symmetry log of sum of exp

Percentage Accurate: 53.8% → 98.7%

Time: 10.5s

Alternatives: 12

Speedup: 1.5×

• could not determine a ground truth

  1. a = 3.432140936672997e+44
  2. b = -1.7842953256659713e-229

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: 53.8% 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

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

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) (/ b (+ 1.0 (exp a))) (log (+ (exp b) (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((exp(b) + exp(a)));
	}
	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 / (1.0d0 + exp(a))
    else
        tmp = log((exp(b) + exp(a)))
    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 / (1.0 + Math.exp(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 = b / (1.0 + math.exp(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(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(exp(b) + exp(a)));
	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 / (1.0 + exp(a));
	else
		tmp = log((exp(b) + exp(a)));
	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[(1.0 + N[Exp[a], $MachinePrecision]), $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 9.1%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   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%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.8%

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

4. Final simplification 78.0%

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

Alternative 2: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) + e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05)
   (/ b (+ 1.0 (exp a)))
   (log (+ (fma (fma (fma 0.16666666666666666 b 0.5) b 1.0) b 1.0) (exp a)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = log((fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) + exp(a)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = log(Float64(fma(fma(fma(0.16666666666666666, b, 0.5), b, 1.0), b, 1.0) + 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], 0.05], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N[(N[(0.16666666666666666 * b + 0.5), $MachinePrecision] * b + 1.0), $MachinePrecision] * b + 1.0), $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 11.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 97.5%

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

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

      \[\leadsto \log \left(e^{a} + \color{blue}{\left(1 + b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \color{blue}{\left(b \cdot \left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) + 1\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \log \left(e^{a} + \left(\color{blue}{\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right)\right) \cdot b} + 1\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \log \left(e^{a} + \color{blue}{\mathsf{fma}\left(1 + b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right), b, 1\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\color{blue}{b \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot b\right) + 1}, b, 1\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot b\right) \cdot b} + 1, b, 1\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot b, b, 1\right)}, b, 1\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot b + \frac{1}{2}}, b, 1\right), b, 1\right)\right) \]

      8. lower-fma.f64 67.2

         \[\leadsto \log \left(e^{a} + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.16666666666666666, b, 0.5\right)}, b, 1\right), b, 1\right)\right) \]

   5. Applied rewrites 67.2%

      \[\leadsto \log \left(e^{a} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, b, 0.5\right), b, 1\right), b, 1\right) + e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{b} + e^{a} \leq 1.5:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (+ (exp b) (exp a)) 1.5) (* 0.5 b) (fma 0.5 b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if ((exp(b) + exp(a)) <= 1.5) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(0.5, b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (Float64(exp(b) + exp(a)) <= 1.5)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(0.5, b, log(2.0));
	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[(N[Exp[b], $MachinePrecision] + N[Exp[a], $MachinePrecision]), $MachinePrecision], 1.5], N[(0.5 * b), $MachinePrecision], N[(0.5 * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (exp.f64 a) (exp.f64 b)) < 1.5

   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. +-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 52.9

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

   5. Applied rewrites 52.9%

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

   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 3.4%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{2} \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 10.9%

       \[\leadsto 0.5 \cdot b \]

   if 1.5 < (+.f64 (exp.f64 a) (exp.f64 b))

   1. Initial program 96.3%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 94.8

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

   5. Applied rewrites 94.8%

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

   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 90.2%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{b}, \log 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{b} + e^{a} \leq 1.5:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0) (/ b (+ 1.0 (exp a))) (log1p (exp b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.0) {
		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) <= 0.0) {
		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) <= 0.0:
		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) <= 0.0)
		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], 0.0], 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) < 0.0

   1. Initial program 9.1%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 100.0

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

   5. Applied rewrites 100.0%

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

   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%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.8%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 66.5

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

   5. Applied rewrites 66.5%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.005208333333333333, 0.125\right), b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05)
   (/ b (+ 1.0 (exp a)))
   (fma (fma (fma (* b b) -0.005208333333333333 0.125) b 0.5) b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = fma(fma(fma((b * b), -0.005208333333333333, 0.125), b, 0.5), b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = fma(fma(fma(Float64(b * b), -0.005208333333333333, 0.125), b, 0.5), b, log(2.0));
	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], 0.05], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(b * b), $MachinePrecision] * -0.005208333333333333 + 0.125), $MachinePrecision] * b + 0.5), $MachinePrecision] * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 11.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 97.5%

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

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 70.5%

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

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 70.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 70.5

