symmetry log of sum of exp

Percentage Accurate: 54.0% → 99.0%

Time: 10.6s

Alternatives: 10

Speedup: 1.5×

• Could not converge on a ground truth

  1. a = -1.0166525877928235e+124
  2. b = -3.838435851378185e+66

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: 54.0% 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: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(a\right) \cdot b}{\mathsf{expm1}\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{b} + e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ (* (expm1 a) b) (expm1 (+ a a)))
   (log (+ (exp b) (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = (expm1(a) * b) / expm1((a + a));
	} else {
		tmp = log((exp(b) + exp(a)));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = (Math.expm1(a) * b) / Math.expm1((a + a));
	} else {
		tmp = Math.log((Math.exp(b) + Math.exp(a)));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = (math.expm1(a) * b) / math.expm1((a + a))
	else:
		tmp = math.log((math.exp(b) + math.exp(a)))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(Float64(expm1(a) * b) / expm1(Float64(a + a)));
	else
		tmp = log(Float64(exp(b) + exp(a)));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 96.8%

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

   9. Applied rewrites 96.8%

      \[\leadsto \mathsf{fma}\left(\frac{{\left(\mathsf{expm1}\left(a \cdot 2\right)\right)}^{-1}}{1}, \color{blue}{\frac{\mathsf{expm1}\left(a\right)}{{b}^{-1}}}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   10. Taylor expanded in b around inf

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

   12. Applied rewrites 97.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 67.9%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(a\right) \cdot b}{\mathsf{expm1}\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{b} + e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{\mathsf{expm1}\left(a\right) \cdot b}{\mathsf{expm1}\left(a + a\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot b + \mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ (* (expm1 a) b) (expm1 (+ a a)))
   (+ (* 0.5 b) (log1p (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = (expm1(a) * b) / expm1((a + a));
	} else {
		tmp = (0.5 * b) + log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = (Math.expm1(a) * b) / Math.expm1((a + a));
	} else {
		tmp = (0.5 * b) + Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = (math.expm1(a) * b) / math.expm1((a + a))
	else:
		tmp = (0.5 * b) + math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(Float64(expm1(a) * b) / expm1(Float64(a + a)));
	else
		tmp = Float64(Float64(0.5 * b) + log1p(exp(a)));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(N[(N[(Exp[a] - 1), $MachinePrecision] * b), $MachinePrecision] / N[(Exp[N[(a + a), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * b), $MachinePrecision] + N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.1%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 97.0

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

   5. Applied rewrites 97.0%

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

   7. Applied rewrites 96.8%

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

   9. Applied rewrites 96.8%

      \[\leadsto \mathsf{fma}\left(\frac{{\left(\mathsf{expm1}\left(a \cdot 2\right)\right)}^{-1}}{1}, \color{blue}{\frac{\mathsf{expm1}\left(a\right)}{{b}^{-1}}}, \mathsf{log1p}\left(e^{a}\right)\right) \]

   10. Taylor expanded in b around inf

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

   12. Applied rewrites 97.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 67.9%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. lower-log1p.f64 N/A

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

      12. lower-exp.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 66.2%

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

Alternative 3: 98.5% 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 53.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-exp.f64 73.9

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

5. Applied rewrites 73.9%

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

6. Final simplification 73.9%

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

Alternative 4: 58.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-exp.f64 73.9

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

5. Applied rewrites 73.9%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 54.2%

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

Alternative 5: 52.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log(((1.0d0 + b) + exp(a)))
end function

Java

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

Python

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

Julia

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

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = log(((1.0 + b) + exp(a)));
end

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b around 0

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

   1. lower-+.f64 51.0

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

5. Applied rewrites 51.0%

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

6. Final simplification 51.0%

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

Alternative 6: 50.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 50.8

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

5. Applied rewrites 50.8%

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

Alternative 7: 49.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	return fma(0.5, b, log(2.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b around 0

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

   1. +-commutative N/A

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

   2. *-rgt-identity N/A

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

   3. associate-*r/ N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. *-rgt-identity N/A

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

   7. lower-/.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-log1p.f64 N/A

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

   12. lower-exp.f64 73.9

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

5. Applied rewrites 73.9%

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

6. Taylor expanded in a around 0

   \[\leadsto \log 2 + \color{blue}{\frac{1}{2} \cdot b} \]
7. Step-by-step derivation

8. Applied rewrites 50.0%

   \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{b}, \log 2\right) \]
9. Add Preprocessing

Alternative 8: 49.5% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in a around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 50.9

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

5. Applied rewrites 50.9%

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

6. Taylor expanded in b around 0

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

8. Applied rewrites 49.3%

   \[\leadsto \mathsf{log1p}\left(1 + b\right) \]
9. Add Preprocessing

Alternative 9: 49.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return log1p(1.0)
end

Wolfram

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

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 49.8%

   \[\leadsto \mathsf{log1p}\left(1\right) \]
9. Add Preprocessing

Alternative 10: 3.2% accurate, 27.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \left(0.125 \cdot a\right) \cdot a \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (* (* 0.125 a) a))

C

assert(a < b);
double code(double a, double b) {
	return (0.125 * a) * a;
}

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 = (0.125d0 * a) * a
end function

Java

assert a < b;
public static double code(double a, double b) {
	return (0.125 * a) * a;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (0.125 * a) * a

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(0.125 * a) * a)
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (0.125 * a) * a;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[(0.125 * a), $MachinePrecision] * a), $MachinePrecision]

Derivation

1. Initial program 53.7%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 50.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), \color{blue}{a}, \log 2\right) \]

9. Taylor expanded in a around inf

   \[\leadsto \frac{1}{8} \cdot {a}^{\color{blue}{2}} \]
10. Step-by-step derivation

11. Applied rewrites 4.1%

    \[\leadsto \left(0.125 \cdot a\right) \cdot a \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024294 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))