symmetry log of sum of exp

Percentage Accurate: 54.1% → 98.1%

Time: 17.8s

Alternatives: 13

Speedup: 2.9×

• could not determine a ground truth

  1. a = 1.010463615383478e+205
  2. b = 1.3025229968750055e-120

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 2e-33) (/ b (+ 1.0 (exp a))) (log1p (+ a (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 2e-33) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p((a + exp(b)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 2e-33) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p((a + Math.exp(b)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 2e-33:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log1p((a + math.exp(b)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 2e-33)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log1p(Float64(a + exp(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], 2e-33], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(a + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 2.0000000000000001e-33

   1. Initial program 10.7%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around inf 100.0%

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

   if 2.0000000000000001e-33 < (exp.f64 a)

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0 61.6%

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

      1. +-commutative 61.6%

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

   5. Simplified 61.6%

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

   6. Taylor expanded in b around inf 61.6%

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

      1. log1p-def 96.8%

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

   8. Simplified 96.8%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 2 \cdot 10^{-33}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(a + e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := 1 + e^{a}\\ \log t_0 + \left(0.5 \cdot \left({b}^{2} \cdot \left(\frac{1}{t_0} + \frac{-1}{{t_0}^{2}}\right)\right) + \frac{b}{t_0}\right) \end{array} \end{array} \]

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))))
   (+
    (log t_0)
    (+
     (* 0.5 (* (pow b 2.0) (+ (/ 1.0 t_0) (/ -1.0 (pow t_0 2.0)))))
     (/ b t_0)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	return log(t_0) + ((0.5 * (pow(b, 2.0) * ((1.0 / t_0) + (-1.0 / pow(t_0, 2.0))))) + (b / t_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
    real(8) :: t_0
    t_0 = 1.0d0 + exp(a)
    code = log(t_0) + ((0.5d0 * ((b ** 2.0d0) * ((1.0d0 / t_0) + ((-1.0d0) / (t_0 ** 2.0d0))))) + (b / t_0))
end function

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	t_0 = 1.0 + exp(a);
	tmp = log(t_0) + ((0.5 * ((b ^ 2.0) * ((1.0 / t_0) + (-1.0 / (t_0 ^ 2.0))))) + (b / t_0));
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]}, N[(N[Log[t$95$0], $MachinePrecision] + N[(N[(0.5 * N[(N[Power[b, 2.0], $MachinePrecision] * N[(N[(1.0 / t$95$0), $MachinePrecision] + N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 47.8%

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

3. Taylor expanded in b around 0 71.7%

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

4. Final simplification 71.7%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

3. Taylor expanded in b around 0 71.6%

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

   1. log1p-def 71.7%

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

5. Simplified 71.7%

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

6. Final simplification 71.7%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-107}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 1e-107) (/ b (+ 1.0 (exp a))) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 1e-107) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log1p(exp(b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 1e-107) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log1p(Math.exp(b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 1e-107:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log1p(math.exp(b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 1e-107)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log1p(exp(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], 1e-107], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[b], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 1e-107

   1. Initial program 10.7%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around inf 100.0%

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

   if 1e-107 < (exp.f64 a)

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0 60.3%

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

      1. log1p-def 60.8%

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

   5. Simplified 60.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 10^{-107}:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.0% 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 47.8%

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

   1. add-sqr-sqrt 46.6%

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

   2. pow2 46.6%

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

   3. metadata-eval 46.6%

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

   4. pow-to-exp 46.6%

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

   5. metadata-eval 46.6%

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

4. Applied egg-rr 46.6%

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

   1. exp-to-pow 46.6%

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

   2. pow2 46.6%

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

   3. log1p-expm1-u 46.6%

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

   4. add-sqr-sqrt 47.8%

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

   5. log1p-udef 47.8%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\log \left(e^{a} + e^{b}\right)\right)\right)} \]

   6. expm1-udef 47.8%

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

   7. add-exp-log 47.8%

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

6. Applied egg-rr 47.8%

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

   1. log1p-def 47.8%

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

   2. associate--l+ 47.9%

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

   3. expm1-def 72.2%

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

8. Simplified 72.2%

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

9. Final simplification 72.2%

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

Alternative 6: 57.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 10.7%

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

   3. Taylor expanded in a around 0 4.1%

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

   4. Taylor expanded in b around 0 5.0%

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

   5. Taylor expanded in b around inf 18.8%

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

   if -1 < a

   1. Initial program 62.9%

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

   3. Taylor expanded in b around 0 59.0%

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

      1. associate-+r+ 59.0%

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

      2. +-commutative 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in a around 0 58.2%

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

4. Final simplification 46.8%

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

Alternative 7: 56.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -250:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot b\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -250.0) (* 0.5 b) (+ (log 2.0) (* 0.5 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -250.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log(2.0) + (0.5 * 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 (a <= (-250.0d0)) then
        tmp = 0.5d0 * b
    else
        tmp = log(2.0d0) + (0.5d0 * b)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -250.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log(2.0) + (0.5 * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -250.0:
		tmp = 0.5 * b
	else:
		tmp = math.log(2.0) + (0.5 * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -250.0)
		tmp = Float64(0.5 * b);
	else
		tmp = Float64(log(2.0) + Float64(0.5 * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -250.0)
		tmp = 0.5 * b;
	else
		tmp = log(2.0) + (0.5 * 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[a, -250.0], N[(0.5 * b), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(0.5 * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -250

