symmetry log of sum of exp

Percentage Accurate: 54.1% → 98.3%

Time: 12.0s

Alternatives: 12

Speedup: 1.4×

• could not determine a ground truth

  1. a = 7.651187167493154e+121
  2. b = 1.5159559041159736e-149

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: 54.1% 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: 98.3% 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 53.1%

   \[\log \left(e^{a} + e^{b}\right) \]
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. *-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 76.4

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

5. Applied rewrites 76.4%

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

6. Final simplification 76.4%

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

Alternative 2: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (log (+ (exp a) (fma b (fma b (fma b 0.16666666666666666 0.5) 1.0) 1.0)))))

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) + fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 1.0)));
	}
	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) + fma(b, fma(b, fma(b, 0.16666666666666666, 0.5), 1.0), 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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(b * N[(b * N[(b * 0.16666666666666666 + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.6%

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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.2%

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

   3. Taylor expanded in b around 0

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 66.7

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

   5. Applied rewrites 66.7%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \mathsf{fma}\left(b, \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.16666666666666666, 0.5\right), 1\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (log (+ (exp a) (fma b (fma b 0.5 1.0) 1.0)))))

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) + fma(b, fma(b, 0.5, 1.0), 1.0)));
	}
	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) + fma(b, fma(b, 0.5, 1.0), 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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[N[(N[Exp[a], $MachinePrecision] + N[(b * N[(b * 0.5 + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.6%

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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.2%

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

   3. Taylor expanded in b around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.6

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

   5. Applied rewrites 67.6%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{a} + \mathsf{fma}\left(b, \mathsf{fma}\left(b, 0.5, 1\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

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) + (b + 1.0)));
	}
	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) + (b + 1.0d0)))
    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) + (b + 1.0)));
	}
	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) + (b + 1.0)))
	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) + Float64(b + 1.0)));
	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) + (b + 1.0)));
	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[(b + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.6%

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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.2%

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

   3. Taylor expanded in b around 0

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

      1. lower-+.f64 66.6

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

   5. Applied rewrites 66.6%

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

4. Final simplification 75.8%

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

Alternative 5: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

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 = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (Math.exp(a) <= 0.0) {
		tmp = b / (Math.exp(a) + 1.0);
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if math.exp(a) <= 0.0:
		tmp = b / (math.exp(a) + 1.0)
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (exp(a) <= 0.0)
		tmp = Float64(b / Float64(exp(a) + 1.0));
	else
		tmp = log1p(exp(a));
	end
	return tmp
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[N[Exp[a], $MachinePrecision], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.6%

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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.2%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

4. Final simplification 76.0%

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

Alternative 6: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 0.125, a, \mathsf{fma}\left(a, 0.5, \log 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (fma (* a 0.125) a (fma a 0.5 (log 2.0)))))

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 = fma((a * 0.125), a, fma(a, 0.5, log(2.0)));
	}
	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 = fma(Float64(a * 0.125), a, fma(a, 0.5, 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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(a * 0.125), $MachinePrecision] * a + N[(a * 0.5 + N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.6%

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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.2%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot a\right) + \log 2} \]

      2. metadata-eval N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \color{blue}{\left(\frac{1}{8} - 0\right)} \cdot a\right) + \log 2 \]

      3. mul0-rgt N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \color{blue}{b \cdot 0}\right) \cdot a\right) + \log 2 \]

      4. metadata-eval N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot \color{blue}{\left(\frac{-1}{8} + \frac{1}{8}\right)}\right) \cdot a\right) + \log 2 \]

      5. distribute-rgt-out N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \color{blue}{\left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)}\right) \cdot a\right) + \log 2 \]

      6. *-commutative N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \color{blue}{a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)}\right) + \log 2 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right), \log 2\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right) + \frac{1}{2}}, \log 2\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - \color{blue}{b \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)}\right) + \frac{1}{2}, \log 2\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - b \cdot \color{blue}{0}\right) + \frac{1}{2}, \log 2\right) \]

      11. mul0-rgt N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - \color{blue}{0}\right) + \frac{1}{2}, \log 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \color{blue}{\frac{1}{8}} + \frac{1}{2}, \log 2\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{8}, \frac{1}{2}\right)}, \log 2\right) \]

      14. lower-log.f64 67.0

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

   8. Applied rewrites 67.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.125, 0.5\right), \log 2\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(\left(a \cdot \frac{1}{8}\right) \cdot a + \frac{1}{2} \cdot a\right)} + \log 2 \]

