symmetry log of sum of exp

Percentage Accurate: 53.2% → 98.4%

Time: 24.1s

Alternatives: 15

Speedup: 2.9×

• could not determine a ground truth

  1. a = 1.6109459652256993e+87
  2. b = -7.579196537717984e+103

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: 53.2% 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: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ 1.0 (exp a)))
   (log1p (+ (exp a) (* b (+ 1.0 (* b 0.5)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p((exp(a) + (b * (1.0 + (b * 0.5)))));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p((Math.exp(a) + (b * (1.0 + (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 / (1.0 + math.exp(a))
	else:
		tmp = math.log1p((math.exp(a) + (b * (1.0 + (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(1.0 + exp(a)));
	else
		tmp = log1p(Float64(exp(a) + Float64(b * Float64(1.0 + Float64(b * 0.5)))));
	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[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(N[Exp[a], $MachinePrecision] + 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) < 0.0

   1. Initial program 7.1%

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

   3. Taylor expanded in b around 0 6.5%

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

      1. associate-+r+ 6.5%

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

      2. *-commutative 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.1%

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

      1. log1p-expm1-u 65.1%

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

      2. expm1-undefine 65.1%

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

      3. add-exp-log 65.1%

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

   4. Applied egg-rr 65.1%

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

   5. Taylor expanded in b around 0 63.4%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{a} + b \cdot \left(1 + 0.5 \cdot b\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := 1 + e^{a}\\ t_1 := \frac{-1}{{t\_0}^{2}}\\ t_2 := \frac{1}{t\_0}\\ \log t\_0 + b \cdot \left(t\_2 + b \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(\left(2 \cdot \frac{1}{{t\_0}^{3}} + t\_2\right) + 3 \cdot t\_1\right)\right) + 0.5 \cdot \left(t\_2 + t\_1\right)\right)\right) \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp a))) (t_1 (/ -1.0 (pow t_0 2.0))) (t_2 (/ 1.0 t_0)))
   (+
    (log t_0)
    (*
     b
     (+
      t_2
      (*
       b
       (+
        (*
         0.16666666666666666
         (* b (+ (+ (* 2.0 (/ 1.0 (pow t_0 3.0))) t_2) (* 3.0 t_1))))
        (* 0.5 (+ t_2 t_1)))))))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	double t_1 = -1.0 / pow(t_0, 2.0);
	double t_2 = 1.0 / t_0;
	return log(t_0) + (b * (t_2 + (b * ((0.16666666666666666 * (b * (((2.0 * (1.0 / pow(t_0, 3.0))) + t_2) + (3.0 * t_1)))) + (0.5 * (t_2 + t_1))))));
}

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) :: t_1
    real(8) :: t_2
    t_0 = 1.0d0 + exp(a)
    t_1 = (-1.0d0) / (t_0 ** 2.0d0)
    t_2 = 1.0d0 / t_0
    code = log(t_0) + (b * (t_2 + (b * ((0.16666666666666666d0 * (b * (((2.0d0 * (1.0d0 / (t_0 ** 3.0d0))) + t_2) + (3.0d0 * t_1)))) + (0.5d0 * (t_2 + t_1))))))
end function

Java

assert a < b;
public static double code(double a, double b) {
	double t_0 = 1.0 + Math.exp(a);
	double t_1 = -1.0 / Math.pow(t_0, 2.0);
	double t_2 = 1.0 / t_0;
	return Math.log(t_0) + (b * (t_2 + (b * ((0.16666666666666666 * (b * (((2.0 * (1.0 / Math.pow(t_0, 3.0))) + t_2) + (3.0 * t_1)))) + (0.5 * (t_2 + t_1))))));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	t_0 = 1.0 + math.exp(a)
	t_1 = -1.0 / math.pow(t_0, 2.0)
	t_2 = 1.0 / t_0
	return math.log(t_0) + (b * (t_2 + (b * ((0.16666666666666666 * (b * (((2.0 * (1.0 / math.pow(t_0, 3.0))) + t_2) + (3.0 * t_1)))) + (0.5 * (t_2 + t_1))))))

