symmetry log of sum of exp

Percentage Accurate: 96.5% → 95.5%

Time: 19.8s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. a = 3835975549838784.5
  2. b = -3.936370951473402e+252

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: 96.5% 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: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 1.000005:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 1.000005) (log (+ (exp a) (+ b 1.0))) (/ b (+ (exp a) 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(b) <= 1.000005) {
		tmp = log((exp(a) + (b + 1.0)));
	} else {
		tmp = b / (exp(a) + 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(b) <= 1.000005d0) then
        tmp = log((exp(a) + (b + 1.0d0)))
    else
        tmp = b / (exp(a) + 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(b) <= 1.000005) {
		tmp = Math.log((Math.exp(a) + (b + 1.0)));
	} else {
		tmp = b / (Math.exp(a) + 1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(b) <= 1.000005:
		tmp = math.log((math.exp(a) + (b + 1.0)))
	else:
		tmp = b / (math.exp(a) + 1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(b) <= 1.000005)
		tmp = log(Float64(exp(a) + Float64(b + 1.0)));
	else
		tmp = Float64(b / Float64(exp(a) + 1.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(b) <= 1.000005)
		tmp = log((exp(a) + (b + 1.0)));
	else
		tmp = b / (exp(a) + 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[b], $MachinePrecision], 1.000005], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 1.00000500000000003

   1. Initial program 97.1%

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

   3. Taylor expanded in b around 0 69.9%

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

      1. associate-+r+ 69.9%

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

      2. +-commutative 69.9%

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

   5. Simplified 69.9%

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

   if 1.00000500000000003 < (exp.f64 b)

   1. Initial program 52.3%

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

   3. Taylor expanded in b around 0 50.6%

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

      1. log1p-define 50.6%

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

   5. Simplified 50.6%

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

   6. Taylor expanded in b around inf 50.1%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 1.000005:\\ \;\;\;\;\log \left(e^{a} + \left(b + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{b} \leq 1.000005:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp b) 1.000005) (log1p (exp a)) (/ b (+ (exp a) 1.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(b) <= 1.000005) {
		tmp = log1p(exp(a));
	} else {
		tmp = b / (exp(a) + 1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(b) <= 1.000005) {
		tmp = Math.log1p(Math.exp(a));
	} else {
		tmp = b / (Math.exp(a) + 1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(b) <= 1.000005:
		tmp = math.log1p(math.exp(a))
	else:
		tmp = b / (math.exp(a) + 1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(b) <= 1.000005)
		tmp = log1p(exp(a));
	else
		tmp = Float64(b / Float64(exp(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[b], $MachinePrecision], 1.000005], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 b) < 1.00000500000000003

   1. Initial program 97.1%

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

   3. Taylor expanded in b around 0 70.6%

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

      1. log1p-define 70.6%

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

   5. Simplified 70.6%

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

   if 1.00000500000000003 < (exp.f64 b)

   1. Initial program 52.3%

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

   3. Taylor expanded in b around 0 50.6%

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

      1. log1p-define 50.6%

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

   5. Simplified 50.6%

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

   6. Taylor expanded in b around inf 50.1%

      \[\leadsto \color{blue}{\frac{b}{1 + e^{a}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} \leq 1.000005:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.2%

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

3. Final simplification 96.2%

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

Alternative 4: 54.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(a \cdot \left(0.5 - b \cdot 0.25\right) + b \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (+ (* a (- 0.5 (* b 0.25))) (* b 0.5)))))

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(2.0) + ((a * (0.5 - (b * 0.25))) + (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) :: tmp
    if (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + ((a * (0.5d0 - (b * 0.25d0))) + (b * 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) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + ((a * (0.5 - (b * 0.25))) + (b * 0.5));
	}
	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(2.0) + ((a * (0.5 - (b * 0.25))) + (b * 0.5))
	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 = Float64(log(2.0) + Float64(Float64(a * Float64(0.5 - Float64(b * 0.25))) + Float64(b * 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) <= 0.0)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + ((a * (0.5 - (b * 0.25))) + (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_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(N[(a * N[(0.5 - N[(b * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 95.9%