      2. lift-log.f64 N/A

         \[\leadsto \left(\left(a \cdot \frac{1}{8}\right) \cdot a + \frac{1}{2} \cdot a\right) + \color{blue}{\log 2} \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(a \cdot \frac{1}{8}\right) \cdot a + \left(\frac{1}{2} \cdot a + \log 2\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \frac{1}{8}, a, \frac{1}{2} \cdot a + \log 2\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{a \cdot \frac{1}{8}}, a, \frac{1}{2} \cdot a + \log 2\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \frac{1}{8}, a, \color{blue}{a \cdot \frac{1}{2}} + \log 2\right) \]

      7. lower-fma.f64 67.0

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

   10. Applied rewrites 67.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot 0.125, a, \mathsf{fma}\left(a, 0.5, \log 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.125, 0.5\right), \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= (exp a) 0.0)
   (/ b (+ (exp a) 1.0))
   (fma a (fma a 0.125 0.5) (log 2.0))))

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 = fma(a, fma(a, 0.125, 0.5), log(2.0));
	}
	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 = fma(a, fma(a, 0.125, 0.5), 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], 0.0], N[(b / N[(N[Exp[a], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(a * N[(a * 0.125 + 0.5), $MachinePrecision] + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 a) < 0.0

   1. Initial program 8.6%

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

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-exp.f64 100.0

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

   8. Applied rewrites 100.0%

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

   if 0.0 < (exp.f64 a)

   1. Initial program 70.2%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot a\right) + \log 2} \]

      2. metadata-eval N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \color{blue}{\left(\frac{1}{8} - 0\right)} \cdot a\right) + \log 2 \]

      3. mul0-rgt N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \color{blue}{b \cdot 0}\right) \cdot a\right) + \log 2 \]

      4. metadata-eval N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot \color{blue}{\left(\frac{-1}{8} + \frac{1}{8}\right)}\right) \cdot a\right) + \log 2 \]

      5. distribute-rgt-out N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \color{blue}{\left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)}\right) \cdot a\right) + \log 2 \]

      6. *-commutative N/A

         \[\leadsto a \cdot \left(\frac{1}{2} + \color{blue}{a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)}\right) + \log 2 \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right), \log 2\right)} \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right) + \frac{1}{2}}, \log 2\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - \color{blue}{b \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)}\right) + \frac{1}{2}, \log 2\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - b \cdot \color{blue}{0}\right) + \frac{1}{2}, \log 2\right) \]

      11. mul0-rgt N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - \color{blue}{0}\right) + \frac{1}{2}, \log 2\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(a, a \cdot \color{blue}{\frac{1}{8}} + \frac{1}{2}, \log 2\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{8}, \frac{1}{2}\right)}, \log 2\right) \]

      14. lower-log.f64 67.0

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

   8. Applied rewrites 67.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{a} \leq 0:\\ \;\;\;\;\frac{b}{e^{a} + 1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \mathsf{fma}\left(a, 0.125, 0.5\right), \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.8% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

   \[\log \left(e^{a} + e^{b}\right) \]
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. *-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 76.4

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

5. Applied rewrites 76.4%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot b + \log 2} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, b, \log 2\right)} \]

   3. lower-log.f64 49.6

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

8. Applied rewrites 49.6%

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

Alternative 9: 49.5% 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 53.1%

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

3. Taylor expanded in a around 0

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

   1. lower-+.f64 N/A

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

   2. lower-exp.f64 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in b around 0

   \[\leadsto \log \color{blue}{\left(2 + b\right)} \]
7. Step-by-step derivation

   1. lower-+.f64 48.9

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

8. Applied rewrites 48.9%

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

9. Final simplification 48.9%

   \[\leadsto \log \left(b + 2\right) \]
10. Add Preprocessing

Alternative 10: 49.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in a around 0

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

8. Applied rewrites 49.1%

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

Alternative 11: 3.2% accurate, 27.6× speedup

Math

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

FPCore

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

C

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

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 = a * (a * 0.125d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot a\right) + \log 2} \]

   2. metadata-eval N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \color{blue}{\left(\frac{1}{8} - 0\right)} \cdot a\right) + \log 2 \]

   3. mul0-rgt N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \color{blue}{b \cdot 0}\right) \cdot a\right) + \log 2 \]

   4. metadata-eval N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot \color{blue}{\left(\frac{-1}{8} + \frac{1}{8}\right)}\right) \cdot a\right) + \log 2 \]