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(1.0 + exp(a))
	t_1 = Float64(-1.0 / (t_0 ^ 2.0))
	t_2 = Float64(1.0 / t_0)
	return Float64(log(t_0) + Float64(b * Float64(t_2 + Float64(b * Float64(Float64(0.16666666666666666 * Float64(b * Float64(Float64(Float64(2.0 * Float64(1.0 / (t_0 ^ 3.0))) + t_2) + Float64(3.0 * t_1)))) + Float64(0.5 * Float64(t_2 + t_1)))))))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	t_0 = 1.0 + exp(a);
	t_1 = -1.0 / (t_0 ^ 2.0);
	t_2 = 1.0 / t_0;
	tmp = log(t_0) + (b * (t_2 + (b * ((0.16666666666666666 * (b * (((2.0 * (1.0 / (t_0 ^ 3.0))) + t_2) + (3.0 * t_1)))) + (0.5 * (t_2 + t_1))))));
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := Block[{t$95$0 = N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / t$95$0), $MachinePrecision]}, N[(N[Log[t$95$0], $MachinePrecision] + N[(b * N[(t$95$2 + N[(b * N[(N[(0.16666666666666666 * N[(b * N[(N[(N[(2.0 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] + N[(3.0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$2 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in b around 0 73.2%

   \[\leadsto \color{blue}{\log \left(1 + e^{a}\right) + b \cdot \left(b \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(\left(2 \cdot \frac{1}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{1 + e^{a}}\right) - 3 \cdot \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + 0.5 \cdot \left(\frac{1}{1 + e^{a}} - \frac{1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + \frac{1}{1 + e^{a}}\right)} \]

4. Final simplification 73.2%

   \[\leadsto \log \left(1 + e^{a}\right) + b \cdot \left(\frac{1}{1 + e^{a}} + b \cdot \left(0.16666666666666666 \cdot \left(b \cdot \left(\left(2 \cdot \frac{1}{{\left(1 + e^{a}\right)}^{3}} + \frac{1}{1 + e^{a}}\right) + 3 \cdot \frac{-1}{{\left(1 + e^{a}\right)}^{2}}\right)\right) + 0.5 \cdot \left(\frac{1}{1 + e^{a}} + \frac{-1}{{\left(1 + e^{a}\right)}^{2}}\right)\right)\right) \]
5. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 7.1%

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

   3. Taylor expanded in b around 0 6.5%

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

      1. associate-+r+ 6.5%

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

      2. *-commutative 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 65.1%

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

      1. log1p-expm1-u 65.1%

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

      2. expm1-undefine 65.1%

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

      3. add-exp-log 65.1%

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

   4. Applied egg-rr 65.1%

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

   5. Taylor expanded in b around 0 62.3%

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

4. Final simplification 72.6%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{1 + e^{a}} \end{array} \]

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return log1p(exp(a)) + (b / (1.0 + exp(a)));
}

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p(math.exp(a)) + (b / (1.0 + math.exp(a)))

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(log1p(exp(a)) + Float64(b / Float64(1.0 + exp(a))))
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[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in b around 0 73.2%

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

   1. log1p-define 73.2%

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

5. Simplified 73.2%

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

Alternative 5: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

   1. log1p-expm1-u 49.2%

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

   2. expm1-undefine 49.2%

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

   3. add-exp-log 49.2%

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

4. Applied egg-rr 49.2%

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

   1. associate--l+ 49.3%

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

   2. expm1-define 74.4%

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

6. Applied egg-rr 74.4%

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

Alternative 6: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.004:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b \cdot \left(1 + b \cdot 0.5\right) + \left(a + 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.004)
   (/ b (+ 1.0 (exp a)))
   (log (+ (* b (+ 1.0 (* b 0.5))) (+ a 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.004) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(((b * (1.0 + (b * 0.5))) + (a + 2.0)));
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 0.004d0) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log(((b * (1.0d0 + (b * 0.5d0))) + (a + 2.0d0)))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.004) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log(((b * (1.0 + (b * 0.5))) + (a + 2.0)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.004:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log(((b * (1.0 + (b * 0.5))) + (a + 2.0)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.004)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(Float64(b * Float64(1.0 + Float64(b * 0.5))) + Float64(a + 2.0)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 0.004)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log(((b * (1.0 + (b * 0.5))) + (a + 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.004], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(b * N[(1.0 + N[(b * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(a + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0040000000000000001

   1. Initial program 8.2%

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

   3. Taylor expanded in b around 0 7.6%

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

      1. associate-+r+ 7.6%

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

      2. *-commutative 7.6%

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

   5. Simplified 7.6%

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

   6. Taylor expanded in b around 0 99.9%

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

      1. log1p-define 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 98.6%

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

   if 0.0040000000000000001 < (exp.f64 a)