   1. Initial program 10.7%

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

   3. Taylor expanded in a around 0 4.1%

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

   4. Taylor expanded in b around 0 5.0%

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

   5. Taylor expanded in b around inf 18.8%

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

   if -250 < a

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0 60.3%

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

   4. Taylor expanded in b around 0 58.5%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -250:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -76:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot b\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -76.0) (/ b (+ 1.0 (exp a))) (+ (log 2.0) (* 0.5 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -76.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log(2.0) + (0.5 * 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 (a <= (-76.0d0)) then
        tmp = b / (1.0d0 + exp(a))
    else
        tmp = log(2.0d0) + (0.5d0 * b)
    end if
    code = tmp
end function

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -76.0) {
		tmp = b / (1.0 + Math.exp(a));
	} else {
		tmp = Math.log(2.0) + (0.5 * b);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -76.0:
		tmp = b / (1.0 + math.exp(a))
	else:
		tmp = math.log(2.0) + (0.5 * b)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -76.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = Float64(log(2.0) + Float64(0.5 * b));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -76.0)
		tmp = b / (1.0 + exp(a));
	else
		tmp = log(2.0) + (0.5 * 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[a, -76.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[2.0], $MachinePrecision] + N[(0.5 * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -76

   1. Initial program 10.7%

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

   3. Taylor expanded in b around 0 100.0%

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

   4. Taylor expanded in b around 0 100.0%

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

      1. log1p-def 100.0%

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

      2. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in b around inf 100.0%

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

   if -76 < a

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0 60.3%

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

   4. Taylor expanded in b around 0 58.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -76:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 + 0.5 \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 10.7%

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

   3. Taylor expanded in a around 0 4.1%

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

   4. Taylor expanded in b around 0 5.0%

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

   5. Taylor expanded in b around inf 18.8%

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

   if -1 < a

   1. Initial program 62.9%

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

   3. Taylor expanded in b around 0 59.0%

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

      1. associate-+r+ 59.0%

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

      2. +-commutative 59.0%

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

   5. Simplified 59.0%

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

   6. Taylor expanded in a around 0 58.2%

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

   7. Taylor expanded in b around 0 59.0%

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

4. Final simplification 47.4%

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

Alternative 10: 56.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -120:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -120.0) (* 0.5 b) (log (+ b 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -120.0) {
		tmp = 0.5 * b;
	} 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 <= (-120.0d0)) then
        tmp = 0.5d0 * b
    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 <= -120.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log((b + 2.0));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -120.0:
		tmp = 0.5 * b
	else:
		tmp = math.log((b + 2.0))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -120.0)
		tmp = Float64(0.5 * b);
	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 <= -120.0)
		tmp = 0.5 * b;
	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, -120.0], N[(0.5 * b), $MachinePrecision], N[Log[N[(b + 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -120

   1. Initial program 10.7%

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

   3. Taylor expanded in a around 0 4.1%

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

   4. Taylor expanded in b around 0 5.0%

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

   5. Taylor expanded in b around inf 18.8%

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

   if -120 < a

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0 60.3%

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

   4. Taylor expanded in b around 0 57.4%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -120:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log \left(b + 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -140:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -140.0) (* 0.5 b) (log1p (+ 1.0 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log1p((1.0 + b));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -140.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p((1.0 + b));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -140.0:
		tmp = 0.5 * b
	else:
		tmp = math.log1p((1.0 + b))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -140.0)
		tmp = Float64(0.5 * b);
	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, -140.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + N[(1.0 + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -140

   1. Initial program 10.7%

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

   3. Taylor expanded in a around 0 4.1%

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

   4. Taylor expanded in b around 0 5.0%

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

   5. Taylor expanded in b around inf 18.8%

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

   if -140 < a

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0 60.3%

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

   4. Taylor expanded in b around 0 57.4%

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

      1. log1p-expm1-u 57.4%

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

      2. expm1-udef 57.4%

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

      3. add-exp-log 57.4%

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

   6. Applied egg-rr 57.4%

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

      1. +-commutative 57.4%

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

      2. associate--l+ 57.4%

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

      3. metadata-eval 57.4%

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

   8. Simplified 57.4%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -140:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -125.0) {
		tmp = 0.5 * b;
	} 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 = 0.5d0 * b
    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 = 0.5 * b;
	} 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 = 0.5 * b
	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(0.5 * b);
	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 = 0.5 * b;
	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[(0.5 * b), $MachinePrecision], N[Log[2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -125

   1. Initial program 10.7%

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

   3. Taylor expanded in a around 0 4.1%

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

   4. Taylor expanded in b around 0 5.0%

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

   5. Taylor expanded in b around inf 18.8%

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

   if -125 < a

   1. Initial program 62.9%

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

   3. Taylor expanded in a around 0 60.3%

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

   4. Taylor expanded in b around 0 58.2%

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

4. Final simplification 46.8%

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

Alternative 13: 12.0% accurate, 101.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

3. Taylor expanded in a around 0 44.0%

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

4. Taylor expanded in b around 0 43.0%

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

5. Taylor expanded in b around inf 8.0%

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

6. Final simplification 8.0%

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

Reproduce

herbie shell --seed 2024011 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))