   5. distribute-rgt-out N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \color{blue}{\left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)}\right) \cdot a\right) + \log 2 \]

   6. *-commutative N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \color{blue}{a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)}\right) + \log 2 \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right), \log 2\right)} \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right) + \frac{1}{2}}, \log 2\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - \color{blue}{b \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)}\right) + \frac{1}{2}, \log 2\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - b \cdot \color{blue}{0}\right) + \frac{1}{2}, \log 2\right) \]

   11. mul0-rgt N/A

       \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - \color{blue}{0}\right) + \frac{1}{2}, \log 2\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(a, a \cdot \color{blue}{\frac{1}{8}} + \frac{1}{2}, \log 2\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{8}, \frac{1}{2}\right)}, \log 2\right) \]

   14. lower-log.f64 49.0

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

8. Applied rewrites 49.0%

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

9. Taylor expanded in a around inf

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

    1. unpow2 N/A

       \[\leadsto \frac{1}{8} \cdot \color{blue}{\left(a \cdot a\right)} \]

    2. associate-*r* N/A

       \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot a\right) \cdot a} \]

    3. *-commutative N/A

       \[\leadsto \color{blue}{a \cdot \left(\frac{1}{8} \cdot a\right)} \]

    4. lower-*.f64 N/A

       \[\leadsto \color{blue}{a \cdot \left(\frac{1}{8} \cdot a\right)} \]

    5. *-commutative N/A

       \[\leadsto a \cdot \color{blue}{\left(a \cdot \frac{1}{8}\right)} \]

    6. lower-*.f64 4.2

       \[\leadsto a \cdot \color{blue}{\left(a \cdot 0.125\right)} \]

11. Applied rewrites 4.2%

    \[\leadsto \color{blue}{a \cdot \left(a \cdot 0.125\right)} \]
12. Add Preprocessing

Alternative 12: 2.6% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

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 = a * 0.5d0
end function

Java

assert a < b;
public static double code(double a, double b) {
	return a * 0.5;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	return a * 0.5

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.1%

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

3. Taylor expanded in b around 0

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

   1. lower-log1p.f64 N/A

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

   2. lower-exp.f64 49.8

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

5. Applied rewrites 49.8%

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

6. Taylor expanded in a around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot a\right) + \log 2} \]

   2. metadata-eval N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \color{blue}{\left(\frac{1}{8} - 0\right)} \cdot a\right) + \log 2 \]

   3. mul0-rgt N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \color{blue}{b \cdot 0}\right) \cdot a\right) + \log 2 \]

   4. metadata-eval N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - b \cdot \color{blue}{\left(\frac{-1}{8} + \frac{1}{8}\right)}\right) \cdot a\right) + \log 2 \]

   5. distribute-rgt-out N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \left(\frac{1}{8} - \color{blue}{\left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)}\right) \cdot a\right) + \log 2 \]

   6. *-commutative N/A

      \[\leadsto a \cdot \left(\frac{1}{2} + \color{blue}{a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)}\right) + \log 2 \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right), \log 2\right)} \]

   8. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right) + \frac{1}{2}}, \log 2\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - \color{blue}{b \cdot \left(\frac{-1}{8} + \frac{1}{8}\right)}\right) + \frac{1}{2}, \log 2\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - b \cdot \color{blue}{0}\right) + \frac{1}{2}, \log 2\right) \]

   11. mul0-rgt N/A

       \[\leadsto \mathsf{fma}\left(a, a \cdot \left(\frac{1}{8} - \color{blue}{0}\right) + \frac{1}{2}, \log 2\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(a, a \cdot \color{blue}{\frac{1}{8}} + \frac{1}{2}, \log 2\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \color{blue}{\mathsf{fma}\left(a, \frac{1}{8}, \frac{1}{2}\right)}, \log 2\right) \]

   14. lower-log.f64 49.0

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

8. Applied rewrites 49.0%

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

9. Taylor expanded in a around 0

   \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{1}{2}}, \log 2\right) \]
10. Step-by-step derivation

11. Applied rewrites 49.2%

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

12. Taylor expanded in a around inf

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

    1. lower-*.f64 7.0

       \[\leadsto \color{blue}{0.5 \cdot a} \]

14. Applied rewrites 7.0%

    \[\leadsto \color{blue}{0.5 \cdot a} \]

15. Final simplification 7.0%

    \[\leadsto a \cdot 0.5 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))