   1. Initial program 65.0%

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

   3. Taylor expanded in b around 0 63.2%

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

      1. associate-+r+ 63.2%

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

      2. *-commutative 63.2%

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

   5. Simplified 63.2%

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

   6. Taylor expanded in a around 0 63.1%

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

      1. +-commutative 63.1%

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

   8. Simplified 63.1%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.004:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(b \cdot \left(1 + b \cdot 0.5\right) + \left(a + 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 3.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 3.9e-32)
   (/ b (+ 1.0 (exp a)))
   (+ (log 2.0) (* b (+ 0.5 (* b 0.125))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 3.9e-32) {
		tmp = b / (1.0 + exp(a));
	} 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) <= 3.9d-32) then
        tmp = b / (1.0d0 + exp(a))
    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) <= 3.9e-32) {
		tmp = b / (1.0 + Math.exp(a));
	} 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) <= 3.9e-32:
		tmp = b / (1.0 + math.exp(a))
	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) <= 3.9e-32)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	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) <= 3.9e-32)
		tmp = b / (1.0 + exp(a));
	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], 3.9e-32], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $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) < 3.9000000000000001e-32

   1. Initial program 7.1%

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

   3. Taylor expanded in b around 0 6.5%

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

      1. associate-+r+ 6.5%

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

      2. *-commutative 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 3.9000000000000001e-32 < (exp.f64 a)

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in b around 0 61.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 3.9 \cdot 10^{-32}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + b \cdot \left(0.5 + b \cdot 0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2.1e-30)
   (/ b (+ 1.0 (exp a)))
   (log (+ 2.0 (* b (+ 1.0 (* b 0.5)))))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2.1e-30) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((2.0 + (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) :: tmp
    if (exp(a) <= 2.1d-30) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log((2.0d0 + (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 tmp;
	if (Math.exp(a) <= 2.1e-30) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log((2.0 + (b * (1.0 + (b * 0.5)))));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2.1e-30:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log((2.0 + (b * (1.0 + (b * 0.5)))))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2.1e-30)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(2.0 + 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)
	tmp = 0.0;
	if (exp(a) <= 2.1e-30)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log((2.0 + (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_] := If[LessEqual[N[Exp[a], $MachinePrecision], 2.1e-30], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(2.0 + 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) < 2.1000000000000002e-30

   1. Initial program 7.1%

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

   3. Taylor expanded in b around 0 6.5%

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

      1. associate-+r+ 6.5%

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

      2. *-commutative 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 2.1000000000000002e-30 < (exp.f64 a)

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in b around 0 62.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(2 + b \cdot \left(1 + b \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (log1p (+ (exp a) (* b (+ 1.0 (* b (+ 0.5 (* b 0.16666666666666666))))))))

C

assert(a < b);
double code(double a, double b) {
	return log1p((exp(a) + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p((Math.exp(a) + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p((math.exp(a) + (b * (1.0 + (b * (0.5 + (b * 0.16666666666666666)))))))

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(Float64(exp(a) + Float64(b * Float64(1.0 + Float64(b * Float64(0.5 + Float64(b * 0.16666666666666666)))))))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[Log[1 + N[(N[Exp[a], $MachinePrecision] + N[(b * N[(1.0 + N[(b * N[(0.5 + N[(b * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 49.2%

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

   1. log1p-expm1-u 49.2%

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

   2. expm1-undefine 49.2%

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

   3. add-exp-log 49.2%

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

4. Applied egg-rr 49.2%

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

5. Taylor expanded in b around 0 72.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{a} + b \cdot \left(1 + b \cdot \left(0.5 + 0.16666666666666666 \cdot b\right)\right)}\right) \]
6. Step-by-step derivation

   1. *-commutative 72.2%

      \[\leadsto \mathsf{log1p}\left(e^{a} + b \cdot \left(1 + b \cdot \left(0.5 + \color{blue}{b \cdot 0.16666666666666666}\right)\right)\right) \]

7. Simplified 72.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{a} + b \cdot \left(1 + b \cdot \left(0.5 + b \cdot 0.16666666666666666\right)\right)}\right) \]
8. Add Preprocessing

Alternative 10: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 3.3e-43) (/ b (+ 1.0 (exp a))) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 3.3e-43) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = (b * 0.5) + log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (exp(a) <= 3.3d-43) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = (b * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 3.3e-43) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 3.3e-43:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 3.3e-43)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = Float64(Float64(b * 0.5) + log(2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (exp(a) <= 3.3e-43)
		tmp = b / (1.0 + exp(a));
	else
		tmp = (b * 0.5) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 3.3e-43], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 3.30000000000000016e-43

   1. Initial program 7.1%

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

   3. Taylor expanded in b around 0 6.5%

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

      1. associate-+r+ 6.5%

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

      2. *-commutative 6.5%

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

   5. Simplified 6.5%

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

   6. Taylor expanded in b around 0 100.0%

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

      1. log1p-define 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in b around inf 100.0%