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 66.9

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

   7. Applied rewrites 66.9%

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

   8. Taylor expanded in b around 0

      \[\leadsto \log 2 + \color{blue}{b \cdot \left(\frac{1}{2} + b \cdot \left(\frac{1}{8} + \frac{-1}{192} \cdot {b}^{2}\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 65.4%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.005208333333333333, 0.125\right), b, 0.5\right), \color{blue}{b}, \log 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(b \cdot b, -0.005208333333333333, 0.125\right), b, 0.5\right), b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05)
   (/ b (+ 1.0 (exp a)))
   (fma (fma 0.125 b 0.5) b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = b / (1.0 + exp(a));
	} else {
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(b / Float64(1.0 + exp(a)));
	else
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	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], 0.05], N[(b / N[(1.0 + N[Exp[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 11.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in b around inf

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

   8. Applied rewrites 97.5%

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

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 70.5%

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

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 70.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 70.5

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 66.9

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

   7. Applied rewrites 66.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 65.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), \color{blue}{b}, \log 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;\frac{b}{1 + e^{a}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), b, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05) (* 0.5 b) (fma (fma 0.125 b 0.5) b (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(fma(0.125, b, 0.5), b, log(2.0));
	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], 0.05], N[(0.5 * b), $MachinePrecision], N[(N[(0.125 * b + 0.5), $MachinePrecision] * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 11.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   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 4.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{2} \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 18.6%

       \[\leadsto 0.5 \cdot b \]

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 70.5%

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

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 70.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 70.5

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 66.9

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

   7. Applied rewrites 66.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 65.3%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, b, 0.5\right), \color{blue}{b}, \log 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 56.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05) (* 0.5 b) (fma (fma 0.125 a 0.5) a (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(fma(0.125, a, 0.5), a, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(fma(0.125, a, 0.5), a, log(2.0));
	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], 0.05], N[(0.5 * b), $MachinePrecision], N[(N[(0.125 * a + 0.5), $MachinePrecision] * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 11.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   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 4.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{2} \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 18.6%

       \[\leadsto 0.5 \cdot b \]

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 67.4

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

   5. Applied rewrites 67.4%

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

   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 66.8%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), \color{blue}{a}, \log 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05) (* 0.5 b) (log1p (+ 1.0 b))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		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 (Math.exp(a) <= 0.05) {
		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 math.exp(a) <= 0.05:
		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 (exp(a) <= 0.05)
		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[N[Exp[a], $MachinePrecision], 0.05], N[(0.5 * b), $MachinePrecision], N[Log[1 + N[(1.0 + b), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 11.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   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 4.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{2} \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 18.6%

       \[\leadsto 0.5 \cdot b \]

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 70.5%

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

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

      7. lower-log.f64 N/A

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

      8. clear-num N/A

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

      9. flip-+ N/A

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

      10. lift-+.f64 N/A

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

      11. lower-/.f64 70.5

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 70.5

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

   4. Applied rewrites 70.5%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. log-rec N/A

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

      3. remove-double-neg N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-exp.f64 66.9

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

   7. Applied rewrites 66.9%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 64.0%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + a\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.05) (* 0.5 b) (log1p (+ 1.0 a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = 0.5 * b;
	} else {
		tmp = log1p((1.0 + a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.05) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p((1.0 + a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.05:
		tmp = 0.5 * b
	else:
		tmp = math.log1p((1.0 + a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(0.5 * b);
	else
		tmp = log1p(Float64(1.0 + 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], 0.05], N[(0.5 * b), $MachinePrecision], N[Log[1 + N[(1.0 + a), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 11.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   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 4.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{2} \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 18.6%

       \[\leadsto 0.5 \cdot b \]

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 66.0%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1 + a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0.05:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= (exp a) 0.05) (* 0.5 b) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (exp(a) <= 0.05) {
		tmp = 0.5 * b;
	} else {
		tmp = log1p(1.0);
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.05) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p(1.0);
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.05:
		tmp = 0.5 * b
	else:
		tmp = math.log1p(1.0)
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.05)
		tmp = Float64(0.5 * b);
	else
		tmp = log1p(1.0);
	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], 0.05], N[(0.5 * b), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.050000000000000003

   1. Initial program 11.9%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   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 4.8%

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

   9. Taylor expanded in b around inf

      \[\leadsto \frac{1}{2} \cdot b \]
   10. Step-by-step derivation

   11. Applied rewrites 18.6%

       \[\leadsto 0.5 \cdot b \]

   if 0.050000000000000003 < (exp.f64 a)

   1. Initial program 70.5%

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

   3. Taylor expanded in b around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-exp.f64 67.4

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 64.8%

      \[\leadsto \mathsf{log1p}\left(1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 12.0% accurate, 50.7× 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 55.6%

   \[\log \left(e^{a} + e^{b}\right) \]
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 75.1

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

5. Applied rewrites 75.1%

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

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 49.5%

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

9. Taylor expanded in b around inf

   \[\leadsto \frac{1}{2} \cdot b \]
10. Step-by-step derivation

11. Applied rewrites 7.3%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))