symmetry log of sum of exp

Percentage Accurate: 53.3% → 100.0%

Time: 10.7s

Alternatives: 12

Speedup: 2.8×

• could not determine a ground truth

  1. a = -1.385117813117035e-152
  2. b = 2.654550442652625e+250

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: 53.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(e^{a} + e^{b}\right) \end{array} \]

FPCore

(FPCore (a b) :precision binary64 (log (+ (exp a) (exp b))))

C

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

Fortran

real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = log((exp(a) + exp(b)))
end function

Java

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

Python

def code(a, b):
	return math.log((math.exp(a) + math.exp(b)))

Julia

function code(a, b)
	return log(Float64(exp(a) + exp(b)))
end

MATLAB

function tmp = code(a, b)
	tmp = log((exp(a) + exp(b)));
end

Wolfram

code[a_, b_] := N[Log[N[(N[Exp[a], $MachinePrecision] + N[Exp[b], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	return Float64(log1p(exp(Float64(a - b))) + b)
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[N[(a - b), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. lift-+.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh-cosh N/A

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

   7. sinh---cosh-rev N/A

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

   8. lift-exp.f64 N/A

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

   9. sinh-+-cosh-rev N/A

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

   10. flip-+ N/A

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

   11. sinh---cosh-rev N/A

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

   12. frac-add N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 48.9%

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

   1. lift-log.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. exp-sum N/A

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

   6. lift-exp.f64 N/A

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

   7. *-rgt-identity N/A

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

   8. lift-*.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. *-lft-identity N/A

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

   11. associate-/r* N/A

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

   12. log-div N/A

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

   13. *-lft-identity N/A

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

   14. lift-exp.f64 N/A

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

6. Applied rewrites 72.3%

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

7. Taylor expanded in a around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 N/A

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

   3. lower-log1p.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower--.f64 72.3

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

9. Applied rewrites 72.3%

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

Alternative 2: 58.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -160:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{-b}\right) + b\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -160.0) (* 0.5 b) (+ (log1p (exp (- b))) b)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -160.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log1p(exp(-b)) + b;
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -160.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p(Math.exp(-b)) + b;
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -160.0:
		tmp = 0.5 * b
	else:
		tmp = math.log1p(math.exp(-b)) + b
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -160.0)
		tmp = Float64(0.5 * b);
	else
		tmp = Float64(log1p(exp(Float64(-b))) + 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[a, -160.0], N[(0.5 * b), $MachinePrecision], N[(N[Log[1 + N[Exp[(-b)], $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -160

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -160 < a

   1. Initial program 67.0%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 62.8%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. exp-sum N/A

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

      6. lift-exp.f64 N/A

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

      7. *-rgt-identity N/A

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

      8. lift-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. *-lft-identity N/A

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

      11. associate-/r* N/A

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

      12. log-div N/A

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

      13. *-lft-identity N/A

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

      14. lift-exp.f64 N/A

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

   6. Applied rewrites 64.4%

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

   7. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. lower-log1p.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 62.6

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

   9. Applied rewrites 62.6%

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

Alternative 3: 57.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -360:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -360.0) (* 0.5 b) (log1p (exp a))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -360.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log1p(exp(a));
	}
	return tmp;
}

Java

assert a < b;
public static double code(double a, double b) {
	double tmp;
	if (a <= -360.0) {
		tmp = 0.5 * b;
	} else {
		tmp = Math.log1p(Math.exp(a));
	}
	return tmp;
}

Python

[a, b] = sort([a, b])
def code(a, b):
	tmp = 0
	if a <= -360.0:
		tmp = 0.5 * b
	else:
		tmp = math.log1p(math.exp(a))
	return tmp

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -360.0)
		tmp = Float64(0.5 * b);
	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[a, -360.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + N[Exp[a], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -360

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -360 < a

   1. Initial program 67.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.8

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

   5. Applied rewrites 63.8%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -360:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(e^{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -132:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, \mathsf{fma}\left(-0.25, b, 0.5\right)\right), a, \mathsf{fma}\left(0.5, b, \log 2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -132.0)
   (* 0.5 b)
   (fma (fma 0.125 a (fma -0.25 b 0.5)) a (fma 0.5 b (log 2.0)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -132.0) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(fma(0.125, a, fma(-0.25, b, 0.5)), a, fma(0.5, b, log(2.0)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -132.0)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(fma(0.125, a, fma(-0.25, b, 0.5)), a, 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[a, -132.0], N[(0.5 * b), $MachinePrecision], N[(N[(0.125 * a + N[(-0.25 * b + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(0.5 * b + N[Log[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -132

