symmetry log of sum of exp

Percentage Accurate: 53.7% → 98.9%

Time: 13.2s

Alternatives: 19

Speedup: 2.8×

• could not determine a ground truth

  1. a = 1.1768115810034397e+45
  2. b = 2.0625270574729547e+20

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: 53.7% 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: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -37:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -37.0) (/ b (+ 1.0 (exp a))) (log (+ (exp a) (exp b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -37.0) {
		tmp = b / (1.0 + exp(a));
	} 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 (a <= (-37.0d0)) then
        tmp = b / (1.0d0 + exp(a))
    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 (a <= -37.0) {
		tmp = b / (1.0 + Math.exp(a));
	} 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 a <= -37.0:
		tmp = b / (1.0 + math.exp(a))
	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 (a <= -37.0)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	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 (a <= -37.0)
		tmp = b / (1.0 + exp(a));
	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[a, -37.0], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -37

   1. Initial program 10.4%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 10.4

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

   4. Applied rewrites 10.4%

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

   5. Taylor expanded in b around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. log-rec N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-log1p.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. associate-*r/ N/A

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

      11. *-rgt-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. lower-exp.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in b around inf

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

   10. Applied rewrites 100.0%

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

   if -37 < a

   1. Initial program 94.9%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} t_0 := 1 + e^{a}\\ \mathsf{fma}\left(b, \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), \frac{1}{t\_0}, \frac{b \cdot -0.5}{{t\_0}^{2}}\right), \mathsf{log1p}\left(e^{a}\right)\right) \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp a))))
   (fma
    b
    (fma (fma b 0.5 1.0) (/ 1.0 t_0) (/ (* b -0.5) (pow t_0 2.0)))
    (log1p (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double t_0 = 1.0 + exp(a);
	return fma(b, fma(fma(b, 0.5, 1.0), (1.0 / t_0), ((b * -0.5) / pow(t_0, 2.0))), log1p(exp(a)));
}

Julia

a, b = sort([a, b])
function code(a, b)
	t_0 = Float64(1.0 + exp(a))
	return fma(b, fma(fma(b, 0.5, 1.0), Float64(1.0 / t_0), Float64(Float64(b * -0.5) / (t_0 ^ 2.0))), log1p(exp(a)))
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[(b * N[(N[(b * 0.5 + 1.0), $MachinePrecision] * N[(1.0 / t$95$0), $MachinePrecision] + N[(N[(b * -0.5), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $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

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f64 N/A

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

5. Applied rewrites 98.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(b, \mathsf{fma}\left(\mathsf{fma}\left(b, 0.5, 1\right), \frac{1}{1 + e^{a}}, \frac{b \cdot -0.5}{{\left(1 + e^{a}\right)}^{2}}\right), \mathsf{log1p}\left(e^{a}\right)\right)} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))