symmetry log of sum of exp

Percentage Accurate: 53.1% → 98.2%

Time: 14.7s

Alternatives: 8

Speedup: 1.5×

• Unsound rule application detected in e-graph. Results may not be sound.

• could not determine a ground truth

  1. a = 1.0627316015288324e-262
  2. b = 1.4052300566925958e+119

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: 53.1% 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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]

2. Taylor expanded in b around 0 71.5%

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

   1. log1p-def 71.5%

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

4. Simplified 71.5%

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

5. Final simplification 71.5%

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

Alternative 2: 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 52.1%

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

   1. add-sqr-sqrt 50.9%

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

   2. log-prod 51.3%

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

3. Applied egg-rr 51.3%

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

   1. log-prod 50.9%

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

   2. rem-square-sqrt 52.1%

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

   3. log1p-expm1 52.1%

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

   4. expm1-def 52.1%

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

   5. rem-exp-log 52.1%

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

   6. associate--l+ 52.2%

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

   7. expm1-def 72.1%

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

5. Simplified 72.1%

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

6. Final simplification 72.1%

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

Alternative 3: 51.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]

2. Taylor expanded in b around 0 50.3%

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

   1. associate-+r+ 50.3%

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

   2. +-commutative 50.3%

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

   3. associate-+l+ 50.3%

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

4. Simplified 50.3%

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

5. Final simplification 50.3%

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

Alternative 4: 57.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \mathsf{log1p}\left(e^{a}\right) + \frac{b}{2} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (+ (log1p (exp a)) (/ b 2.0)))

C

assert(a < b);
double code(double a, double b) {
	return log1p(exp(a)) + (b / 2.0);
}

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log1p(Math.exp(a)) + (b / 2.0);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log1p(math.exp(a)) + (b / 2.0)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(log1p(exp(a)) + Float64(b / 2.0))
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := N[(N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision] + N[(b / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]

2. Taylor expanded in b around 0 71.5%

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

   1. log1p-def 71.5%

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

4. Simplified 71.5%

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

5. Taylor expanded in a around 0 53.7%

   \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{\color{blue}{2}} \]

6. Final simplification 53.7%

   \[\leadsto \mathsf{log1p}\left(e^{a}\right) + \frac{b}{2} \]

Alternative 5: 50.1% 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 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]

2. Taylor expanded in b around 0 50.4%

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

   1. log1p-def 50.5%

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

4. Simplified 50.5%

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

5. Final simplification 50.5%

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

Alternative 6: 48.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

assert a < b;
public static double code(double a, double b) {
	return (b * 0.5) + Math.log(2.0);
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return (b * 0.5) + math.log(2.0)

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(Float64(b * 0.5) + log(2.0))
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp = code(a, b)
	tmp = (b * 0.5) + log(2.0);
end

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]

2. Taylor expanded in a around 0 50.3%

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

3. Taylor expanded in b around 0 49.6%

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

4. Final simplification 49.6%

   \[\leadsto b \cdot 0.5 + \log 2 \]

Alternative 7: 48.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log((b + 2.0d0))
end function

Java

assert a < b;
public static double code(double a, double b) {
	return Math.log((b + 2.0));
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log((b + 2.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]

2. Taylor expanded in a around 0 50.3%

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

3. Taylor expanded in b around 0 48.8%

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

4. Final simplification 48.8%

   \[\leadsto \log \left(b + 2\right) \]

Alternative 8: 47.9% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log(2.0d0)
end function

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	return math.log(2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

   \[\log \left(e^{a} + e^{b}\right) \]

2. Taylor expanded in a around 0 50.3%

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

3. Taylor expanded in b around 0 49.3%

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

4. Final simplification 49.3%

   \[\leadsto \log 2 \]

Reproduce

herbie shell --seed 2023187 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))