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -132 < a

   1. Initial program 67.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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. +-commutative 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 63.7

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

   5. Applied rewrites 63.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto \log 2 + \color{blue}{\left(\frac{1}{2} \cdot b + a \cdot \left(\left(\frac{1}{2} + a \cdot \left(\frac{1}{8} - \left(\frac{-1}{8} \cdot b + \frac{1}{8} \cdot b\right)\right)\right) - \frac{1}{4} \cdot b\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.1%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.125, a, \mathsf{fma}\left(-0.25, b, 0.5\right)\right), \color{blue}{a}, \mathsf{fma}\left(0.5, b, \log 2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -132:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, \mathsf{fma}\left(-0.25, b, 0.5\right)\right), a, \mathsf{fma}\left(0.5, b, \log 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 56.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -2.6:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.005208333333333333, 0.125\right), a, 0.5\right), a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -2.6)
   (* 0.5 b)
   (fma (fma (fma (* a a) -0.005208333333333333 0.125) a 0.5) a (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -2.6) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(fma(fma((a * a), -0.005208333333333333, 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 (a <= -2.6)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(fma(fma(Float64(a * a), -0.005208333333333333, 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[a, -2.6], N[(0.5 * b), $MachinePrecision], N[(N[(N[(N[(a * a), $MachinePrecision] * -0.005208333333333333 + 0.125), $MachinePrecision] * a + 0.5), $MachinePrecision] * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -2.60000000000000009

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -2.60000000000000009 < a

   1. Initial program 67.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 63.2%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.6:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a \cdot a, -0.005208333333333333, 0.125\right), a, 0.5\right), a, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -132:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, a, 1\right), a, 1\right) + \left(1 + b\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -132.0) (* 0.5 b) (log (+ (fma (fma 0.5 a 1.0) a 1.0) (+ 1.0 b)))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -132.0) {
		tmp = 0.5 * b;
	} else {
		tmp = log((fma(fma(0.5, a, 1.0), a, 1.0) + (1.0 + b)));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -132.0)
		tmp = Float64(0.5 * b);
	else
		tmp = log(Float64(fma(fma(0.5, a, 1.0), a, 1.0) + 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[a, -132.0], N[(0.5 * b), $MachinePrecision], N[Log[N[(N[(N[(0.5 * a + 1.0), $MachinePrecision] * a + 1.0), $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -132

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -132 < a

   1. Initial program 67.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 60.9

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in a around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 61.9

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

   11. Applied rewrites 61.9%

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

Alternative 7: 56.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -132:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -132.0) (* 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 (a <= -132.0) {
		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 (a <= -132.0)
		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[a, -132.0], 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 a < -132

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -132 < a

   1. Initial program 67.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.8

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

   5. Applied rewrites 63.8%

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

   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 63.2%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -132:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.125, a, 0.5\right), a, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.8% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: a and b should be sorted in increasing order before calling this function.
real(8) function code(a, b)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (a <= (-1.0d0)) then
        tmp = 0.5d0 * b
    else
        tmp = log(((1.0d0 + a) + (1.0d0 + b)))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.0)
		tmp = Float64(0.5 * b);
	else
		tmp = log(Float64(Float64(1.0 + a) + Float64(1.0 + b)));
	end
	return tmp
end

MATLAB

a, b = num2cell(sort([a, b])){:}
function tmp_2 = code(a, b)
	tmp = 0.0;
	if (a <= -1.0)
		tmp = 0.5 * b;
	else
		tmp = log(((1.0 + a) + (1.0 + b)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: a and b should be sorted in increasing order before calling this function.
code[a_, b_] := If[LessEqual[a, -1.0], N[(0.5 * b), $MachinePrecision], N[Log[N[(N[(1.0 + a), $MachinePrecision] + N[(1.0 + b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -1 < a

