symmetry log of sum of exp

Percentage Accurate: 54.3% → 98.7%

Time: 12.1s

Alternatives: 16

Speedup: 2.8×

• could not determine a ground truth

  1. a = 2.7823489640384805e-71
  2. b = 7.583896877354766e+169

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: 54.3% 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: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = -log((1.0 / (exp(a) + exp(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 (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = -log((1.0d0 / (exp(a) + exp(b))))
    end if
    code = tmp
end function

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = -math.log((1.0 / (math.exp(a) + math.exp(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(exp(a) + 1.0));
	else
		tmp = Float64(-log(Float64(1.0 / Float64(exp(a) + exp(b)))));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 11.5%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 98.5%

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

      1. lift-log.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

      5. log-rec N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\log \left(\frac{e^{a} - e^{b}}{e^{a} \cdot e^{a} - e^{b} \cdot e^{b}}\right)\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\log \left(\frac{e^{a} - e^{b}}{e^{a} \cdot e^{a} - e^{b} \cdot e^{b}}\right)\right)} \]

      7. lower-log.f64 N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\log \left(\frac{e^{a} - e^{b}}{e^{a} \cdot e^{a} - e^{b} \cdot e^{b}}\right)}\right) \]

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

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

4. Final simplification 99.2%

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

Alternative 2: 98.7% accurate, 0.7× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (exp(a) + 1.0);
	} else {
		tmp = log((exp(a) + exp(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 (exp(a) <= 0.0d0) then
        tmp = b / (exp(a) + 1.0d0)
    else
        tmp = log((exp(a) + exp(b)))
    end if
    code = tmp
end function

Java

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

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log((math.exp(a) + math.exp(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(exp(a) + 1.0));
	else
		tmp = log(Float64(exp(a) + exp(b)));
	end
	return tmp
end

MATLAB

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

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 9.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. *-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-exp.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 97.4%

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

4. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024228 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))