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

   if 3.30000000000000016e-43 < (exp.f64 a)

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in b around 0 61.6%

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

      1. *-commutative 61.6%

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

   6. Simplified 61.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -135:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -135.0) (* b 0.5) (+ (* b 0.5) (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -135.0) {
		tmp = b * 0.5;
	} else {
		tmp = (b * 0.5) + log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-135.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = (b * 0.5d0) + log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -135.0) {
		tmp = b * 0.5;
	} else {
		tmp = (b * 0.5) + Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -135.0:
		tmp = b * 0.5
	else:
		tmp = (b * 0.5) + math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -135.0)
		tmp = Float64(b * 0.5);
	else
		tmp = Float64(Float64(b * 0.5) + log(2.0));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -135.0)
		tmp = b * 0.5;
	else
		tmp = (b * 0.5) + log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -135.0], N[(b * 0.5), $MachinePrecision], N[(N[(b * 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -135

   1. Initial program 7.1%

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

   3. Taylor expanded in a around 0 3.9%

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

   4. Taylor expanded in b around 0 4.1%

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

      1. *-commutative 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in b around inf 18.8%

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

   if -135 < a

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in b around 0 61.6%

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

      1. *-commutative 61.6%

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

   6. Simplified 61.6%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -135:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;b \cdot 0.5 + \log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -130:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -130.0) (* b 0.5) (log1p (+ 1.0 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -130.0) {
		tmp = b * 0.5;
	} else {
		tmp = log1p((1.0 + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -130.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log1p((1.0 + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -130.0:
		tmp = b * 0.5
	else:
		tmp = math.log1p((1.0 + b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -130.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log1p(Float64(1.0 + b));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -130.0], N[(b * 0.5), $MachinePrecision], N[Log[1 + N[(1.0 + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -130

   1. Initial program 7.1%

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

   3. Taylor expanded in a around 0 3.9%

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

   4. Taylor expanded in b around 0 4.1%

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

      1. *-commutative 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in b around inf 18.8%

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

   if -130 < a

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in b around 0 60.7%

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

      1. log1p-expm1-u 60.7%

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

      2. expm1-undefine 60.7%

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

      3. add-exp-log 60.7%

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

   6. Applied egg-rr 60.7%

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

      1. +-commutative 60.7%

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

      2. associate--l+ 60.7%

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

      3. metadata-eval 60.7%

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

   8. Simplified 60.7%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -130:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -170:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -170.0) (* b 0.5) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -170.0) {
		tmp = b * 0.5;
	} else {
		tmp = log((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 (a <= (-170.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log((b + 2.0d0))
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -170.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -170.0:
		tmp = b * 0.5
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -170.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(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 (a <= -170.0)
		tmp = b * 0.5;
	else
		tmp = log((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[a, -170.0], N[(b * 0.5), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -170

   1. Initial program 7.1%

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

   3. Taylor expanded in a around 0 3.9%

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

   4. Taylor expanded in b around 0 4.1%

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

      1. *-commutative 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in b around inf 18.8%

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

   if -170 < a

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in b around 0 60.7%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -170:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -125:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -125.0) (* b 0.5) (log 2.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -125.0) {
		tmp = b * 0.5;
	} else {
		tmp = log(2.0);
	}
	return tmp;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-125.0d0)) then
        tmp = b * 0.5d0
    else
        tmp = log(2.0d0)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -125.0) {
		tmp = b * 0.5;
	} else {
		tmp = Math.log(2.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -125.0:
		tmp = b * 0.5
	else:
		tmp = math.log(2.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -125.0)
		tmp = Float64(b * 0.5);
	else
		tmp = log(2.0);
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -125.0)
		tmp = b * 0.5;
	else
		tmp = log(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -125.0], N[(b * 0.5), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -125

   1. Initial program 7.1%

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

   3. Taylor expanded in a around 0 3.9%

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

   4. Taylor expanded in b around 0 4.1%

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

      1. *-commutative 4.1%

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

   6. Simplified 4.1%

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

   7. Taylor expanded in b around inf 18.8%

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

   if -125 < a

   1. Initial program 65.1%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in b around 0 61.3%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -125:\\ \;\;\;\;b \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 12.0% accurate, 101.0× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return b * 0.5;
}

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = b * 0.5d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return b * 0.5;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return b * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.2%

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

3. Taylor expanded in a around 0 46.7%

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

4. Taylor expanded in b around 0 45.9%

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

   1. *-commutative 45.9%

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

6. Simplified 45.9%

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

7. Taylor expanded in b around inf 7.5%

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

8. Final simplification 7.5%

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

Reproduce

herbie shell --seed 2024088 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))