   1. Initial program 67.0%

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

   3. Taylor expanded in a around 0

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in b around 0

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

      1. lower-+.f64 60.9

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

   8. Applied rewrites 60.9%

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

   9. Taylor expanded in a around 0

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

       1. lower-+.f64 61.5

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

   11. Applied rewrites 61.5%

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

Alternative 9: 56.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, a, \log 2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b)
 :precision binary64
 (if (<= a -1.36) (* 0.5 b) (fma 0.5 a (log 2.0))))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.36) {
		tmp = 0.5 * b;
	} else {
		tmp = fma(0.5, a, log(2.0));
	}
	return tmp;
}

Julia

a, b = sort([a, b])
function code(a, b)
	tmp = 0.0
	if (a <= -1.36)
		tmp = Float64(0.5 * b);
	else
		tmp = fma(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[a, -1.36], N[(0.5 * b), $MachinePrecision], N[(0.5 * a + N[Log[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.3600000000000001

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -1.3600000000000001 < a

   1. Initial program 67.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.36:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, a, \log 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.2% accurate, 2.8× speedup

Math

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

FPCore

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

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -1.0) {
		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 (a <= -1.0) {
		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 a <= -1.0:
		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 (a <= -1.0)
		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[a, -1.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + N[(1.0 + a), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -1 < a

   1. Initial program 67.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.7%

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

4. Final simplification 52.9%

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

Alternative 11: 55.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} [a, b] = \mathsf{sort}([a, b])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -132:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: a and b should be sorted in increasing order before calling this function.
(FPCore (a b) :precision binary64 (if (<= a -132.0) (* 0.5 b) (log1p 1.0)))

C

assert(a < b);
double code(double a, double b) {
	double tmp;
	if (a <= -132.0) {
		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 (a <= -132.0) {
		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 a <= -132.0:
		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 (a <= -132.0)
		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[a, -132.0], N[(0.5 * b), $MachinePrecision], N[Log[1 + 1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -132

   1. Initial program 5.9%

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

      1. lift-+.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh-cosh N/A

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

      7. sinh---cosh-rev N/A

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

      8. lift-exp.f64 N/A

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

      9. sinh-+-cosh-rev N/A

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

      10. flip-+ N/A

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

      11. sinh---cosh-rev N/A

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

      12. frac-add N/A

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

      13. lower-/.f64 N/A

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

   4. Applied rewrites 0.1%

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

   5. Taylor expanded in a around 0

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

      1. div-add N/A

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

      2. *-inverses N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. exp-neg N/A

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

      6. remove-double-div N/A

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

      7. lower-exp.f64 3.8

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

   7. Applied rewrites 3.8%

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

   8. Taylor expanded in b around 0

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

   10. Applied rewrites 3.8%

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

   11. Taylor expanded in b around inf

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

   13. Applied rewrites 18.8%

       \[\leadsto 0.5 \cdot b \]

   if -132 < a

   1. Initial program 67.0%

      \[\log \left(e^{a} + e^{b}\right) \]
   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 63.8

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

   5. Applied rewrites 63.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 62.2%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -132:\\ \;\;\;\;0.5 \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(1\right)\\ \end{array} \]
5. 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 53.4%

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

   1. lift-+.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh-cosh N/A

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

   7. sinh---cosh-rev N/A

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

   8. lift-exp.f64 N/A

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

   9. sinh-+-cosh-rev N/A

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

   10. flip-+ N/A

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

   11. sinh---cosh-rev N/A

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

   12. frac-add N/A

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

   13. lower-/.f64 N/A

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

4. Applied rewrites 48.9%

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

5. Taylor expanded in a around 0

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

   1. div-add N/A

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

   2. *-inverses N/A

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

   3. +-commutative N/A

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

   4. lower-log1p.f64 N/A

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

   5. exp-neg N/A

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

   6. remove-double-div N/A

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

   7. lower-exp.f64 50.1

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

7. Applied rewrites 50.1%

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

8. Taylor expanded in b around 0

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

10. Applied rewrites 48.9%

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

11. Taylor expanded in b around inf

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

13. Applied rewrites 6.8%

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

Reproduce

herbie shell --seed 2024343 
(FPCore (a b)
  :name "symmetry log of sum of exp"
  :precision binary64
  (log (+ (exp a) (exp b))))