right) \]

      7. metadata-eval 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 - b \cdot \color{blue}{0}\right)\right) \]

      8. mul0-rgt 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + a \cdot \left(0.125 - \color{blue}{0}\right)\right) \]

      9. metadata-eval 64.7%

         \[\leadsto \log 2 + a \cdot \left(0.5 + a \cdot \color{blue}{0.125}\right) \]

   8. Simplified 64.7%

      \[\leadsto \color{blue}{\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-187}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot \left(0.5 + a \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 5e-7)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (+ (* b 0.5) (* a 0.5)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 5e-7) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + ((b * 0.5) + (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 (exp(a) <= 5d-7) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + ((b * 0.5d0) + (a * 0.5d0))
    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-7) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + ((b * 0.5) + (a * 0.5));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 5e-7:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + ((b * 0.5) + (a * 0.5))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 5e-7)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(Float64(b * 0.5) + 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 (exp(a) <= 5e-7)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + ((b * 0.5) + (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[N[Exp[a], $MachinePrecision], 5e-7], 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 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 4.99999999999999977e-7

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if 4.99999999999999977e-7 < (exp.f64 a)

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.8%

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

      1. log1p-define 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in a around 0 64.8%

      \[\leadsto \color{blue}{\log 2 + \left(0.5 \cdot b + a \cdot \left(0.5 - 0.25 \cdot b\right)\right)} \]

   7. Taylor expanded in b around 0 64.8%

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

      1. *-commutative 64.8%

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

   9. Simplified 64.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(b \cdot 0.5 + a \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 97.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -42:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + a \cdot \left(1 + a \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -42.0) (/ b (+ (exp a) 1.0)) (log (+ 2.0 (* a (+ 1.0 (* a 0.5)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -42.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((2.0 + (a * (1.0 + (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 <= (-42.0d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((2.0d0 + (a * (1.0d0 + (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 <= -42.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log((2.0 + (a * (1.0 + (a * 0.5)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -42.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((2.0 + (a * (1.0 + (a * 0.5)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -42.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log(Float64(2.0 + Float64(a * Float64(1.0 + 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 <= -42.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log((2.0 + (a * (1.0 + (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, -42.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + N[(a * N[(1.0 + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -42

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if -42 < a

   1. Initial program 68.9%

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

   3. Taylor expanded in a around 0 67.9%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{b} + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 67.9%

         \[\leadsto \log \color{blue}{\left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)} \]

      2. *-commutative 67.9%

         \[\leadsto \log \left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + \color{blue}{a \cdot 0.16666666666666666}\right)\right)\right) \]

   5. Simplified 67.9%

      \[\leadsto \log \color{blue}{\left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)} \]

   6. Taylor expanded in b around 0 64.7%

      \[\leadsto \color{blue}{\log \left(2 + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)} \]

   7. Taylor expanded in a around 0 64.5%

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

      1. *-commutative 64.5%

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

   9. Simplified 64.5%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -42:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + a \cdot \left(1 + a \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.35) (/ b 2.0) (+ (log 2.0) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.35) {
		tmp = b / 2.0;
	} else {
		tmp = log(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 <= (-1.35d0)) then
        tmp = b / 2.0d0
    else
        tmp = log(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 <= -1.35) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log(2.0) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.35:
		tmp = b / 2.0
	else:
		tmp = math.log(2.0) + (a * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.35)
		tmp = Float64(b / 2.0);
	else
		tmp = Float64(log(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 <= -1.35)
		tmp = b / 2.0;
	else
		tmp = log(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, -1.35], N[(b / 2.0), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3500000000000001

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   7. Taylor expanded in a around 0 18.8%

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

   if -1.3500000000000001 < a

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.1%

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

      1. log1p-define 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in a around 0 64.2%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 97.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.35) (/ b (+ (exp a) 1.0)) (+ (log 2.0) (* a 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.35) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(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 <= (-1.35d0)) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(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 <= -1.35) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (a * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -1.35:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (a * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.35)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(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 <= -1.35)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(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, -1.35], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(a * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3500000000000001

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   if -1.3500000000000001 < a

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.1%

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

      1. log1p-define 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in a around 0 64.2%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + a \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.7% accurate, 2.8× speedup

Math

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

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

C

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   7. Taylor expanded in a around 0 18.8%

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

   if -1 < a

   1. Initial program 68.9%

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

   3. Taylor expanded in a around 0 67.9%

      \[\leadsto \log \color{blue}{\left(1 + \left(e^{b} + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ 67.9%

         \[\leadsto \log \color{blue}{\left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)} \]

      2. *-commutative 67.9%

         \[\leadsto \log \left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + \color{blue}{a \cdot 0.16666666666666666}\right)\right)\right) \]

   5. Simplified 67.9%

      \[\leadsto \log \color{blue}{\left(\left(1 + e^{b}\right) + a \cdot \left(1 + a \cdot \left(0.5 + a \cdot 0.16666666666666666\right)\right)\right)} \]

   6. Taylor expanded in b around 0 64.7%

      \[\leadsto \color{blue}{\log \left(2 + a \cdot \left(1 + a \cdot \left(0.5 + 0.16666666666666666 \cdot a\right)\right)\right)} \]

   7. Taylor expanded in a around 0 64.1%

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

4. Final simplification 53.1%

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

Alternative 16: 56.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -130:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -130.0) (/ b 2.0) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -130.0) {
		tmp = b / 2.0;
	} 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 <= (-130.0d0)) then
        tmp = b / 2.0d0
    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 <= -130.0) {
		tmp = b / 2.0;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -130.0:
		tmp = b / 2.0
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -130.0)
		tmp = Float64(b / 2.0);
	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 <= -130.0)
		tmp = b / 2.0;
	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, -130.0], N[(b / 2.0), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -130

   1. Initial program 16.1%

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

   3. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 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 100.0%

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

   7. Taylor expanded in a around 0 18.8%

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

   if -130 < a

   1. Initial program 68.9%

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

   3. Taylor expanded in b around 0 65.1%

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

      1. log1p-define 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in a around 0 63.0%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -130:\\ \;\;\;\;\frac{b}{2}\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 12.0% accurate, 101.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return 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 = b / 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in b around 0 74.1%

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

   1. log1p-define 74.1%

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

5. Simplified 74.1%

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

6. Taylor expanded in b around inf 26.9%

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

7. Taylor expanded in a around 0 7.2%

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

8. Final simplification 7.2%

   \[\leadsto \frac{b}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024077 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))