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

   3. Taylor expanded in b around 0 12.3%

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

      1. log1p-define 12.3%

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

   5. Simplified 12.3%

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

   6. Taylor expanded in b around inf 12.3%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 96.4%

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

   3. Taylor expanded in b around 0 60.8%

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

      1. log1p-define 60.8%

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

   5. Simplified 60.8%

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

   6. Taylor expanded in a around 0 60.4%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + \left(a \cdot \left(0.5 - b \cdot 0.25\right) + b \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.8)
   (/ b (+ (exp a) 1.0))
   (+ (log 2.0) (* b (+ 0.5 (* b 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.8) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (b * (0.5 + (b * 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) <= 0.8d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * (0.5d0 + (b * 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) <= 0.8) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * (0.5 + (b * 0.125)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.8:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * (0.5 + (b * 0.125)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.8)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * Float64(0.5 + Float64(b * 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) <= 0.8)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (b * (0.5 + (b * 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], 0.8], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * N[(0.5 + N[(b * 0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.80000000000000004

   1. Initial program 95.9%

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

   3. Taylor expanded in b around 0 12.3%

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

      1. log1p-define 12.3%

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

   5. Simplified 12.3%

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

   6. Taylor expanded in b around inf 12.3%

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

   if 0.80000000000000004 < (exp.f64 a)

   1. Initial program 96.4%

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

   3. Taylor expanded in a around 0 93.9%

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

      1. log1p-define 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in b around 0 59.9%

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

      1. +-commutative 59.9%

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

      2. unpow2 59.9%

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

      3. associate-*r* 59.9%

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

      4. distribute-rgt-out 59.9%

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

      5. *-commutative 59.9%

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

   8. Simplified 59.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.8) (/ b (+ (exp a) 1.0)) (+ (log 2.0) (* b 0.5))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.8) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log(2.0) + (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) :: tmp
    if (exp(a) <= 0.8d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log(2.0d0) + (b * 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) <= 0.8) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log(2.0) + (b * 0.5);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.8:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log(2.0) + (b * 0.5)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.8)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = Float64(log(2.0) + Float64(b * 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) <= 0.8)
		tmp = b / (exp(a) + 1.0);
	else
		tmp = log(2.0) + (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_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.8], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.80000000000000004

   1. Initial program 95.9%

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

   3. Taylor expanded in b around 0 12.3%

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

      1. log1p-define 12.3%

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

   5. Simplified 12.3%

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

   6. Taylor expanded in b around inf 12.3%

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

   if 0.80000000000000004 < (exp.f64 a)

   1. Initial program 96.4%

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

   3. Taylor expanded in a around 0 93.9%

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

      1. log1p-define 93.9%

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

   5. Simplified 93.9%

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

   6. Taylor expanded in b around 0 59.9%

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

      1. *-commutative 59.9%

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

   8. Simplified 59.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.8:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.2% 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 96.2%

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

3. Taylor expanded in a around 0 69.1%

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

   1. log1p-define 69.1%

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

5. Simplified 69.1%

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

6. Taylor expanded in b around 0 44.6%

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

   1. *-commutative 44.6%

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

8. Simplified 44.6%

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

9. Final simplification 44.6%

   \[\leadsto \log 2 + b \cdot 0.5 \]
10. Add Preprocessing

Alternative 8: 50.0% 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 96.2%

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

3. Taylor expanded in b around 0 69.1%

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

   1. associate-+r+ 69.1%

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

   2. +-commutative 69.1%

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

5. Simplified 69.1%

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

6. Taylor expanded in a around 0 43.7%

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

7. Final simplification 43.7%

   \[\leadsto \log \left(b + 2\right) \]
8. Add Preprocessing

Alternative 9: 49.4% 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 96.2%

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

3. Taylor expanded in a around 0 69.1%

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

   1. log1p-define 69.1%

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

5. Simplified 69.1%

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

6. Taylor expanded in b around 0 44.6%

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

7. Final simplification 44.6%

   \[\leadsto \log 2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))